A255687 a(n) = n*(n + 1)*(7*n + 11)/6.

0, 6, 25, 64, 130, 230, 371, 560, 804, 1110, 1485, 1936, 2470, 3094, 3815, 4640, 5576, 6630, 7809, 9120, 10570, 12166, 13915, 15824, 17900, 20150, 22581, 25200, 28014, 31030, 34255, 37696, 41360, 45254, 49385, 53760, 58386, 63270, 68419, 73840, 79540, 85526

Offset: 0

Author

Luce ETIENNE, Mar 02 2015

Keywords

• nonn
• easy

Comments

This sequence gives the number of triangles of all sizes in (3*n^2+2*n)-polyiamonds in a pentagonal or heptagonal configuration.

Also sum of 2*n*(n+1)*(n+2)/3 triangles oriented in one direction and n*(n+1)^2/2 oriented in the opposite direction.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

First bisection of A212977.

Partial sums of A179986.

Cf. A000292, A006002, A007584, A255211.

Programs

Magma

[n*(n+1)*(7*n+11)/6: n in [0..50]]; // Bruno Berselli, Mar 02 2015

Maple

A255687:=n->n*(n+1)*(7*n+11)/6: seq(A255687(n), n=0..50); # Wesley Ivan Hurt, Mar 03 2015

Mathematica

Table[n (n + 1) (7 n + 11)/6, {n, 0, 50}] (* Bruno Berselli, Mar 02 2015 *)
LinearRecurrence[{4,-6,4,-1},{0,6,25,64},50] (* Harvey P. Dale, Jul 17 2015 *)

PARI

vector(50, n, n--; n*(n+1)*(7*n+11)/6)

PARI

concat(0, Vec(x*(x+6)/(x-1)^4 + O(x^100))) \\ Colin Barker, Mar 02 2015

Sage

[n*(n+1)*(7*n+11)/6 for n in (0..50)] # Bruno Berselli, Mar 02 2015

Formula

a(n) = (1/2)*(Sum_{j=0..n} (n+1-j)*(3*n-j) + Sum_{j=0..n-1} (n-j)*(3*n+1-3*j)).

From Colin Barker, Mar 02 2015: (Start)

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

G.f.: x*(x + 6)/(x - 1)^4. (End)

a(n) = -A007584(-n-1). - Bruno Berselli, Mar 02 2015

From Elmo R. Oliveira, Aug 18 2025: (Start)

E.g.f.: exp(x)*x*(36 + 39*x + 7*x^2)/6.

a(n) = A212977(2*n). (End)