A325657 a(n) = (1/2)*(-1 + (-1)^n)*(n-1) + n^2.

0, 1, 4, 7, 16, 21, 36, 43, 64, 73, 100, 111, 144, 157, 196, 211, 256, 273, 324, 343, 400, 421, 484, 507, 576, 601, 676, 703, 784, 813, 900, 931, 1024, 1057, 1156, 1191, 1296, 1333, 1444, 1483, 1600, 1641, 1764, 1807, 1936, 1981, 2116, 2163, 2304, 2353, 2500, 2551

Offset: 0

Author

Stefano Spezia, May 13 2019

Keywords

• nonn
• easy

Comments

For n > 0, a(n) is the n-th element of the diagonal of the triangle A325655. Equivalently, a(n) is the element M_{n,1} of the matrix M(n) whose permanent is A322277(n).

Links

• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000290, A054569, A317614, A322277, A325655.

Programs

GAP

Flat(List([0..55], n->(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2));

Magma

[(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2: n in [0..55]];

Maple

a:=n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): seq(a(n), n=0..55);

Mathematica

Table[(1/2)*(- 1+(-1)^n)*(n-1)+n^2,{n,0,55}]

PARI

a(n) = (1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2;

Formula

O.g.f.: (-1 - 3*x - x^2 - 3*x^3)/((-1 + x)^3*(1+x)^2).

E.g.f.: (1/2)*exp(-x)*(-1 - x + exp(2*x)*(1 + x + 2*x^2)).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4

a(n) = n^2 if n is even.

a(n) = n^2 - n + 1 if n is odd.