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

Author

Henry Bottomley, May 13 2001

Keywords

• nonn
• easy

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).

Programs

Mathematica

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

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

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