Displaying 1-10 of 21 results found.
1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65, 67, 67, 69, 69, 71, 71, 73
COMMENTS
The number of rounds in a round-robin tournament with n competitors. - A. Timothy Royappa, Aug 13 2011 When partitioning a convex n-gon by all the diagonals, the maximum number of sides in resulting polygons is 2*floor(n/2)+1 = a(n-1) (from Moscow Olympiad problem 1950). - Tanya Khovanova, Apr 06 2008 The inverse values of the coefficients in the series expansion of f(x) = (1/2)*(1+x)*log((1+x)/(1-x)) lead to this sequence; cf. A098557. - Johannes W. Meijer, Nov 12 2009 A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - Michael Somos, May 29 2013 For n > 0, also the chromatic number of the (n+1)-triangular (Johnson) graph. - Eric W. Weisstein, Nov 17 2017 a(n-1), for n >= 1, is also the upper bound a_{up}(b), where b = 2*n + 1, in the first (top) row of the complete coach system Sigma(b) of Hilton and Pedersen [H-P]. All odd numbers <= a_{up}(b) of the smallest positive restricted residue system of b appear once in the first rows of the c(2*n+1) = A135303(n) coaches. If b is an odd prime a_{up}(b) is the maximum. See a comment in the proof of the quasi-order theorem of H-P, on page 263 ["Furthermore, every possible a_i < b/2 ..."]. For an example see below. - Wolfdieter Lang, Feb 19 2020 Satisfies the nested recurrence a(n) = a(a(n-2)) + 2*a(n-a(n-1)) with a(0) = a(1) = 1. Cf. A004001. - Peter Bala, Aug 30 2022 The binomial transform is 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560,.. (see A057711). - R. J. Mathar, Feb 25 2023
REFERENCES
Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, 3rd printing 2012, pp. (260-281).
FORMULA
a(n) = 2*floor(n/2) + 1.
G.f.: (1 + x + x^2 + x^3)/(1 - x^2)^2. - Paul Barry, Oct 14 2005 a(n) = 2*a(n-2) - a(n-4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 3. - Philippe Deléham, Nov 03 2008 a(n) = R(n, -2), where R(n, x) is the n-th row polynomial of A211955. a(n) = (-1)^n + 2*Sum_{k = 1..n} (-1)^(n - k - 2)*4^(k-1)*binomial(n+k, 2*k). Cf. A084159. - Peter Bala, May 01 2012 G.f.: ( 1 + x^2 ) / ( (1 + x)*(x - 1)^2 ). - R. J. Mathar, Jul 12 2016 a(-n) = -a(n-1).
EXAMPLE
G.f. = 1 + x + 3*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 9*x^9 + ...
Complete coach system for (a composite) b = 2*n + 1 = 33: Sigma(33) ={[1; 5], [5, 7, 13; 2, 1, 2]} (the first two rows are here 1 and 5, 7, 13), a_{up}(33) = a(15) = 15. But 15 is not in the reduced residue system modulo 33, so the maximal (odd) a number is 13. For the prime b = 31, a_{up}(31) = a(14) = 15 appears as maximum of the first rows. - Wolfdieter Lang, Feb 19 2020
MATHEMATICA
With[{c=2*Range[0, 40]+1}, Riffle[c, c]] (* Harvey P. Dale, Jan 02 2020 *)
PROG
(Haskell)
a109613 = (+ 1) . (* 2) . (`div` 2)
a109613_list = 1 : 1 : map (+ 2) a109613_list
(Sage) def a(n) : return( len( CuspForms( Gamma0( 5), 2*n + 4, prec=1). basis())); # Michael Somos, May 29 2013 (Scala) ((1 to 49) by 2) flatMap { List.fill(2)(_) } // Alonso del Arte, Sep 11 2019
The even numbers repeated.
+10
65
0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 46, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 58, 60, 60, 62, 62, 64, 64, 66, 66, 68, 68, 70, 70, 72, 72
COMMENTS
a(n) is also the binary rank of the complete graph K(n). - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 6, a(n) is the number of (0,1) n X n matrices A <= P^(-1)+I+P having exactly two 1's in every row and column with perA=2. - Vladimir Shevelev, Apr 12 2010 a(n+2) is the number of symmetry allowed, linearly independent terms at n-th order in the series expansion of the (E+A)xe vibronic perturbation matrix, H(Q) (cf. Eisfeld & Viel). - Bradley Klee, Jul 21 2015 For n > 1, also the chromatic number of the n X n white bishop graph. - Eric W. Weisstein, Nov 17 2017 For n > 2, also the maximum vertex degree of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 23 2018 For n >= 2, a(n+2) gives the minimum weight of a Boolean function of algebraic degree at most n-2 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019
REFERENCES
C. D. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001, page 181. - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009
V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19.
FORMULA
a(n) = 2*floor(n/2).
G.f.: 2*x^2/((-1+x)^2*(1+x)).
a(n) + a(n+1) + 2 - 2*n = 0.
a(n) = n - 1/2 + (-1)^n/2.
a(n) = a(n-1) + a(n-2) - a(n-3). - R. J. Mathar, Feb 19 2010 a(n) = a(a(n-1)) + a(n-a(n-1)) for n>2. - Nathan Fox, Jul 24 2016 a(b(n)) = b(n) + ((-1)^b(n) - 1)/2 for any sequence b(n) of offset 0.
a(a(n)) = a(n), idempotent.
a(n)*a(n+1)/2 = A007590(n), also equals partial sums of a(n).
MAPLE
spec := [S, {S=Union(Sequence(Prod(Z, Z)), Prod(Sequence(Z), Sequence(Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
With[{ev=2Range[0, 40]}, Riffle[ev, ev]] (* Harvey P. Dale, May 08 2021 *)
PROG
(Haskell)
a052928 = (* 2) . flip div 2
a052928_list = 0 : 0 : map (+ 2) a052928_list
CROSSREFS
Cf. A000034, A000124, A004001, A004526, A005843, A007590, A008619, A008794, A032766, A064455, A099392, A109613, A118266, A123684, A124356, A192442, A289187, A342819. For n >= 3, A329822(n) gives the minimum weight of a Boolean function of algebraic degree at most n-3 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
Removed duplicate of recurrence; corrected original recurrence and g.f. against offset - R. J. Mathar, Feb 19 2010
Multiples of 2 and of 3 interleaved: a(2n-1) = 2n, a(2n) = 3n.
+10
13
2, 3, 4, 6, 6, 9, 8, 12, 10, 15, 12, 18, 14, 21, 16, 24, 18, 27, 20, 30, 22, 33, 24, 36, 26, 39, 28, 42, 30, 45, 32, 48, 34, 51, 36, 54, 38, 57, 40, 60, 42, 63, 44, 66, 46, 69, 48, 72, 50, 75, 52, 78, 54, 81, 56, 84, 58, 87, 60, 90, 62, 93, 64, 96, 66, 99, 68, 102
FORMULA
Pair(2*n, 3*n).
G.f.: x*(2+3*x)/(1-x^2)^2.
a(n) = (5*n+(n-2)*(-1)^n+2)/4.
a(n) = 2*a(n-2) - a(n-4) = a(n-2) + A010693(n-1).
MATHEMATICA
With[{r = Range[50]}, Riffle[2*r, 3*r]] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {2, 3, 4, 6}, 100] (* Paolo Xausa, Feb 09 2024 *)
PROG
(Magma) &cat[[2*n, 3*n]: n in [1..34]]; // Bruno Berselli, Sep 25 2011 (Haskell)
import Data.List (transpose)
a195013 n = a195013_list !! (n-1)
a195013_list = concat $ transpose [[2, 4 ..], [3, 6 ..]]
CROSSREFS
Cf. A111712 (partial sums of this sequence prepended with 1).
Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*binomial(n,k), 0 <= k <= n.
+10
7
2, 3, 3, 5, 8, 5, 9, 18, 18, 9, 17, 40, 48, 40, 17, 33, 90, 120, 120, 90, 33, 65, 204, 300, 320, 300, 204, 65, 129, 462, 756, 840, 840, 756, 462, 129, 257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257, 513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513
COMMENTS
A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))* A007318(n, k). When (p, q) = (2, 1) this sequence is obtained. Some related triangles are:
(p, q) = (3, 3) : 2* A038221(n,k). (End)
FORMULA
T(n, k) = (2^(n-k) + 2^k)* A007318(n, k). T(n, n-k) = T(n, k) (symmetry).
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1). Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = n+1 + A107920(n+1). (End)
EXAMPLE
Triangle begins as:
2;
3, 3;
5, 8, 5;
9, 18, 18, 9;
17, 40, 48, 40, 17;
33, 90, 120, 120, 90, 33;
65, 204, 300, 320, 300, 204, 65;
129, 462, 756, 840, 840, 756, 462, 129;
257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257;
513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513;
1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700, 5140, 1025;
MATHEMATICA
T[n_, m_]:= (2^(n-m) + 2^m)*Binomial[n, m];
Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten
PROG
(Magma)
A154690:= func< n, k | (2^(n-k)+2^k)*Binomial(n, k) >;
(Python)
from sage.all import *
def A154690(n, k): return (pow(2, n-k)+pow(2, k))*binomial(n, k) print(flatten([[ A154690(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025
Triangle read by rows: T(n, k) = (2^(n-k)*3^k + 2^k*3^(n-k))*binomial(n, k).
+10
7
2, 5, 5, 13, 24, 13, 35, 90, 90, 35, 97, 312, 432, 312, 97, 275, 1050, 1800, 1800, 1050, 275, 793, 3492, 7020, 8640, 7020, 3492, 793, 2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315, 6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817
FORMULA
Sum_{k=0..n} (-1)^k*T(n, k) = A010673(n+1). Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A088137(n+1) + A000225(n+1). (End)
EXAMPLE
Triangle begins
2;
5, 5;
13, 24, 13;
35, 90, 90, 35;
97, 312, 432, 312, 97;
275, 1050, 1800, 1800, 1050, 275;
793, 3492, 7020, 8640, 7020, 3492, 793;
2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315;
6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064, 6817;
MAPLE
(2^(n-m)*3^m+2^m*3^(n-m))*binomial(n, m) ;
end proc:
MATHEMATICA
p=2; q=3;
T[n_, m_]= (p^(n-m)*q^m + p^m*q^(n-m))*Binomial[n, m];
Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten
PROG
(Magma)
A154692:= func< n, k | (2^(n-k)*3^k + 2^k*3^(n-k))*Binomial(n, k) >;
(Python)
from sage.all import *
def A154692(n, k): return (pow(2, n-k)*pow(3, k)+pow(2, k)*pow(3, n-k))*binomial(n, k) print(flatten([[ A154692(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025
Sum over the divisors d = 2 and/or 3 of n.
+10
6
0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3, 2, 0, 5, 0, 2, 3
COMMENTS
Periodic with period {0,2,3,2,0,5}.
FORMULA
a(n) = Sum_{d|n, d=2 or d=3} d.
a(n+6) = a(n).
a(n) = -a(n-1) + a(n-3) + a(n-4).
G.f.: -x*(2+5*x+5*x^2) / ( (x-1)*(1+x)*(1+x+x^2) ).
MATHEMATICA
Table[Total@ Select[Divisors@ n, 2 <= # <= 3 &], {n, 120}] (* or *)
Table[Total[Divisors@ n /. {d_ /; d < 2 -> Nothing, d_ /; d > 3 -> Nothing} ], {n, 120}] (* Michael De Vlieger, Feb 07 2016 *) Flatten[Table[{0, 2, 3, 2, 0, 5}, {16}]] (* Amiram Eldar, Aug 03 2024 *)
PROG
(PARI) a(n) = sumdiv(n, d, d*((d==2) || (d==3))); \\ Michel Marcus, Feb 07 2016 (PARI) a(n) = [0, 2, 3, 2, 0, 5][(n-1) % 6 + 1]; \\ Amiram Eldar, Aug 03 2024
EXTENSIONS
Replaced recurrence by a shorter one; added keyword:less - R. J. Mathar, May 28 2010
0, 2, 4, 6, 8, 10, 12, 22, 16, 18, 20, 14, 24, 26, 28, 30, 64, 46, 36, 58, 40, 66, 76, 34, 48, 70, 52, 54, 56, 38, 60, 74, 32, 42, 68, 50, 72, 62, 44, 78, 80, 82, 84, 190, 136, 90, 172, 118, 192, 226, 100, 138, 208, 154, 108, 166, 112, 174, 220, 94, 120, 202, 148, 198, 184, 130
FORMULA
Other identities. For all n >= 0:
PROG
(Python)
from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
f=factorint(n)
return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
Square array read by antidiagonals upwards in which each new term is the least nonnegative integer distinct from its neighbors.
+10
5
0, 1, 2, 0, 3, 0, 1, 2, 1, 2, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2
COMMENTS
In the square array we have that:
Odd-indexed antidiagonals give the initial terms of A010674. Even-indexed antidiagonals give the initial terms of A000034. This is also a triangle read by rows in which each new term is the least nonnegative integer distinct from its neighbors.
In the triangle we have that:
Odd-indexed diagonals give A010673. Even-indexed diagonals give A010684. Odd-indexed rows give the initial terms of A010674. Even-indexed rows give the initial terms of A000034. Odd-indexed antidiagonals give the initial terms of A010673. Even-indexed antidiagonals give the initial terms of A010684.
FORMULA
G.f.: 3*x/(1-x^2) - Sum_{k>=0} (2*x^(2*k^2+3*k+1)-x^(2*k^2+5*k+3))/(1+x).
G.f. as triangle: x*(1+2*y+3*x*y)/((1-x^2*y^2)*(1-x^2)). (End)
EXAMPLE
The corner of the square array begins:
0, 2, 0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, ...
0, 2, 0, 2, ...
1, 3, 1, ...
0, 2, ...
1, ...
...
The sequence written as a triangle begins:
0;
1, 2;
0, 3, 0;
1, 2, 1, 2;
0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2, 1, 2;
...
MAPLE
ListTools:-Flatten([seq([[0, 3]$i, 0, [1, 2]$(i+1)], i=0..10)]); # Robert Israel, Nov 14 2016
MATHEMATICA
Table[Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)
Number of toothpicks and D-toothpicks added at n-th stage to the H-toothpick structure of A182838.
+10
4
0, 1, 2, 4, 4, 4, 6, 10, 8, 4, 6, 12, 16, 14, 14, 22, 16, 4, 6, 12, 16, 16, 20, 32, 36, 22, 14, 28, 42, 40, 36, 50, 32, 4, 6, 12, 16, 16, 20, 32, 36, 24
COMMENTS
The "word" of this cellular automaton is "ab".
Apart from the initial zero the structure of the irregular triangle is as shown below:
a,b;
a,b;
a,b,a,b;
a,b,a,b,a,b,a,b;
a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
...
Columns "a" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only toothpicks (of length 1).
Columns "b" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only D-toothpicks (of length sqrt(2)).
An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.
Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612. For further information about the word of cellular automata see A296612. It appears that the right border of the irregular triangle gives the even powers of 2. (End)
EXAMPLE
The nonzero terms can write as an irregular triangle as shown below:
1, 2;
4, 4;
4, 6, 10, 8;
4, 6, 12, 16, 14, 14, 22, 16;
4, 6, 12, 16, 16, 20, 32, 36, 22, 14, 28, 42, 40, 36, 50, 32;
...
(End)
Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.
+10
4
1, 3, 3, 7, 11, 23, 43, 87, 171, 343, 683, 1367, 2731, 5463, 10923, 21847, 43691, 87383, 174763, 349527, 699051, 1398103, 2796203, 5592407, 11184811, 22369623, 44739243, 89478487, 178956971, 357913943, 715827883, 1431655767, 2863311531, 5726623063
COMMENTS
Initialized with a single black (ON) cell at stage zero.
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
FORMULA
a(n) = (3 - 2*(-1)^n + 2^(1+n))/3.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)
a(n) = a(n-1) + 2* A078008(n-1). (End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=14; stages=128;
rule=IntegerDigits[code, 2, 10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]][[i]], Range[i, 2*i-1]], 2], {i, 1, stages-1}]
LinearRecurrence[{2, 1, -2}, {1, 3, 3}, 32] (* or *)
CoefficientList[ Series[(1 + x - 4x^2)/(1 - 2x - x^2 + 2x^3), {x, 0, 31}], x] (* Robert G. Wilson v, Nov 05 2016 *)
PROG
(Magma) I:=[1, 3, 3]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Nov 06 2016
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