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

Author

Clark Kimberling, May 24 2023

Keywords

• nonn
• easy

Links

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

Crossrefs

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

Range of terms: A363284\{0}.

Programs

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}]

PARI

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

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