A271391 Expansion of (1 + x + 2*x^2 + 6*x^3 + x^4 + x^5)/(1 - x^2)^3.

1, 1, 5, 9, 13, 25, 25, 49, 41, 81, 61, 121, 85, 169, 113, 225, 145, 289, 181, 361, 221, 441, 265, 529, 313, 625, 365, 729, 421, 841, 481, 961, 545, 1089, 613, 1225, 685, 1369, 761, 1521, 841, 1681, 925, 1849, 1013, 2025, 1105, 2209, 1201, 2401, 1301, 2601, 1405

Offset: 0

Author

Ilya Gutkovskiy, Apr 06 2016

Keywords

• nonn
• easy

Comments

Centered square numbers alternating with odd squares.

Examples

			Illustration of initial terms:
                                                      o
                            o       o o o o o       o o o
          o     o o o     o o o     o o o o o     o o o o o
o   o   o o o   o o o   o o o o o   o o o o o   o o o o o o o
          o     o o o     o o o     o o o o o     o o o o o
                            o       o o o o o       o o o
                                                      o
0   1     2       3         4           5             6
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A001844, A016754.

Programs

Maple

a:=series((1+x+2*x^2+6*x^3+x^4+x^5)/(1-x^2)^3,x=0,55): seq(coeff(a,x,n),n=0..54); # Paolo P. Lava, Mar 27 2019

Mathematica

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 1, 5, 9, 13, 25}, 55]
Table[(3 n^2 + 2 n + 2 + (-1)^n (-n^2 + 2 n + 2))/4, {n, 0, 54}]

PARI

x='x+O('x^99); Vec((1+x+2*x^2+6*x^3+x^4+x^5)/(1-x^2)^3) \\ Altug Alkan, Apr 06 2016

Formula

E.g.f.: ((2 + x*(2 + x))*cosh(x) + x*(3 + 2*x)*sinh(x))/2.

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

a(n) = (3*n^2 + 2*n + 2 + (-1)^n*(-n^2 + 2*n + 2))/4.

a(2n) = A001844(n). a(2n+1) = (2n+1)^2.