A323724 a(n) = n*(2*(n - 2)*n + (-1)^n + 3)/4.

0, 0, 2, 6, 20, 40, 78, 126, 200, 288, 410, 550, 732, 936, 1190, 1470, 1808, 2176, 2610, 3078, 3620, 4200, 4862, 5566, 6360, 7200, 8138, 9126, 10220, 11368, 12630, 13950, 15392, 16896, 18530, 20230, 22068, 23976, 26030, 28158, 30440, 32800, 35322, 37926, 40700

Offset: 0

Author

Stefano Spezia, Jan 25 2019

Keywords

• nonn
• easy

Comments

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

All the terms of this sequence are even numbers (A005843), but do not end with 4.

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. A000290, A000982, A004526, A005843, A005997, A317614, A322277, A323723, A325516.

Programs

GAP

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

Magma

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

Maple

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

Mathematica

a[n_] := 1/2 (-1 + n)^2 n - (-1 + n) Floor[n/2] + 2 Floor[n/2]^2; Array[a, 50, 0];
Table[n (2 (n - 2) n + (-1)^n + 3)/4, {n, 0, 50}] (* Bruno Berselli, Feb 06 2019 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,0,2,6,20,40},50] (* Harvey P. Dale, Jan 13 2024 *)

Maxima

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

PARI

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

PARI

T(i,j,n) = if (i %2, j + n*(i-1), n*i - j + 1);
a(n) = sum(k=1, n-1, T(k,k+1,n)); \\ Michel Marcus, Feb 06 2019

Python

[int((1/2)*(-1 + n)**2*n - (-1 + n)*int(n/2) + 2*(int(n/2))**2) for n in range(0,50)]

Formula

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

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

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

a(n) = (1/2)*(-1 + n)^2*n - (-1 + n)*A004526(n) + 2*A000290(A004526(n)).

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

a(n) = 2*A004526(n*A000982(n-1)). [Found by Christian Krause's LODA miner] - Stefano Spezia, Dec 12 2021

a(n) = 2*A005997(n-1) for n >= 2. - Hugo Pfoertner, Dec 13 2021

Extensions

Definition by Bruno Berselli, Feb 06 2019