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 A137726 #21 Feb 16 2025 08:33:07
%S A137726 2,2,8,9,12,16,23,29,41,56,79,110,158,225,325,469,682,991,1446,2110,
%T A137726 3085,4511,6603,9666,14157,20736,30380,44511,65223,95575,140060,
%U A137726 205253,300800,440828,646051,946817,1387613,2033628,2980411,4367986,6401578,9381949,13749897,20151433,29533342
%N A137726 Number of sequences of length n with elements {-2,-1,+1,+2}, counted up to simultaneous reversal and negation, such that the sum of elements of the whole sequence but of no proper subsequence equals 0 modulo n. For n>=4, the number of Hamiltonian (undirected) cycles on the circulant graph C_n(1,2).
%C A137726 For n>1, the number of circular permutations (counted up to rotations and reversals) of {0, 1,...,n-1} such that the distance between every two adjacent elements is -2,-1,1,or 2 modulo n.
%H A137726 G. C. Greubel, <a href="/A137726/b137726.txt">Table of n, a(n) for n = 1..1000</a>
%H A137726 Mordecai J. Golin and Yiu Cho Leung, <a href="http://www.cse.ust.hk/tcsc/RR/2004-02.ps.gz">Unhooking Circulant Graphs: A Combinatorial Method for Counting Spanning Trees, Hamiltonian Cycles and other Parameters</a>. Technical report HKUST-TCSC-2004-02.
%H A137726 Spoj, <a href="https://www.spoj.pl/problems/JZPCIR/">Problem 7709: Jumping Zippy</a>
%H A137726 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantGraph.html">Circulant Graph</a>
%H A137726 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,0,-1,1).
%F A137726 For even n>=4, a(n) = n + 3*A000930(n) - 2*A000930(n-1); for odd n>=3, a(n) = n + 1 + 3*A000930(n) - 2*A000930(n-1).
%F A137726 For n>8, a(n) = 2*a(n-1) - a(n-3) - a(n-5) + a(n-6) or a(n) = a(n-1) + a(n-2) - a(n-5) - 2.
%F A137726 a(n) = A137725(n) / 2.
%F A137726 G.f.: -x*(x^7+2*x^5-4*x^4-5*x^3+4*x^2-2*x+2)/((x-1)^2*(x+1)*(x^3+x-1)). - _Colin Barker_, Aug 22 2012
%t A137726 Rest[CoefficientList[Series[-x*(x^7 + 2*x^5 - 4*x^4 - 5*x^3 + 4*x^2 - 2*x + 2)/((x - 1)^2*(x + 1)*(x^3 + x - 1)), {x, 0, 50}], x]] (* _G. C. Greubel_, Apr 27 2017 *)
%o A137726 (PARI) x='x+O('x^50); Vec(-x*(x^7 + 2*x^5 - 4*x^4 - 5*x^3 + 4*x^2 - 2*x + 2)/((x - 1)^2*(x + 1)*(x^3 + x - 1))) \\ _G. C. Greubel_, Apr 27 2017
%Y A137726 Cf. A069241, A124353, A137725.
%K A137726 nonn,easy
%O A137726 1,1
%A A137726 _Max Alekseyev_, Feb 08 2008
%E A137726 Formulae corrected by _Max Alekseyev_, Nov 03 2010