A256666 a(n) = ( 2*n*(2*n^2 + 11*n + 26) - (-1)^n + 1 )/16.

0, 5, 14, 29, 51, 82, 123, 176, 242, 323, 420, 535, 669, 824, 1001, 1202, 1428, 1681, 1962, 2273, 2615, 2990, 3399, 3844, 4326, 4847, 5408, 6011, 6657, 7348, 8085, 8870, 9704, 10589, 11526, 12517, 13563, 14666, 15827, 17048, 18330, 19675, 21084, 22559

Offset: 0

Author

Luce ETIENNE, Apr 07 2015

Keywords

• nonn
• easy

Comments

Consider a grid of small triangles of side 1 forming polygon with side n*(n+3): a(n) is the number of equilateral triangles of side length >=1 in this figure that are oriented with the sides of figure.

This sequence gives the number of triangles of all sizes in a ((n^2+3*n))-iamond with a 3*(2*n-1)-gon n>=1.

Equals (1/2)*Sum_{i=0..n-1} (n-i)*(n+1-i) + (-3 + (1/8)*Sum_{j=0..(2*n+3+(-1)^n)/4} (2*n+5-(-1)^n-4*j)*(2 n+5+(-1)^n-4*j) ) numbers of triangles in a direction and in the opposite direction.

It is also a way (3 stages) to surround triangular n^2-iamonds by 3*n triangles side 1: in first stage we obtain A045947, in second stage A248851, in third stage this sequence.

Examples

			From third comment: a(0)=0, a(1)=1+4, a(2)=4+10, a(3)=10+19, a(4)=20+31, a(5)=35+47, a(6)=56+67.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Luce ETIENNE, Illustration of initial terms.
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A002623, A028552, A045947, A117143, A164004, A248851.

Programs

Magma

[(4*n^3+22*n^2+52*n+1-(-1)^n)/16: n in [0..50]]; // Vincenzo Librandi, Apr 08 2015

Mathematica

Table[(4 n^3 + 22 n^2 + 52 n + 1 - (-1)^n)/16, {n, 0, 50}] (* Vincenzo Librandi, Apr 08 2015 *)
LinearRecurrence[{3,-2,-2,3,-1},{0,5,14,29,51},50] (* Harvey P. Dale, Aug 18 2020 *)

PARI

concat(0, Vec(x*(2*x^3-3*x^2-x+5)/((x-1)^4*(x+1)) + O(x^100))) \\ Colin Barker, Apr 07 2015

Formula

a(n) = 2*A248851(n) - A045947(n) + A004526(n+1).

G.f.: x*(2*x^3-3*x^2-x+5) / ((x-1)^4*(x+1)). - Colin Barker, Apr 07 2015

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4. - Colin Barker, Apr 07 2015