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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.

%I A160790 #19 Jun 13 2015 00:53:12
%S A160790 0,1,2,4,7,10,16,20,30,35,50,56,77,84,112,120,156,165,210,220,275,286,
%T A160790 352,364,442,455,546,560,665,680,800,816,952,969,1122,1140,1311,1330,
%U A160790 1520,1540,1750,1771,2002,2024,2277,2300,2576,2600,2900,2925,3250,3276,3627,3654,4032,4060,4466,4495,4930,4960,5425
%N 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.
%H A160790 Nathaniel Johnston, <a href="/A160790/b160790.txt">Table of n, a(n) for n = 0..5000</a>
%H A160790 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F A160790 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).
%F A160790 G.f.:  -x*(-1-x+x^2) / ( (1+x)^3*(x-1)^4 ).
%F A160790 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
%p A160790 A160791 := proc(n) if type(n,'odd') then ceil(n/2) ; else A000217(n/2) ; end if; end proc:
%p A160790 A160790 := proc(n) if n = 0 then 0; else add(A160791(i),i=0..n) ; end if; end proc:
%p A160790 seq(A160790(n),n=0..60) ;
%t A160790 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 *)
%o A160790 (PARI) Vec(-x*(-1-x+x^2) / ( (1+x)^3*(x-1)^4 ) + O(x^80)) \\ _Michel Marcus_, Apr 01 2015
%Y A160790 Cf. A160791, A160792.
%K A160790 easy,nonn
%O A160790 0,3
%A A160790 _Omar E. Pol_, May 29 2009
%E A160790 Edited by _Omar E. Pol_, Feb 08 2010