A094834 Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 9 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = 3, s(2n+1) = 6.

1, 5, 21, 82, 308, 1131, 4096, 14705, 52497, 186733, 662630, 2347680, 8309143, 29388368, 103895601, 367187437, 1297452581, 4583924154, 16193659132, 57204089987, 202065531888, 713750040577, 2521114546457, 8905002445437

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+1) counts (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n+1, s(0) = j, s(2n+1) = k.

This sequence is the odd bisection of A188048. - John Blythe Dobson, Jun 20 2015

Links

Table of n, a(n) for n=1..24.

Index entries for linear recurrences with constant coefficients, signature (6,-9,1).

Formula

a(n) = (2/9)*Sum_{r=1..8} sin(r*Pi/3)*sin(2*r*Pi/3)*(2*cos(r*Pi/9))^(2n+1).

a(n) = 6*a(n-1) - 9*a(n-2) + a(n-3).

G.f.: x(-1+x)/(-1 + 6x - 9x^2 + x^3).

a(n) = A094829(n+1) - A094829(n). - R. J. Mathar, Nov 15 2019

Mathematica

CoefficientList[Series[(x - 1)/(- 1 + 6 x - 9 x^2 + x^3), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 21 2015 *)
LinearRecurrence[{6, -9, 1}, {1, 5, 21}, 30] (* Harvey P. Dale, Dec 27 2019 *)

Prog

(Magma) I:=[1, 5, 21]; [n le 3 select I[n] else 6*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 21 2015

Crossrefs

Sequence in context: A346225 A051196 A273454 * A303647 A147504 A217783

Adjacent sequences: A094831 A094832 A094833 * A094835 A094836 A094837

Keyword

nonn,easy

Author

Herbert Kociemba, Jun 13 2004

Status

approved