Displaying 1-10 of 16 results found.
Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.
+10
49
0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11
COMMENTS
A general formula for polygonal numbers is P(n,k) = (n-2)*(k-1)*k/2 + k, where P(n,k) is the k-th n-gonal number.
The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - Johannes W. Meijer, Apr 29 2011
FORMULA
T(n,k) = n*(k-1)*k/2+k.
G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)
EXAMPLE
The square array of nonnegatives together with polygonal numbers begins:
=========================================================
....................... A A . . A A A A
....................... 0 0 . . 0 0 1 1
....................... 0 0 . . 1 1 3 3
....................... 0 0 . . 6 7 9 9
....................... 0 0 . . 9 3 6 6
....................... 0 1 . . 5 2 0 0
....................... 4 2 . . 7 9 6 7
=========================================================
Nonnegatives . A001477: 0, 1, 2, 3, 4, 5, 6, 7, ... Triangulars .. A000217: 0, 1, 3, 6, 10, 15, 21, 28, ... Squares ...... A000290: 0, 1, 4, 9, 16, 25, 36, 49, ... Pentagonals .. A000326: 0, 1, 5, 12, 22, 35, 51, 70, ... Hexagonals ... A000384: 0, 1, 6, 15, 28, 45, 66, 91, ... Heptagonals .. A000566: 0, 1, 7, 18, 34, 55, 81, 112, ... Octagonals ... A000567: 0, 1, 8, 21, 40, 65, 96, 133, ... 9-gonals ..... A001106: 0, 1, 9, 24, 46, 75, 111, 154, ... 10-gonals .... A001107: 0, 1, 10, 27, 52, 85, 126, 175, ... 11-gonals .... A051682: 0, 1, 11, 30, 58, 95, 141, 196, ... 12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217, ... ...
=========================================================
The column with the numbers 2, 3, 4, 5, 6, ... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15, ... is formed by the positive members of A008585.
MAPLE
T:= (n, k)-> n*(k-1)*k/2+k:
MATHEMATICA
T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)
PROG
(Python)
def A139600Row(n):
x, y = 1, 1
yield 0
while True:
yield x
x, y = x + y + n, y + n
for n in range(8):
R = A139600Row(n)
(Magma)
T:= func< n, k | k*(n*(k-1)+2)/2 >;
A139600:= func< n, k | T(n-k, k) >;
(SageMath)
def T(n, k): return k*(n*(k-1)+2)/2
def A139600(n, k): return T(n-k, k)
CROSSREFS
A formal extension negative n is in A326728. Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - Johannes W. Meijer, Apr 29 2011 Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
Period 4: repeat [0, 0, 1, 1].
+10
33
0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1
COMMENTS
Decimal expansion of 1/909.
Lexicographically earliest de Bruijn sequence for n = 2 and k = 2.
Except for first term, binary expansion of the decimal number 1/10 = 0.000110011001100110011... in base 2. - Benoit Cloitre, May 18 2002 Content of #2 binary placeholder when n is converted from decimal to binary. a(n) = n*(n-1)/2 mod 2. Example: a(7) = 1 since 7 in binary is 1 -1- 1 and (7*6/2) mod 2 = 1. - Anne M. Donovan (anned3005(AT)aol.com), Sep 15 2003
Expansion in any base b of 1/((b-1)*(b^2+1)) = 1/(b^3-b^2+b-1). E.g., 1/5 in base 2, 1/20 in base 3, 1/51 in base 4, etc. - Franklin T. Adams-Watters, Nov 07 2006 Except for first term, parity of the triangular numbers A000217. - Omar E. Pol, Jan 17 2012 Except for first term, more generally: 1) Parity of the k-polygonal numbers, if k is odd (Cf. A139600, A139601). 2) Parity of the generalized k-gonal numbers, for even k >= 6. - Omar E. Pol, Feb 05 2012
FORMULA
G.f.: x^2*(1 + x)/(1 - x^4).
a(n) = 1/2 - cos(Pi*n/2)/2 - sin(Pi*n/2)/2.
a(n) = a(n-1) - a(n-2) + a(n-3) for n > 2. (End)
a(n+2) = Sum_{k=0..n} b(k), with b(k) = A056594(k) (partial sums of S(n,x) Chebyshev polynomials at x=0). a(n) = -a(n-2) + 1, for n >= 2 with a(0) = a(1) = 0.
G.f.: x^2/((1 - x)*(1 + x^2)) = x^2/(1 - x + x^2 - x^3).
a(n) = 1/2 - sin((2n+1)*Pi/4)/sqrt(2).
a(n) = 1/2 - cos((2n-1)*Pi/4)/sqrt(2). (End)
Euler transform of length 4 sequence [1, -1, 0, 1]. - Michael Somos, Feb 28 2014 a(n) = a(n-4) for n > 3.
EXAMPLE
G.f. = x^2 + x^3 + x^6 + x^7 + x^10 + x^11 + x^14 + x^15 + x^18 + x^19 + ...;
1/909 = 0.001100110011001 ...
MATHEMATICA
a[ n_] := Mod[ Quotient[ n, 2], 2]; (* Michael Somos, Feb 28 2014 *) LinearRecurrence[{1, -1, 1}, {0, 0, 1}, 100] (* Ray Chandler, Aug 25 2015 *) CoefficientList[Series[x^2(1+x)/(1-x^4), {x, 0, 100}], x] (* Vincenzo Librandi, Dec 31 2015 *) (1-(-1)^Binomial[Range[0, 100], 2])/2 (* G. C. Greubel, Apr 03 2019 *)
PROG
(PARI) x='x+O('x^99); concat([0, 0], Vec(x^2/(1-x+x^2-x^3))) \\ Altug Alkan, Apr 04 2016
Rectangular array T(k,n) of polygonal numbers, by antidiagonals.
+10
20
1, 3, 1, 6, 4, 1, 10, 9, 5, 1, 15, 16, 12, 6, 1, 21, 25, 22, 15, 7, 1, 28, 36, 35, 28, 18, 8, 1, 36, 49, 51, 45, 34, 21, 9, 1, 45, 64, 70, 66, 55, 40, 24, 10, 1, 55, 81, 92, 91, 81, 65, 46, 27, 11, 1, 66, 100, 117, 120, 112, 96, 75, 52, 30, 12, 1, 78, 121, 145, 153, 148, 133, 111
COMMENTS
The antidiagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3. By rows, the sequence beginning (1, N, ...) is the binomial transform of (1, (N-1), (N-2), 0, 0, 0, ...); and is the second partial sum of (1, (N-2), (N-2), (N-2), ...). Example: The sequence (1, 4, 9, 16, 25, ...) is the binomial transform of (1, 3, 2, 0, 0, 0, ...) and the second partial sum of (1, 2, 2, 2, ...). - Gary W. Adamson, Aug 23 2015
REFERENCES
Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 76 at p. 189.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.
FORMULA
T(n, k) = n*binomial(k, 2) + k = A057145(n+2,k). 2*T(n, k) = T(n+r, k) + T(n-r, k), where r = 0, 1, 2, 3, ..., n-1 (see table in Example field). - Bruno Berselli, Dec 19 2014 G.f.: x*y*(1 - x + x*y)/((1 - x)^2*(1 - y)^3).
G.f. of k-th column: k*(1 + k - 2*x)*x/(2*(1 - x)^2). (End)
EXAMPLE
First 6 rows:
=========================================
n\k| 1 2 3 4 5 6 7
---|-------------------------------------
1 | 1 3 6 10 15 21 28 ... ( A000217, triangular numbers) 2 | 1 4 9 16 25 36 49 ... ( A000290, squares) 3 | 1 5 12 22 35 51 70 ... ( A000326, pentagonal numbers) 4 | 1 6 15 28 45 66 91 ... ( A000384, hexagonal numbers) 5 | 1 7 18 34 55 81 112 ... ( A000566, heptagonal numbers) 6 | 1 8 21 40 65 96 133 ... ( A000567, octagonal numbers) ...
The array formed by the complements: A183225.
MATHEMATICA
t[n_, k_] := n*Binomial[k, 2] + k; Table[ t[k, n - k + 1], {n, 12}, {k, n}] // Flatten
PROG
(Magma) T:=func<h, i | h*Binomial(i, 2)+i>; [T(k, n-k+1): k in [1..n], n in [1..12]]; // Bruno Berselli, Dec 19 2014
Number of ways to represent n as a polygonal number.
+10
13
1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 2, 2, 2, 4, 1, 1, 2, 2, 1, 2, 1, 1, 4, 2, 1, 2, 2, 1, 3, 2, 1, 2, 3, 1, 2, 2, 1, 2, 1, 1, 2, 3, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 4, 2, 1, 2, 2, 1, 2, 2, 1, 2, 3, 2, 2, 2, 2, 3, 1, 1, 2, 3
COMMENTS
Since n is always n-gonal number, a(n) >= 1.
Conjecture: Every positive integer appears in the sequence.
Records of 2, 3, 4, 5, ... are reached at n = 6, 15, 36, 225, 561, 1225, ... see A063778. [ R. J. Mathar, Aug 15 2010]
REFERENCES
J. J. Tattersall, Elementary Number Theory in Nine chapters, 2nd ed (2005), Cambridge Univ. Press, page 22 Problem 26, citing Wertheim (1897)
FORMULA
G.f.: x * Sum_{k>=2} x^k / (1 - x^(k*(k + 1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 09 2020
MAPLE
local ii, a, n, s, m ;
ii := 2*p ;
a := 0 ;
for n in numtheory[divisors](ii) do
if n > 2 then
s := ii/n ;
if (s-2) mod (n-1) = 0 then
a := a+1 ;
end if;
end if;
end do:
return a;
MATHEMATICA
nn = 100; t = Table[0, {nn}]; Do[k = 2; While[p = k*((n - 2) k - (n - 4))/2; p <= nn, t[[p]]++; k++], {n, 3, nn}]; t (* T. D. Noe, Apr 13 2011 *) Table[Length[Intersection[Divisors[2 n - 2] + 1, Divisors[2 n]]] - 1, {n, 3, 100}] (* Jonathan Sondow, May 09 2014 *)
PROG
(PARI) a(n) = sum(i=3, n, ispolygonal(n, i)); \\ Michel Marcus, Jul 08 2014 (Python)
from sympy import divisors
def a(n):
i=2*n
x=0
for d in divisors(i):
if d>2:
s=i/d
if (s - 2)%(d - 1)==0: x+=1
CROSSREFS
Cf. A129654, A139601, A090428, A176949, A176948, A176774, A176744, A176747, A176775, A175873, A176874.
Integers n for which f = (4^n - 2^n + 8n^2 - 2) / (2n * (2n + 1)) is an integer.
+10
10
3, 11, 23, 83, 131, 179, 191, 239, 243, 251, 359, 419, 431, 443, 491, 659, 683, 719, 743, 891, 911, 1019, 1031, 1103, 1223, 1439, 1451, 1499, 1511, 1539, 1559, 1583, 1811, 1931, 2003, 2039, 2063, 2211, 2339, 2351, 2399, 2459, 2511, 2543, 2699, 2819, 2903
COMMENTS
Superset of A002515; 2n + 1 is prime. A recursive search for members of this sequence results in the infinite series of very large primes A145918. Most members of this sequence are also prime, but five members less than 10000 are composite: .. . 243 = 3^5
.. . 891 = 3^4 * 11
. . 1539 = 3^4 * 19
. . 2211 = 3 * 11 * 67
. . 2511 = 3^4 * 31
The polygonal number with f sides of length 2n + 1 is (2^n - 1)(2^(n - 1)).
The average difference between successive composite terms gradually increases, remaining near their order of magnitude. Roughly 3% of all primes less than 20 billion belong to this sequence or the 2n + 1 sequence. The interval between composite terms 12228632879 and 13169544651 contains 1113606 primes, accounting for 2.75% of the primes in the interval and 1.42% of the primes between 24457265759 and 26339089303.
Prime factors are most often congruent to 3 (mod 4), but some factors are congruent to 1 (mod 4), especially when a term has an even number of not necessarily distinct factors. The most common factor is 3, and often a large power of 3 is a divisor. 5, 7, 13, and 17 are never factors.
The ones digit of composite terms is most often 1, and becomes progressively more likely to be 1. It is never 5. It cannot be 7, because 2n + 1 would then be divisible by 5. The lack of solutions with n divisible by 5 appears crucial to the consistent primality of 2n + 1.
The tens digit is odd if the ones digit is 1 or 9; it is even if the ones digit is 3. This is a consequence of congruence to 3 (mod 4).
The most common least significant two digits of composite terms are 51.
The least significant digits of prime terms do not follow an obvious distribution.
This is the simplest and possibly most productive member of a family of similar sequences defined by f = (s + 8n^2 - 2) / (2n * (2n + 1)), where s is pronic. For these sequences, 2n + 1 is dominated by primes.
=====================================
Large sequences of consecutive primes
=====================================
. Initial term Primes Predecessor Successor Gap
. ---------------------------------------------------------------
. 1529648303 157285 1529648231 1639846391 110198160
. 3832649339 473045 3832647111 4193496803 360849692
. 5897103683 411434 5897102751 6223464171 326361420
. 6543227423 445293 6543226251 6899473631 356247380
. 8126586971 913506 8126586711 8871331491 744744780
. 9533381219 689395 9533380131 10103115231 569735100
. 11576086883 708712 11576086731 12171829419 595742688
. 12228633251 1113606 12228632879 13169544651 940911772
. 21315457451 2328623 21315457251 23375077119 2059619868
(End)
EXAMPLE
ngon(f, k) = k * (f * (k - 1) / 2 - k + 2)
. . . 3 = (4^3 - 2^3 + 8 * 9 - 2) / (6 * 7)
. . . . = (2 * 28 + 70) / 42
. . 126 = (2 * 28 + 70)
.. . 28 = (2^3 - 1) * 2^2
. . . . = 126 - 70 - 28
. . . . = 7 * (18 - 10 - 4)
. . . . = 7 * (3 * 6 - 3 * 3 - 5)
. . . . = 7 * (3 * 3 - 7 + 2)
.. 8287 = (4^11 - 2^11 + 8 * 121 - 2) / (22 * 23)
. . . . = (2 * 2096128 + 966) / 506
4193222 = (2 * 2096128 + 966)
2096128 = (2^11 - 1) * 2^10
. . . . = 4193222 - 2096128 - 966
. . . . = 23 * (182314 - 91136 - 42)
. . . . = 23 * (8287 * 22 - 8287 * 11 - 21)
. . . . = 23 * (8287 * 11 - 23 + 2)
Coincidentally, 8287 = 129 * 64 + 31 = 257 * 32 + 63 is prime, and may be the largest value of f that is.
1031 = 257 * 4 + 3 and 2063 = 1031 * 2 + 1 are both members of this sequence, 4127 = 2063 * 2 + 1 is prime, and 8287 = (4127 + 16) * 2 + 1.
PROG
(Haskell)
a158034 n = a158034_list !! (n-1)
a158034_list = [x | x <- [1..],
(4^x - 2^x + 8*x^2 - 2) `mod` (2*x*(2*x + 1)) == 0]
CROSSREFS
Cf. A145918 (exponential Sophie Germain primes) Cf. A046318, A139876 (related to composite members 243, 891, 1539, and 2511) Cf. A142291 (prime sequence 257, 1031, 2063, 4127)
a(n) is the smallest solution x to A176774(x)=n; a(n)=0 if this equation has no solution.
+10
9
3, 4, 5, 0, 7, 8, 24, 27, 11, 33, 13, 14, 42, 88, 17, 165, 19, 20, 60, 63, 23, 69, 72, 26, 255, 160, 29, 87, 31, 32, 315, 99, 102, 208, 37, 38, 114, 805, 41, 123, 43, 44, 132, 268, 47, 696, 475, 50, 150, 304, 53, 159, 162, 56, 168, 340, 59, 177, 61, 62, 615, 1309, 192, 388
COMMENTS
A greedy inverse function to A176774. Conjecture: For every n >= 4, except for n=6, there exists an n-gonal number N which is not k-gonal for 3 <= k < n.
This means that the sequence contains only one 0: a(6)=0. For n=6 it is easy to prove that every hexagonal number ( A000384) is also triangular ( A000217), i.e., N does not exist. - Vladimir Shevelev, Apr 30 2010
FORMULA
a(p) = p if p is any odd prime.
EXAMPLE
For n=9, 24 is a nonagonal number, but not an octagonal number, heptagonal number, hexagonal number, etc. The smaller nonagonal number 9 is also a square number. Thus, a(9) = 24. - Michael B. Porter, Jul 16 2016
MAPLE
A139601 := proc(k, n) option remember ; n/2*( (k-2)*n-k+4) ; end proc:
A176774 := proc(n) option remember ; local k, m, pol ; for k from 3 do for m from 0 do pol := A139601(k, m) ; if pol = n then return k ; elif pol > n then break; end if; end do: end do: end proc:
A176948 := proc(n) if n = 6 then 0; else for x from 3 do if A176774(x)= n then return x ; end if; end do: end if; end proc:
MATHEMATICA
A176774[n_] := A176774[n] = (m = 3; While[Reduce[k >= 1 && n == k (k (m - 2) - m + 4)/2, k, Integers] == False, m++]; m); a[6] = 0; a[p_?PrimeQ] := p; a[n_] := (x = 3; While[ A176774[x] != n, x++]; x); Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 3, 100}] (* Jean-François Alcover, Sep 04 2016 *)
Irregular triangle read by rows: T(n,k) is the sum of the subparts of the ziggurat diagram of n (described in A347186) that arise from the (2*k-1)-th double-staircase of the double-staircases diagram of n (described in A335616), n >= 1, k >= 1, and the first element of column k is in row A000384(k).
+10
9
1, 4, 6, 16, 12, 36, 1, 20, 0, 64, 0, 30, 6, 90, 0, 42, 0, 144, 17, 56, 0, 156, 0, 72, 34, 1, 256, 0, 0, 90, 0, 0, 324, 10, 0, 110, 0, 0, 400, 0, 8, 132, 70, 0, 342, 0, 0, 156, 0, 0, 576, 121, 0, 182, 0, 25, 462, 0, 0, 210, 102, 0, 784, 0, 0, 1, 240, 0, 0, 0, 900, 24, 52, 0, 272, 0, 0, 0
COMMENTS
Conjecture 1: the number of nonzero terms in row n equals A082647(n). Conjecture 2: column k lists positive integers interleaved with 2*k+2 zeros.
The subparts of the ziggurat diagram are the polygons formed by the cells that are under the staircases.
The connection of the subparts of the ziggurat diagram with the polygonal numbers is as follows:
The area under a double-staircase labeled with the number j is equal to the m-th (j+2)-gonal number plus the (m-1)-th (j+2)-gonal number, where m is the number of steps on one side of the ladder from the base to the top.
The area under a simple-staircase labeled with the number j is equal to the m-th (j+2)-gonal number, where m is the number of steps.
So the k-th column of the triangle is related to the (2*k+1)-gonal numbers, for example:
For the calculation of column 1 we use triangular numbers A000217. For the calculation of column 2 we use pentagonal numbers A000326. For the calculation of column 3 we use heptagonal numbers A000566. For the calculation of column 4 we use enneagonal numbers A001106. And so on.
More generally, for the calculation of column k we use the (2*k+1)-gonal numbers.
For further information about the ziggurat diagram see A347186.
EXAMPLE
Triangle begins:
n / k 1 2 3 4
------------------------------
1 | 1;
2 | 4;
3 | 6;
4 | 16;
5 | 12;
6 | 36, 1;
7 | 20, 0;
8 | 64, 0;
9 | 30, 6;
10 | 90, 0;
11 | 42, 0;
12 | 144, 17;
13 | 56, 0;
14 | 156, 0;
15 | 72, 34, 1;
16 | 256, 0, 0;
17 | 90, 0, 0;
18 | 324, 10, 0;
19 | 110, 0 0;
20 | 400, 0, 8;
21 | 132, 70, 0;
22 | 342, 0, 0;
23 | 156, 0, 0;
24 | 576, 121, 0;
25 | 182, 0, 25;
26 | 462, 0, 0;
27 | 210, 102, 0;
28 | 784, 0, 0, 1;
...
For n = 15 the calculation of the 15th row of the triangle (in accordance with the geometric algorithm described in A347186) is as follows: Stage 1 (Construction):
We draw the diagram called "double-staircases" with 15 levels described in A335616. Then we label the five double-staircases (j = 1..5) as shown below:
_
_| |_
_| _ |_
_| | | |_
_| _| |_ |_
_| | _ | |_
_| _| | | |_ |_
_| | | | | |_
_| _| _| |_ |_ |_
_| | | _ | | |_
_| _| | | | | |_ |_
_| | _| | | |_ | |_
_| _| | | | | |_ |_
_| | | _| |_ | | |_
_| _| _| | _ | |_ |_ |_
|_ _ _ _ _ _ _ _|_ _ _|_ _|_|_|_|_ _|_ _ _|_ _ _ _ _ _ _ _|
1 2 3 4 5
.
Stage 2 (Debugging):
We remove the fourth double-staircase as it does not have at least one step at level 1 of the diagram starting from the base, as shown below:
_
_| |_
_| _ |_
_| | | |_
_| _| |_ |_
_| | _ | |_
_| _| | | |_ |_
_| | | | | |_
_| _| _| |_ |_ |_
_| | | | | |_
_| _| | | |_ |_
_| | _| |_ | |_
_| _| | | |_ |_
_| | | | | |_
_| _| _| _ |_ |_ |_
|_ _ _ _ _ _ _ _|_ _ _|_ _ _|_|_ _ _|_ _ _|_ _ _ _ _ _ _ _|
1 2 3 5
.
Stage 3 (Annihilation):
We delete the second double-staircase and the steps of the first double-staircase that are just above the second double-staircase.
The new diagram has two double-staircases and two simple-staircases as shown below:
_
| |
_ | | _
_| | _| |_ | |_
_| | | | | |_
_| | | | | |_
_| | _| |_ | |_
_| | | | | |_
_| | | | | |_
_| | _| _ |_ | |_
|_ _ _ _ _ _ _ _|_ _ _|_ _ _|_|_ _ _|_ _ _|_ _ _ _ _ _ _ _|
1 3 5
.
The diagram is called "ziggurat of 15".
Now we calculate the area (or the number of cells) under the staircases with multiplicity using polygonal numbers as shown below:
The area under the staircase labeled 1 is equal to A000217(8) = 36. There is a pair of these staircases, so T(15,1) = 2*36 = 72. The area under the double-staircase labeled 3 is equal to A000326(4) + A000326(3) = 22 + 12 = 34, so T(15,2) = 34. The area under the double-staircase labeled 5 is equal to A000566(1) + A000566(0) = 1 + 0 = 1, so T(15,3) = 1. Therefore the 15th row of the triangle is [72, 34, 1].
CROSSREFS
Cf. A347529 (analog for the symmetric representation of sigma). Cf. A000217, A000326, A000384, A000566, A001106, A082647, A139600, A139601, A237270, A237271, A237593, A279387, A279388, A279391, A335616, A346875.
Numbers that have exactly 26 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.
+10
6
1559439365121, 2468046593376, 7760419091425
EXAMPLE
a(1): 1559439365121 has representations P(n,k) = P(3, 519813121708)=P(6, 103962624343)=P(9, 43317760144)=P(11, 28353443004)=P(18, 10192414153)=P(27, 4442847196)=P(33, 2953483648)=P(57, 977092336)=P(66, 727011361)=P(69, 664722664)=P(81, 481308448)=P(86, 426659199)=P(129, 188885584)=P(131, 183140268)=P(171, 107288572)=P(209, 71744544)=P(237, 55761976)=P(414, 18240979)=P(473, 13969968)=P(513, 11874388)=P(711, 6178324)=P(729, 5876784)=P(1881, 881968)=P(3537, 249376)=P(16899, 10924)=P(720981, 8).
a(2): 2468046593376 has representations P(n,k) = P(3, 822682197793)=P(6, 164536439560)=P(12, 37394645356)=P(18, 16131023488)=P(24, 8942197804)=P(26, 7593989520)=P(39, 3330697159)=P(42, 2866488496)=P(56, 1602627660)=P(72, 965589436)=P(84, 707988124)=P(96, 541238290)=P(116, 370021980)=P(126, 313402744)=P(392, 32204796)=P(416, 28591830)=P(576, 14903665)=P(647, 11809911)=P(783, 8061483)=P(936, 5640220)=P(1827, 1479601)=P(2912, 582306)=P(4302, 266776)=P(5823, 145603)=P(7056, 99160)=P(145551, 235).
a(3): 7760419091425 has representations P(n,k) = P(5, 776041909144)=P(7, 369543766260)=P(10, 172453757589)=P(13, 99492552456)=P(19, 45382567788)=P(25, 25868063640)=P(35, 13042721164)=P(37, 11652280920)=P(49, 6598995828)=P(55, 5225871444)=P(65, 3730970719)=P(82, 2336771785)=P(143, 764347396)=P(145, 743335164)=P(154, 658723293)=P(205, 371134344)=P(290, 185190769)=P(325, 147396376)=P(475, 68935548)=P(1225, 10351368)=P(1378, 8179601)=P(1729, 5194893)=P(2755, 2045644)=P(7585, 269814)=P(1969825, 6)=P(3939649, 3).
Square array T(n,k) = (n - 2)*(k - 1)*k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.
+10
5
0, 0, 1, 0, 1, 0, 0, 1, 1, -3, 0, 1, 2, 0, -8, 0, 1, 3, 3, -2, -15, 0, 1, 4, 6, 4, -5, -24, 0, 1, 5, 9, 10, 5, -9, -35, 0, 1, 6, 12, 16, 15, 6, -14, -48, 0, 1, 7, 15, 22, 25, 21, 7, -20, -63, 0, 1, 8, 18, 28, 35, 36, 28, 8, -27, -80, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, -35, -99, 0, 1, 10, 24, 40, 55, 66
COMMENTS
Note that the formula gives several kinds of numbers, for example:
Row 0 gives 0 together with A258837. Row 1 gives 0 together with A080956. Row 2 gives A001477, the nonnegative numbers. For n >= 3, row n gives the n-gonal numbers (see Crossrefs section).
EXAMPLE
Array begins:
------------------------------------------------------------------------
n\k Numbers Seq. No. 0 1 2 3 4 5 6 7 8
------------------------------------------------------------------------
0 ............ ( A258837): 0, 1, 0, -3, -8, -15, -24, -35, -48, ... 1 ............ ( A080956): 0, 1, 1, 0, -2, -5, -9, -14, -20, ... 2 Nonnegatives A001477: 0, 1, 2, 3, 4, 5, 6, 7, 8, ... 3 Triangulars A000217: 0, 1, 3, 6, 10, 15, 21, 28, 36, ... 4 Squares A000290: 0, 1, 4, 9, 16, 25, 36, 49, 64, ... 5 Pentagonals A000326: 0, 1, 5, 12, 22, 35, 51, 70, 92, ... 6 Hexagonals A000384: 0, 1, 6, 15, 28, 45, 66, 91, 120, ... 7 Heptagonals A000566: 0, 1, 7, 18, 34, 55, 81, 112, 148, ... 8 Octagonals A000567: 0, 1, 8, 21, 40, 65, 96, 133, 176, ... 9 9-gonals A001106: 0, 1, 9, 24, 46, 75, 111, 154, 204, ... 10 10-gonals A001107: 0, 1, 10, 27, 52, 85, 126, 175, 232, ... 11 11-gonals A051682: 0, 1, 11, 30, 58, 95, 141, 196, 260, ... 12 12-gonals A051624: 0, 1, 12, 33, 64, 105, 156, 217, 288, ... 13 13-gonals A051865: 0, 1, 13, 36, 70, 115, 171, 238, 316, ... 14 14-gonals A051866: 0, 1, 14, 39, 76, 125, 186, 259, 344, ... 15 15-gonals A051867: 0, 1, 15, 42, 82, 135, 201, 280, 372, ... ...
CROSSREFS
Column 2 gives A001477, which coincides with the row numbers. Row 0 gives 0 together with A258837. Row 1 gives 0 together with A080956. Row 2 gives A001477, the same as column 2. For n >= 3, row n gives the n-gonal numbers: A000217 (n=3), A000290 (n=4), A000326 (n=5), A000384 (n=6), A000566 (n=7), A000567 (n=8), A001106 (n=9), A001107 (n=10), A051682 (n=11), A051624 (n=12), A051865 (n=13), A051866 (n=14), A051867 (n=15), A051868 (n=16), A051869 (n=17), A051870 (n=18), A051871 (n=19), A051872 (n=20), A051873 (n=21), A051874 (n=22), A051875 (n=23), A051876 (n=24), A255184 (n=25), A255185 (n=26), A255186 (n=27), A161935 (n=28), A255187 (n=29), A254474 (n=30). Cf. A303301 (similar table but with generalized polygonal numbers).
Numbers that have exactly 5 representations as a k-gonal number, P(n,k) = n*((k-2)*n - (k-4))/2, k and n >= 3.
+10
5
561, 1485, 1701, 2016, 2556, 2601, 2850, 3025, 3060, 3256, 3321, 4186, 4761, 4851, 5226, 5320, 5565, 5841, 6175, 6216, 6336, 6525, 6670, 7425, 7821, 7840, 8001, 8100, 8625, 8646, 9730, 9856, 9945, 9976, 10116, 10296, 10450, 10585, 11025, 11305, 11340, 12025, 12090
EXAMPLE
561 has representations P(3, 188)=P(6, 39)=P(11, 12)=P(17, 6)=P(33, 3).
1485 has representations P(3, 496)=P(5, 150)=P(9, 43)=P(15, 16)=P(54, 3).
1701 has representations P(3, 568)=P(6, 115)=P(9, 49)=P(18, 13)=P(21, 10).
PROG
(PARI) isok(n) = sum(k=3, n-1, ispolygonal(n, k)) == 5; \\ Michel Marcus, Oct 29 2018 (PARI) is(n) = my(d=divisors(n<<1)); sum(i=2, #d, k=2*(d[i]^2 - 2 * d[i] + n) / (d[i] - 1) / d[i]; k == k\1 && min(d[i], k) >=3) == 5 \\ David A. Corneth, Oct 29 2018
CROSSREFS
Cf. A275256, A057145, A063778, A129654, A139601, A177029, A195527, A195528, A321157, A321158, A321159, A321160, A320943.
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