A005314 -id:A005314 - cp's OEIS frontend

A225800 Triangle of rising diagonals of A011973 (with rows displayed as centered text).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 5, 4, 1, 1, 7, 6, 1, 1, 1, 9, 8, 6, 3, 1, 1, 11, 10, 15, 10, 1, 1, 13, 12, 28, 21, 1, 1, 1, 15, 14, 45, 36, 10, 4, 1, 1, 17, 16, 66, 55, 35, 20, 1, 1, 19, 18, 91, 78, 84, 56, 1, 1, 1, 21, 20, 120, 105, 165, 120, 15, 5, 1

Offset: 1

John Molokach, Jul 27 2013

Row sums are A227300.

			Triangle begins:
  1;
  1,  1;
  1,  1;
  1,  1,  1;
  1,  1,  3,  2;
  1,  1,  5,  4;
  1,  1,  7,  6,  1;
  1,  1,  9,  8,  6,  3;
  1,  1, 11, 10, 15, 10;
  1,  1, 13, 12, 28, 21,  1;

Cf. A011973, A000045, A227300, A005314, A224838.

Table[Binomial[2 n - k - 2 - 3 Floor[k/2], Floor[k/2]], {n, 1, 25}, {k, 0, Floor[(2 n - 1)/3]}] (* John Molokach, Jul 29 2013 *)

r(n) = binomial(2n-k-2-3*floor(k/2), floor(k/2)), k = 0..floor((2n-1)/3). - John Molokach, Jul 29 2013

A301772 Number of odd chordless cycles in the n-antiprism graph.

Original entry on oeis.org

0, 2, 0, 2, 8, 2, 24, 16, 48, 92, 100, 310, 344, 808, 1344, 2102, 4480, 6462, 13092, 21662, 37488, 69904, 113652, 212844, 359856, 636402, 1134068, 1937072, 3493120, 6012746, 10639264, 18706394, 32550976, 57727738, 100407848, 177116816, 310493720, 543717148

Offset: 0

Eric W. Weisstein, Mar 26 2018

Sequence extended to a(0)-a(3) using the formula/recurrence (actual 3-antiprism count is 0).

Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Chordless Cycle
Index entries for linear recurrences with constant coefficients, signature (0, 2, 2, -1, 2, -1).

Table[(RootSum[-1 + #1 - 2 #1^2 + #1^3 &, #1^n &] - RootSum[-1 + #1 + 2 #1^2 + #1^3 &, #1^n &])/2, {n, 0, 20}]
LinearRecurrence[{0, 2, 2, -1, 2, -1}, {0, 2, 0, 2, 8, 2}, 20]
CoefficientList[Series[2 x (1 - x^2 + 2 x^3)/(1 - 2 x^2 - 2 x^3 + x^4 - 2 x^5 + x^6), {x, 0, 20}], x]

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).
G.f.: 2*x*(1 - x^2 + 2*x^3)/( (x^3-x^2-2*x-1)*(x^3-x^2+2*x-1)).
2*a(n) = -3*A077990(n) -4*A077990(n-1)-A077990(n-2) +3*A005314(n+1) -4*A005314(n)+A005314(n-1). - R. J. Mathar, Feb 25 2024

cp's OEIS Frontend

A225800 Triangle of rising diagonals of A011973 (with rows displayed as centered text).

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