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

0,3

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).

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

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 *)

Prog

(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]];
(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)]

Crossrefs

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

Sequence in context: A202963 A130315 A087150 * A214307 A087134 A378027

Adjacent sequences: A323721 A323722 A323723 * A323725 A323726 A323727

Keyword

nonn,easy

Author

Stefano Spezia, Jan 25 2019

Extensions

Definition by Bruno Berselli, Feb 06 2019

Status

approved