A363267 Squares (A000290) alternating with centered squares (A001844).

1, 1, 4, 5, 9, 13, 16, 25, 25, 41, 36, 61, 49, 85, 64, 113, 81, 145, 100, 181, 121, 221, 144, 265, 169, 313, 196, 365, 225, 421, 256, 481, 289, 545, 324, 613, 361, 685, 400, 761, 441, 841, 484, 925, 529, 1013, 576, 1105, 625, 1201, 676, 1301, 729, 1405, 784

Offset: 1

Author

Clark Kimberling, May 24 2023

Keywords

• nonn

Comments

This is a linear recurrence sequence. If the terms are arranged in nondecreasing order, the result, A363319, is linearly recurrent. If the terms are arranged in increasing order, so that there are no duplicates, the result, A363282, is not linearly recurrent.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A000290, A001844, A123596, A363268, A363269, A363282, A363319.

Programs

Mathematica

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

Formula

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).

G.f.: x*(-1 - x - x^2 - 2 x^3 - x^5)/(-1 + x^2)^3.

a(n+1) = n/2+3*n^2/8+3/4+(-1)^n*(1/4+n/2-n^2/8). - R. J. Mathar, Jun 15 2023