A007068 a(n) = a(n-1) + (3+(-1)^n)*a(n-2)/2.

1, 3, 4, 10, 14, 34, 48, 116, 164, 396, 560, 1352, 1912, 4616, 6528, 15760, 22288, 53808, 76096, 183712, 259808, 627232, 887040, 2141504, 3028544, 7311552, 10340096, 24963200, 35303296, 85229696, 120532992, 290992384, 411525376, 993510144, 1405035520, 3392055808

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy
• nice

Comments

First row of spectral array W(sqrt 2).

Row sums of the square of the matrix with general term binomial(floor(n/2),n-k). - Paul Barry, Feb 14 2005

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Aviezri S. Fraenkel and Clark Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discr. Math. 126 (1-3) (1994) 137-149.
• Sean A. Irvine, Walks on Graphs.
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Crossrefs

Cf. A062112, A001882, A007070, A007052, A000034.

Programs

Haskell

a007068 n = a007068_list !! (n-1)
a007068_list = 1 : 3 : zipWith (+)
   (tail a007068_list) (zipWith (*) a000034_list a007068_list)
-- Reinhard Zumkeller, Jan 21 2012

Mathematica

RecurrenceTable[{a[1]==1,a[2]==3,a[n]==a[n-1]+(3+(-1)^n) a[n-2]/2},a[n],{n,40}] (* Harvey P. Dale, Nov 12 2012 *)

Formula

a(2n+1) = a(2n)+a(2n-1); a(2n) = a(2n-1)+2*a(2n-2); same recurrence (mod parity) as A001882. - Len Smiley, Feb 05 2001

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(floor(n/2), n-j)*C(floor(j/2), j-k). - Paul Barry, Feb 14 2005

a(n) = 4*a(n-2)-2*a(n-4). G.f.: -x*(1+x)*(2*x^2-2*x-1)/(1-4*x^2+2*x^4). a(2n+1)=A007070(n). a(2n)=A007052(n). [R. J. Mathar, Aug 17 2009]

a(n) = a(n-1) + a(n-2) * A000034(n-1). [Reinhard Zumkeller, Jan 21 2012]

Extensions

Better description and more terms from Olivier Gérard, Jun 05 2001