A322462 - OEIS

%I #50 Feb 14 2019 14:24:01

%S 0,1,12,13,36,37,72,73,120,121,180,181,252,253,336,337,432,433,540,

%T 541,660,661,792,793,936,937,1092,1093,1260,1261,1440,1441,1632,1633,

%U 1836,1837,2052,2053,2280,2281,2520,2521,2772,2773,3036,3037,3312,3313,3600

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

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

%H Colin Barker, <a href="/A322462/b322462.txt">Table of n, a(n) for n = 0..1000</a>

%H Hans G. Oberlack, <a href="/A322462/a322462.png">Triangle spiral line 0-1-12-13</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

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

%F G.f.: -x*(x^3-x^2+11*x+1)/((x+1)^2*(x-1)^3). - _Alois P. Heinz_, Dec 09 2018

%F 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

%e a(0) = 0

%e a(1) = a(1 - 1) + 1 = 0 + 1

%e a(2) = (3/2) * 2 * (2 + 2) = 3 * 4 = 12

%e a(3) = a(3 - 1) + 1 = 12 + 1 = 13

%e a(4) = (3/2) * 4*(4 + 2) = 3 * 2 * 6 = 6 * 6 = 36

%e a(5) = a(4) + 1 = 36 + 1 = 37.

%p 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

%t 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 *)

%o (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

%Y Cf. A049598.

%K nonn,easy

%O 0,3

%A _Hans G. Oberlack_, Dec 09 2018

%E Examples added by _Hans G. Oberlack_, Dec 20 2018