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

0,3

Comments

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

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).

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A049598.

Sequence in context: A106323 A177796 A342942 * A037304 A041298 A041689

Adjacent sequences: A322459 A322460 A322461 * A322463 A322464 A322465

Keyword

nonn,easy

Author

Hans G. Oberlack, Dec 09 2018

Extensions

Examples added by Hans G. Oberlack, Dec 20 2018

Status

approved