A094297 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 2.

1, 3, 7, 18, 46, 120, 316, 840, 2248, 6048, 16336, 44256, 120160, 326784, 889792, 2424960, 6613120, 18043392, 49247488, 134450688, 367134208, 1002645504, 2738510848, 7480215552, 20433258496, 55818559488, 152486858752

Offset: 1

Author

Herbert Kociemba, Jun 02 2004

Keywords

• easy
• nonn

Comments

In general, a(n,m,j,k) = (2/m)*Sum_{r=1..m-1) sin(j*r*Pi/m)*sin(k*r*Pi/m)*(1+2*cos(Pi*r/m))^n is the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = j, s(n) = k.

Links

• Robert Munafo, Sequences Related to Floretions
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-4).

Crossrefs

First differences of A038508.

Formula

a(n) = (1/3)*Sum_{k=1..5} sin(Pi*k/3)^2*(1+2*cos(Pi*k/6))^n or a(n) = (2^n + (1-sqrt(3))^n + (1 + sqrt(3))^n)/4.

(a(n)) seems to be given by tesseq(- 2'i + 2'j + 2'k - 2i' + 2j' + 2k' - 2'ii' + 2'jj' - 'kk' - 2.5'ik' - 1.5'jk' - 2.5'ki' - 1.5'kj' - e) (disregarding signs) - Creighton Dement, Nov 17 2004

G.f.: ( 1-x-3*x^2 )*x / ( (2*x-1)*(2*x^2+2*x-1) ). - R. J. Mathar, Sep 11 2019

4*a(n) = 2^n + 2*A026150(n). - R. J. Mathar, Oct 25 2022