A094667 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.

1, 4, 14, 48, 165, 572, 2001, 7056, 25042, 89320, 319793, 1148184, 4131009, 14885468, 53697270, 193862592, 700312381, 2530902676, 9149426897, 33083393640, 119645675898, 432748165304, 1565346866889, 5662560013488, 20484930829825, 74108882866612, 268111981441886

Offset: 1

Author

Herbert Kociemba, Jun 14 2004

Keywords

• nonn
• easy

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1791
• G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.
• László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
• Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Programs

Maple

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

Mathematica

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 *)

Formula

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).

a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4).

G.f.: x*(-1+x)*(-1+3*x)/(1-8*x+21*x^2-20*x^3+5*x^4).

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