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 A031164 #34 Jul 20 2025 11:30:18
%S A031164 1,4,15,40,99,212,429,800,1430,2424,3978,6288,9690,14520,21318,30624,
%T A031164 43263,60060,82225,110968,148005,195052,254475,328640,420732,533936,
%U A031164 672452,840480,1043460,1286832,1577532,1922496,2330445
%N A031164 Irreducible Euler sums of weight 8 and depth 10+2n.
%C A031164 a(n-9)=number of aperiodic necklaces (Lyndon words) with 8 black beads and n-8 white beads.
%H A031164 David J. Broadhurst, <a href="http://arXiv.org/abs/hep-th/9604128">On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory</a>, arXiv:hep-th/9604128, 1996.
%H A031164 David Broadhurst and Xavier Roulleau, <a href="https://arxiv.org/abs/2502.19523">Number of partitions of modular integers</a>, arXiv:2502.19523 [math.NT], 2025. See p. 19.
%H A031164 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%H A031164 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-12,17,8,-28,8,17,-12,-2,4,-1).
%F A031164 G.f.: (1+x^2)/((1-x)*(1-x^2))^4
%F A031164 a(n) = [C(n+8,7)-(n%2)*C((n+7)/2,3)]/8, where C = binomial, n%2 = parity of n (=1 if odd, 0 else). - _M. F. Hasler_, May 02 2009
%F A031164 a(0)=1, a(1)=4, a(2)=15, a(3)=40, a(4)=99, a(5)=212, a(6)=429, a(7)=800, a(8)=1430, a(9)=2424, a(10)=3978, a(11)=6288, a(n) = 4*a(n-1)-2*a(n-2)-12*a(n-3)+17*a(n-4)+8*a(n-5)-28*a(n-6)+8*a(n-7)+17*a(n-8)-12*a(n-9)- 2*a(n-10)+4*a(n-11)-a(n-12). - _Harvey P. Dale_, Jun 20 2011
%F A031164 G.f.: ((-1+x)^-8-(-1+x^2)^-4)/(8*x). - _Herbert Kociemba_, Oct 16 2016
%t A031164 Table[(Binomial[n+8,7]-If[OddQ[n],1,0]Binomial[(n+7)/2,3])/8,{n,0,40}] (* or *) CoefficientList[Series[(1+x^2)/((1-x)^8 (1+x)^4),{x,0,40}],x] (* _Harvey P. Dale_, Jun 20 2011 *)
%o A031164 (PARI) A031164(n)=(binomial(n+8,7)-if(n%2,binomial(n\2+4,3)))>>3 \\ _M. F. Hasler_, May 02 2009
%Y A031164 Cf. A000031, A001037, A051168. Row 8 in A245558.
%Y A031164 Cf. A032094. - _M. F. Hasler_, May 02 2009
%K A031164 nonn,easy
%O A031164 0,2
%A A031164 _David Broadhurst_