A094821 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 5.

1, 4, 15, 54, 190, 660, 2276, 7816, 26776, 91600, 313104, 1069728, 3653728, 12477504, 42606656, 145479808, 496722304, 1695962368, 5790470400, 19770087936, 67499673088, 230459040768, 786837865472, 2686435477504, 9172070373376, 31315418927104, 106917551738880, 365039402655744, 1246322574254080

Offset

1,2

Comments

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..1875

Index entries for linear recurrences with constant coefficients, signature (6,-10,4).

Formula

a(n) = (1/4)*Sum_{r=1..7} sin(3*r*Pi/8)*sin(5*r*Pi/8)*(2*cos(r*Pi/8))^(2n).

a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3), n >= 4.

G.f.: x*(1-x)^2 / ( (1-2*x)*(1-4*x+2*x^2) ).

4*a(n) = 2*A007052(n) - 2^n. - R. J. Mathar, Nov 14 2019

E.g.f.: (exp(2*x)*(2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) - 1) - 1)/4. - Stefano Spezia, Apr 25 2023

Mathematica

Rest@ CoefficientList[Series[x (1-x)^2/((1 - 2 x) (1 -4 x +2 x^2)), {x, 0, 35}], x] (* Michael De Vlieger, Feb 12 2022 *)

Prog

(Magma)
I:=[1, 4, 15]; [n le 3 select I[n] else 6*Self(n-1) -10*Self(n-2) +4*Self(n-3): n in [1..50]]; // G. C. Greubel, Mar 03 2026
(SageMath)
A007052= BinaryRecurrenceSequence(4, -2, 1, 3)
def A094821(n): return (A007052(n) - 2^(n-1))//2
print([A094821(n) for n in range(1, 51)]) # G. C. Greubel, Mar 03 2026

Crossrefs

Cf. A007052.

Sequence in context: A090326 A291032 A006234 * A071723 A001559 A002311

Adjacent sequences: A094818 A094819 A094820 * A094822 A094823 A094824

Keyword

nonn,easy

Author

Herbert Kociemba, Jun 12 2004

Status

approved