A060995 Number of routes of length 2n on the sides of an octagon from a point to opposite point.

0, 2, 8, 28, 96, 328, 1120, 3824, 13056, 44576, 152192, 519616, 1774080, 6057088, 20680192, 70606592, 241065984, 823050752, 2810071040, 9594182656, 32756588544, 111837988864, 381838778368, 1303679135744

Offset

1,2

Comments

Also the 2nd row in the 2-shuffle Phi_2(W(sqrt(2)) of the Fraenkel-Kimberling publication. [R. J. Mathar, Aug 17 2009].

First differences of A056236. - Jeremy Gardiner, Aug 11 2013

Links

Harry J. Smith, Table of n, a(n) for n=1,...,200

Tomislav Doslic, I. Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea 11 (2016) 255-276.

International Mathematical Olympiad, 1979 Problem 6

A. S. Fraenkel, C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discr. Math. 126 (1-3) (1994) 137-149. [From R. J. Mathar, Aug 17 2009]

Index entries for linear recurrences with constant coefficients, signature (4, -2).

Formula

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

a(n) = (2 + sqrt(2))^(n-1)/sqrt(2) - (2-sqrt(2))^(n-1)/sqrt(2).

a(n) = 4*a(n-1)-2*a(n-2).

a(n) = 2*A007070(n-2)

G.f.: G(0)/(2*x) - 1/x, where G(k)= 1 + 1/( 1 - 4*x^2/(4*x^2 + 2*(1-2*x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013

Mathematica

LinearRecurrence[{4, -2}, {0, 2}, 40] (* Harvey P. Dale, Mar 03 2012 *)

Prog

(PARI) { for (n=1, 200, if (n>2, a=4*a1 - 2*a2; a2=a1; a1=a, if (n==1, a=a2=0, a=a1=2)); write("b060995.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 16 2009
(Sage) [(lucas_number2(n, 4, 2)-lucas_number2(n-1, 4, 2)) for n in range(0, 24)] # Zerinvary Lajos, Nov 10 2009

Crossrefs

Sequence in context: A090426 A279193 A280279 * A106731 A318010 A291383

Adjacent sequences: A060992 A060993 A060994 * A060996 A060997 A060998

Keyword

nonn

Author

Henry Bottomley, May 13 2001

Status

approved