A007068 a(n) = a(n-1) + (3+(-1)^n)*a(n-2)/2. (Formerly M2360)

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

Offset

1,2

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

A. S. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discr. Math. 126 (1-3) (1994) 137-149.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

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]

a(n) = (1-1/2*(-1)^n)*((2-sqrt(2))^(1/4*(2*n+(-1)^n-1))+(2+sqrt(2))^(1/4*(2*n+(-1)^n-1)))+(1/4*sqrt(2)*(-1)^n-3/4*sqrt(2))*((2-sqrt(2))^(1/4*(2*n+(-1)^n-1))-(2+sqrt(2))^(1/4*(2*n+(-1)^n-1))). - Paolo P. Lava, Feb 13 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 *)

Prog

(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

Crossrefs

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

Sequence in context: A025084 A134512 A106523 * A121720 A056515 A056516

Adjacent sequences: A007065 A007066 A007067 * A007069 A007070 A007071

Keyword

nonn,easy,nice,changed

Author

N. J. A. Sloane, Mira Bernstein

Extensions

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

Status

approved