A323723 a(n) = (-2 - (-1)^n*(-2 + n) + n + 2*n^3)/4.

0, 0, 4, 14, 32, 64, 108, 174, 256, 368, 500, 670, 864, 1104, 1372, 1694, 2048, 2464, 2916, 3438, 4000, 4640, 5324, 6094, 6912, 7824, 8788, 9854, 10976, 12208, 13500, 14910, 16384, 17984, 19652, 21454, 23328, 25344, 27436, 29678, 32000, 34480, 37044, 39774

Offset: 0

Author

Stefano Spezia, Jan 25 2019

Keywords

• nonn
• easy

Comments

For n > 1, a(n) is the subdiagonal sum of the matrix M(n) whose permanent is A322277(n).

All the terms of this sequence are even numbers (A005843).

Links

• Stefano Spezia, Table of n, a(n) for n = 0..10000
• Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A000982, A004526, A005843, A033431, A033999, A317614, A322277, A323724, A325655.

Programs

GAP

Flat(List([0..50], n -> (-2-(-1)^n*(-2+n)+n+2*n^3)/4));

Magma

[(-2-(-1)^n*(-2+n)+n+2*n^3)/4: n in [0..50]];

Maple

a:=n->(-2 - (-1)^n*(-2 + n) + n + 2*n^3)/4: seq(a(n), n=0..50);

Mathematica

a[n_]:=(6 + n + n^3 + 12 Floor[1/2 (-3 + n)] + 4 Floor[1/2 (-3 + n)]^2 - 2 (1 + n) Floor[1/2 (-1 + n)])/2; Array[a,50,0]

Maxima

makelist((-2-(-1)^n*(-2+n)+n+2*n^3)/4, n, 0, 50);

PARI

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

Python

[(-2-(-1)**n*(-2+n)+n+2*n**3)/4 for n in range(50)]

Formula

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

E.g.f.: (1/4)*exp(-x)*(2 + x)*(1 + exp(2*x)*(-1 + 2*x + 2* x^2)).

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5.

a(n) = (6 + n + n^3 + 12*floor((n - 3)/2) + 4*floor((n - 3)/2)^2 - 2*(1 + n)*floor((n - 1)/2))/2.

a(n) = (-2 - A033999(n)*(-2 + n) + n + A033431(n))/4.

a(n) = n^3/2 for even n; a(n) = (n - 1)*(n^2 + n + 2)/2 otherwise. - Bruno Berselli, Feb 06 2019

a(n) = 2*A004526(n*A000982(n)). [Found by Christian Krause's LODA miner] - Stefano Spezia, Dec 12 2021