A160790 Vertex number of a rectangular spiral. The first differences (A160791) are the edge lengths of the spiral. The distances between two nearest edges, that are parallel to the initial edge, are the natural numbers.

0, 1, 2, 4, 7, 10, 16, 20, 30, 35, 50, 56, 77, 84, 112, 120, 156, 165, 210, 220, 275, 286, 352, 364, 442, 455, 546, 560, 665, 680, 800, 816, 952, 969, 1122, 1140, 1311, 1330, 1520, 1540, 1750, 1771, 2002, 2024, 2277, 2300, 2576, 2600, 2900, 2925, 3250, 3276, 3627, 3654, 4032, 4060, 4466, 4495, 4930, 4960, 5425

Offset: 0

Author

Omar E. Pol, May 29 2009

Keywords

• easy
• nonn

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Crossrefs

Cf. A160791, A160792.

Programs

Maple

A160791 := proc(n) if type(n,'odd') then ceil(n/2) ; else A000217(n/2) ; end if; end proc:
A160790 := proc(n) if n = 0 then 0; else add(A160791(i),i=0..n) ; end if; end proc:
seq(A160790(n),n=0..60) ;

Mathematica

Table[(2*n + 3 + (-1)^n)*(2*n + 3 - 3*(-1)^n)*(2*n + 15 + 5*(-1)^n)/ 384, {n, 0, 60}] (* Michael De Vlieger, Mar 31 2015 *)

PARI

Vec(-x*(-1-x+x^2) / ( (1+x)^3*(x-1)^4 ) + O(x^80)) \\ Michel Marcus, Apr 01 2015

Formula

a(n) = +a(n-1) +3*a(n-2) -3*a(n-3) -3*a(n-4) +3*a(n-5) +a(n-6) -a(n-7).

G.f.: -x*(-1-x+x^2) / ( (1+x)^3*(x-1)^4 ).

a(n) = (2*n+3+(-1)^n)*(2*n+3-3*(-1)^n)*(2*n+15+5*(-1)^n)/384. - Luce ETIENNE, Mar 31 2015

Extensions

Edited by Omar E. Pol, Feb 08 2010