This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A117571 #52 Oct 11 2024 16:11:15
%S A117571 1,1,3,4,4,6,7,7,9,10,10,12,13,13,15,16,16,18,19,19,21,22,22,24,25,25,
%T A117571 27,28,28,30,31,31,33,34,34,36,37,37,39,40,40,42,43,43,45,46,46,48,49,
%U A117571 49,51,52,52,54,55,55,57,58,58,60,61,61,63,64,64,66,67,67,69,70,70,72
%N A117571 Expansion of (1+2*x^2)/((1-x)*(1-x^3)).
%C A117571 Row sums of A116948.
%C A117571 Place n+2 equally-spaced points around a circle, labeled 0,1,2,...,n+1. For each i = 0..n+1 such that 2i != i mod n+2, draw an (undirected) chord from i to 2i mod n+2. Then a(n) is the number of distinct chords. - _Kival Ngaokrajang_, May 13 2016 (Edited by _N. J. A. Sloane_, Jun 23 2016)
%C A117571 From _Gus Wiseman_, Apr 19 2019: (Start)
%C A117571 Also the number of integer partitions of n + 2 with 1 fewer distinct multiplicities than (not necessarily distinct) parts. These are partitions of the form (x,x), (x,y), (x,x,y), or (x,y,y). For example, the a(0) = 1 through a(8) = 9 partitions are the following. The Heinz numbers of these partitions are given by A325270.
%C A117571 (11) (21) (22) (32) (33) (43) (44) (54) (55)
%C A117571 (31) (41) (42) (52) (53) (63) (64)
%C A117571 (211) (221) (51) (61) (62) (72) (73)
%C A117571 (311) (411) (322) (71) (81) (82)
%C A117571 (331) (332) (441) (91)
%C A117571 (511) (422) (522) (433)
%C A117571 (611) (711) (442)
%C A117571 (622)
%C A117571 (811)
%C A117571 (End)
%H A117571 Kival Ngaokrajang, <a href="/A117571/a117571.pdf">Illustration of initial terms</a>
%H A117571 Burkard Polster, <a href="https://www.youtube.com/watch?v=qhbuKbxJsk8">Times Tables, Mandelbrot and the Heart of Mathematics</a>, Mathologer video (2015).
%H A117571 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A117571 G.f.: (1+2*x^2)/((1-x)*(1-x^3)).
%F A117571 a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.
%F A117571 a(n) = cos(2*Pi*n/3+Pi/6)/sqrt(3)-sin(2*Pi*n/3+Pi/6)/3+(3n+2)/3.
%F A117571 a(n) = Sum_{k=0..n} 2*A001045(L((n-k+2)/3)) where L(j/p) is the Legendre symbol of j and p.
%F A117571 a(n) = 1 + floor((n+1)/3) + floor(2*n/3). - _Wesley Ivan Hurt_, Jul 25 2016
%F A117571 a(n) = n+sign((n-1) mod 3). - _Wesley Ivan Hurt_, Sep 25 2017
%p A117571 A117571:=n->1 + floor(2*n/3) + floor((n+1)/3): seq(A117571(n), n=0..100); # _Wesley Ivan Hurt_, Jul 25 2016
%t A117571 CoefficientList[Series[(1 + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 71}], x] (* _Michael De Vlieger_, May 13 2016 *)
%o A117571 (Magma) [1 + Floor(2*n/3) + Floor((n+1)/3) : n in [0..100]]; // _Wesley Ivan Hurt_, Jul 25 2016
%Y A117571 Cf. A001045, A116948, A273724.
%Y A117571 Cf. A090858, A127002, A323055, A325242, A325243, A325244, A325270.
%K A117571 easy,nonn
%O A117571 0,3
%A A117571 _Paul Barry_, Mar 29 2006