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%I #38 Oct 14 2022 16:32:00
%S 1,4,14,48,165,572,2001,7056,25042,89320,319793,1148184,4131009,
%T 14885468,53697270,193862592,700312381,2530902676,9149426897,
%U 33083393640,119645675898,432748165304,1565346866889,5662560013488,20484930829825,74108882866612,268111981441886
%N Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 1, s(2n+1) = 4.
%H Michael De Vlieger, <a href="/A094667/b094667.txt">Table of n, a(n) for n = 1..1791</a>
%H G. Dresden and Y. Li, <a href="https://arxiv.org/abs/2210.04322">Periodic Weighted Sums of Binomial Coefficients</a>, arXiv:2210.04322 [math.NT], 2022.
%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-21,20,-5).
%F a(n) = (1/5)*Sum_{r=1..9} sin(r*Pi/10)*sin(2*r*Pi/5)*(2*cos(r*Pi/10))^(2*n+1).
%F a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4).
%F G.f.: x*(-1+x)*(-1+3*x)/(1-8*x+21*x^2-20*x^3+5*x^4).
%F a(n) = Sum_{k=0..n} binomial(2*n,n+k)*(k|20), where (k|20) represents the Kronecker symbol. - _Greg Dresden_, Oct 09 2022
%p with(NumberTheory): a := n -> add(binomial(2*n, n+k)*KroneckerSymbol(k, 20), k = 0..n): seq(a(n), n = 1..28); # _Peter Luschny_, Oct 14 2022
%t Rest@ CoefficientList[Series[x (-1 + x)*(-1 + 3*x)/(1 - 8 x + 21 x^2 - 20 x^3 + 5 x^4), {x, 0, 24}], x] (* _Michael De Vlieger_, Aug 04 2021 *)
%K nonn,easy
%O 1,2
%A _Herbert Kociemba_, Jun 14 2004
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