Displaying 1-10 of 11 results found.
1, 15, 29, 43, 57, 71, 85, 99, 113, 127, 141, 155, 169, 183, 197, 211, 225, 239, 253, 267, 281, 295, 309, 323, 337, 351, 365, 379, 393, 407, 421, 435, 449, 463, 477, 491, 505, 519, 533, 547, 561, 575, 589, 603, 617, 631, 645, 659, 673, 687, 701, 715, 729
COMMENTS
Binomial transform of (1, 14, 0, 0, 0, ...).
FORMULA
a(n) = 14*n + 1.
G.f.: (1+13*x)/(1-x)^2.
E.g.f.: exp(x)*(1 + 14*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)
EXAMPLE
a(2) = 29 = 2*14 + 1.
a(2) = 29 = (1, 2, 1) dot (1, 14, 0) = (1 + 28 + 0).
Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.
+0
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).
Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k.
+0
16
0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 5, 9, 10, 0, 1, 6, 12, 16, 15, 0, 1, 7, 15, 22, 25, 21, 0, 1, 8, 18, 28, 35, 36, 28, 0, 1, 9, 21, 34, 45, 51, 49, 36, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 0, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66
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. - Omar E. Pol, Dec 21 2008
FORMULA
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009 t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
Sum_{k=0..n} t(n, k) = A006522(n+2). Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n). Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)
EXAMPLE
The square array of polygonal numbers begins:
========================================================
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, And so on ..............................................
========================================================
MATHEMATICA
T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)
PROG
(Magma)
T:= func< n, k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n, k | T(n-k, k) >;
(SageMath)
def T(n, k): return k*((n+1)*(k-1)+2)/2
def A139601(n, k): return T(n-k, k)
CROSSREFS
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).
a(n) = n*(n+1)*(14*n-11)/6.
+0
4
0, 1, 17, 62, 150, 295, 511, 812, 1212, 1725, 2365, 3146, 4082, 5187, 6475, 7960, 9656, 11577, 13737, 16150, 18830, 21791, 25047, 28612, 32500, 36725, 41301, 46242, 51562, 57275, 63395, 69936, 76912, 84337, 92225, 100590, 109446, 118807, 128687
COMMENTS
Generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=7.
In fact, the sequence is related to A001106 by a(n) = n* A001106(n) - Sum_{k=0..n-1} A001106(k) and this is the case d=7 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. Also 16-gonal (or hexadecagonal) pyramidal numbers.
Inverse binomial transform of this sequence: 0, 1, 15, 14, 0, 0 (0 continued). (End)
REFERENCES
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. [From Bruno Berselli, Feb 13 2014]
LINKS
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
FORMULA
a(n) = Sum_{i=0..n-1} (n-i)*(14*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014 E.g.f.: x*(6 + 45*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 17, 62}, 50] (* Vincenzo Librandi, Mar 01 2012 *)
PROG
(PARI) vector(40, n, n*(n-1)*(14*n-25)/6) \\ G. C. Greubel, Aug 30 2019 (Magma) [n*(n+1)*(14*n-11)/6: n in [0..40]] // G. C. Greubel, Aug 30 2019 (Sage) [n*(n+1)*(14*n-11)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019 (GAP) List([0..40], n-> n*(n+1)*(14*n-11)/6); # G. C. Greubel, Aug 30 2019
CROSSREFS
Cf. similar sequences listed in A237616.
25-gonal numbers: a(n) = n*(23*n-21)/2.
+0
12
0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, 8722, 9367, 10035, 10726, 11440, 12177, 12937, 13720, 14526, 15355, 16207, 17082, 17980
COMMENTS
If b(n,k) = n*((k-2)*n-(k-4))/2 is n-th k-gonal number, then b(n,k) = A000217(n) + (k-3)* A000217(n-1) (see Deza in References section, page 21, where the formula is attributed to Bachet de Méziriac). Also, b(n,k) = b(n,k-1) + A000217(n-1) (see Deza and Picutti in References section, page 20 and 137 respectively, where the formula is attributed to Nicomachus). Some examples: This is the case k=25.
REFERENCES
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (23rd row of the table).
E. Picutti, Sul numero e la sua storia, Feltrinelli Economica (1977), pages 131-147.
FORMULA
G.f.: x*(-1 - 22*x)/(-1 + x)^3.
Product_{n>=2} (1 - 1/a(n)) = 23/25. - Amiram Eldar, Jan 22 2021
MATHEMATICA
Table[n (23 n - 21)/2, {n, 40}]
PROG
(Magma) k:=25; [n*((k-2)*n-(k-4))/2: n in [0..40]]; // Bruno Berselli, Apr 10 2015
CROSSREFS
Cf. k-gonal numbers: A000217 (k=3), A000290 (k=4), A000326 (k=5), A000384 (k=6), A000566 (k=7), A000567 (k=8), A001106 (k=9), A001107 (k=10), A051682 (k=11), A051624 (k=12), A051865 (k=13), A051866 (k=14), A051867 (k=15), A051868 (k=16), A051869 (k=17), A051870 (k=18), A051871 (k=19), A051872 (k=20), A051873 (k=21), A051874 (k=22), A051875 (k=23), A051876 (k=24), this sequence (k=25), A255185 (k=26), A255186 (k=27), A161935 (k=28), A255187 (k=29), A254474 (k=30).
30-gonal pyramidal numbers: a(n) = n*(n+1)*(28*n-25)/6.
+0
2
0, 1, 31, 118, 290, 575, 1001, 1596, 2388, 3405, 4675, 6226, 8086, 10283, 12845, 15800, 19176, 23001, 27303, 32110, 37450, 43351, 49841, 56948, 64700, 73125, 82251, 92106, 102718, 114115, 126325, 139376, 153296, 168113, 183855, 200550, 218226, 236911, 256633
REFERENCES
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (28th row of the table).
FORMULA
G.f.: x*(1 + 27*x)/(1 - x)^4.
MATHEMATICA
Table[n (n + 1) (28 n - 25)/6, {n, 0, 40}]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 31, 118}, 40] (* Vincenzo Librandi, Apr 08 2015 *)
CROSSREFS
Cf. similar sequences listed in A237616.
Integers of the form m*(m + 6)/7.
+0
32
0, 1, 13, 16, 40, 45, 81, 88, 136, 145, 205, 216, 288, 301, 385, 400, 496, 513, 621, 640, 760, 781, 913, 936, 1080, 1105, 1261, 1288, 1456, 1485, 1665, 1696, 1888, 1921, 2125, 2160, 2376, 2413, 2641, 2680, 2920, 2961, 3213, 3256, 3520, 3565, 3841, 3888, 4176, 4225, 4525, 4576
COMMENTS
Nonnegative values of m are listed in A047274. Also, numbers h such that 7*h + 9 is a square.
Equivalently, numbers of the form i*(7*i - 6) with i = 0, 1, -1, 2, -2, 3, -3, ...
Infinitely many squares belong to this sequence.
Generalized 16-gonal (or hexadecagonal) numbers. See the third comment. - Omar E. Pol, Jun 06 2018 Exponents in expansion of Product_{n >= 1} (1 + x^(14*n-13))*(1 + x^(14*n-1))*(1 - x^(14*n)) = 1 + x + x^13 + x^16+ x^40 + .... - Peter Bala, Dec 10 2020
FORMULA
O.g.f.: x^2*(1 + 12*x + x^2)/((1 + x)^2*(1 - x)^3).
E.g.f.: (5*(2*x + 1)*exp(-x) + (14*x^2 - 5)*exp(x))/8.
a(n) = (14*(n-1)*n - 5*(2*n-1)*(-1)^n - 5)/8.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n >= 6. - Wesley Ivan Hurt, Dec 18 2020 Sum_{n>=2} 1/a(n) = (7 + 6*Pi*cot(Pi/7))/36. - Amiram Eldar, Feb 28 2022
EXAMPLE
88 is in the sequence because 88 = 22*(22+6)/7 or also 88 = 4*(7*4-6).
MATHEMATICA
Select[Table[(n(n+6))/7, {n, 0, 200}], IntegerQ] (* Harvey P. Dale, Sep 20 2022 *)
PROG
(Sage)
h = lambda m: m*(m+6)/7
return [h(m) for m in (0..len) if h(m) in ZZ]
(Magma) [t: m in [0..200] | IsIntegral(t) where t is m*(m+6)/7];
CROSSREFS
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), this sequence (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Square array T(n,k) = (n - 2)*(k - 1)*k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.
+0
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).
Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).
+0
0
0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0
COMMENTS
\c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
r\
_0 0 1 0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195 A067998_1 0 1 1 0 -2 -5 -9 -14 -20 -27 -35 -44 -54 -65 -77 -90 A080956_2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A001477_3 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 A000217_4 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 A000290_5 0 1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 A000326_6 0 1 6 15 28 45 66 91 120 153 190 231 276 325 378 435 A000384_7 0 1 7 18 34 55 81 112 148 189 235 286 342 403 469 540 A000566_8 0 1 8 21 40 65 96 133 176 225 280 341 408 481 560 645 A000567_9 0 1 9 24 46 75 111 154 204 261 325 396 474 559 651 750 A00110610 0 1 10 27 52 85 126 175 232 297 370 451 540 637 742 855 A00110711 0 1 11 30 58 95 141 196 260 333 415 506 606 715 833 960 A05168212 0 1 12 33 64 105 156 217 288 369 460 561 672 793 924 1065 A05162413 0 1 13 36 70 115 171 238 316 405 505 616 738 871 1015 1170 A05186514 0 1 14 39 76 125 186 259 344 441 550 671 804 949 1106 1275 A05186615 0 1 15 42 82 135 201 280 372 477 595 726 870 1027 1197 1380 A051867...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.
FORMULA
P(r, c) = (r - 2)(c(c-1)/2) + c.
MATHEMATICA
Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten
CROSSREFS
Cf. A317302 (the same array) but read by ascending antidiagonals. Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ; Columns (maybe missing some leading terms: A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
32-gonal numbers: a(n) = n*(15*n-14).
+0
1
0, 1, 32, 93, 184, 305, 456, 637, 848, 1089, 1360, 1661, 1992, 2353, 2744, 3165, 3616, 4097, 4608, 5149, 5720, 6321, 6952, 7613, 8304, 9025, 9776, 10557, 11368, 12209, 13080, 13981, 14912, 15873, 16864, 17885, 18936, 20017, 21128, 22269, 23440, 24641, 25872
FORMULA
G.f.: x*(1 + 29*x)/(1 - x)^3.
E.g.f.: exp(x)*(x + 15*x^2).
MATHEMATICA
Table[n (15 n - 14), {n, 30}]
Search completed in 0.013 seconds
|