A363269 Positive squares (A000290) alternating with positive square pyramidal numbers (A000330).

1, 1, 4, 5, 9, 14, 16, 30, 25, 55, 36, 91, 49, 140, 64, 204, 81, 285, 100, 385, 121, 506, 144, 650, 169, 819, 196, 1015, 225, 1240, 256, 1496, 289, 1785, 324, 2109, 361, 2470, 400, 2870, 441, 3311, 484, 3795, 529, 4324, 576, 4900, 625, 5525, 676, 6201, 729

Offset

1,3

Links

Table of n, a(n) for n=1..53.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Formula

a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8).

G.f.: x*(1 + x + x^3 - x^4)/(-1 + x^2)^4.

E.g.f.: ((18*x + 6*x^2)*cosh(x) + (6 + 6*x + 6*x^2 + x^3)*sinh(x))/24. - Stefano Spezia, Jun 10 2023

48*a(n) = (n+1) * (n^2 +(-1)^n*n^2 -4*(-1)^n*n +8*n -6*(-1)^n +6). - R. J. Mathar, Jun 22 2023

Mathematica

c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, c[n - 2] + c[n - 1]]
Table[c[n], {n, 1, 120}]

Prog

(PARI) a(n) = if(n%2, (n+1)^2/4, n*(n+1)*(n+2)/24); \\ Kevin Ryde, Jun 10 2023

Crossrefs

Cf. A000290, A000330, A123596, A363267, A363268.

Range of terms: A363284\{0}.

Sequence in context: A038099 A243166 A363284 * A120740 A274282 A347553

Adjacent sequences: A363266 A363267 A363268 * A363270 A363271 A363272

Keyword

nonn,easy

Author

Clark Kimberling, May 24 2023

Status

approved