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A224811 Number of subsets of {1,2,...,n-8} without differences equal to 2, 4, 6 or 8.

%I A224811 #20 Sep 04 2024 09:57:11
%S A224811 1,1,1,1,1,1,1,1,1,2,4,6,9,12,16,20,25,30,36,48,64,88,121,165,225,300,
%T A224811 400,520,676,884,1156,1530,2025,2700,3600,4800,6400,8480,11236,14840,
%U A224811 19600,25900,34225,45325,60025,79625,105625,140075,185761,246101,326041,431676,571536,756756,1002001,1327326,1758276,2329782,3087049,4090296,5419584
%N A224811 Number of subsets of {1,2,...,n-8} without differences equal to 2, 4, 6 or 8.
%C A224811 Number of permutations (p(1), p(2), ..., p(n)) satisfying -k <= p(i)-i <= r and p(i)-i in the set I, i=1..n, with k=2, r=8, I={-2,0,8}.
%H A224811 G. C. Greubel, <a href="/A224811/b224811.txt">Table of n, a(n) for n = 0..1000</a>
%H A224811 Michael A. Allen, <a href="https://arxiv.org/abs/2209.01377">On a Two-Parameter Family of Generalizations of Pascal's Triangle</a>, arXiv:2209.01377 [math.CO], 2022.
%H A224811 Vladimir Baltic, <a href="http://pefmath.etf.rs/vol4num1/AADM-Vol4-No1-119-135.pdf">On the number of certain types of strongly restricted permutations</a>, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135
%H A224811 <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 1, -1, 1, -1, 1, 2, -1, 1, 0, 0, -2, 1, -2, 0, 0, -1, 0, 0, 0, 0, 1).
%F A224811 a(n) = a(n-1) +a(n-5) -a(n-6) +a(n-7) -a(n-8) +a(n-9) +2*a(n-10) -a(n-11) +a(n-12) -2*a(n-15) +a(n-16) -2*a(n-17) -a(n-20) +a(n-25).
%F A224811 G.f.: (1-x^10-x^5-x^7+x^15) / ( (1-x) *(1+x) *(x^2-x+1) *(x^3+x^2-1) *(x^6-x^2-1) *(x^12+x^10+x^8+2*x^6+x^4+1) ).
%F A224811 a(2*k) = (A003520(k))^2,
%F A224811 a(2*k+1) = A003520(k) * A003520(k+1)
%t A224811 CoefficientList[Series[(1 - x^10 - x^5 - x^7 + x^15)/((1 - x)*(1 + x)*(x^2 - x + 1)*(x^3 + x^2 - 1)*(x^6 - x^2 - 1)*(x^12 + x^10 + x^8 + 2*x^6 + x^4 + 1)), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 28 2017 *)
%o A224811 (PARI) x='x+O('x^50); Vec((1-x^10-x^5-x^7+x^15)/((1-x)*(1+x)*(x^2-x+1)*( x^3+x^2-1)*(x^6-x^2-1)*(x^12+x^10+x^8+2*x^6+x^4+1) )) \\ _G. C. Greubel_, Oct 28 2017
%Y A224811 Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A217694,A224808-A224810.
%K A224811 nonn
%O A224811 0,10
%A A224811 _Vladimir Baltic_, May 18 2013