A322462 Numbers on the 0-1-12 line in a spiral on a grid of equilateral triangles.

0, 1, 12, 13, 36, 37, 72, 73, 120, 121, 180, 181, 252, 253, 336, 337, 432, 433, 540, 541, 660, 661, 792, 793, 936, 937, 1092, 1093, 1260, 1261, 1440, 1441, 1632, 1633, 1836, 1837, 2052, 2053, 2280, 2281, 2520, 2521, 2772, 2773, 3036, 3037, 3312, 3313, 3600

Offset: 0

Author

Hans G. Oberlack, Dec 09 2018

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 1, 12, ... in the triangle spiral.

Examples

			a(0) = 0
a(1) = a(1 - 1) + 1 = 0 + 1
a(2) = (3/2) * 2 * (2 + 2) = 3 * 4 = 12
a(3) = a(3 - 1) + 1 = 12 + 1 = 13
a(4) = (3/2) * 4*(4 + 2) = 3 * 2 * 6 = 6 * 6 = 36
a(5) = a(4) + 1 = 36 + 1 = 37.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Hans G. Oberlack, Triangle spiral line 0-1-12-13
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A049598.

Programs

Maple

seq(coeff(series(-x*(x^3-x^2+11*x+1)/((x+1)^2*(x-1)^3),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Dec 19 2018

Mathematica

a[0] = 0; a[n_] := a[n] = If[OddQ[n], a[n - 1] + 1, 3/2*n*(n + 2)]; Array[a, 50, 0] (* Amiram Eldar, Dec 09 2018 *)

PARI

concat(0, Vec(x*(1 + 11*x - x^2 + x^3) / ((1 - x)^3*(1 + x)^2) + O(x^40))) \\ Colin Barker, Dec 09 2018

Formula

a(n) = (3/2)*n*(n+2) = A049598(n/2) if n even, a(n) = a(n-1)+1 if n odd.

G.f.: -x*(x^3-x^2+11*x+1)/((x+1)^2*(x-1)^3). - Alois P. Heinz, Dec 09 2018

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4. - Colin Barker, Dec 09 2018

Extensions

Examples added by Hans G. Oberlack, Dec 20 2018