A007068 - OEIS

%I M2360 #54 Aug 03 2024 02:29:53

%S 1,3,4,10,14,34,48,116,164,396,560,1352,1912,4616,6528,15760,22288,

%T 53808,76096,183712,259808,627232,887040,2141504,3028544,7311552,

%U 10340096,24963200,35303296,85229696,120532992,290992384,411525376

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

%C First row of spectral array W(sqrt 2).

%C Row sums of the square of the matrix with general term binomial(floor(n/2),n-k). - _Paul Barry_, Feb 14 2005

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

%H Reinhard Zumkeller, <a href="/A007068/b007068.txt">Table of n, a(n) for n = 1..1000</a>

%H Aviezri S. Fraenkel and Clark Kimberling, <a href="https://doi.org/10.1016/0012-365X(94)90259-3">Generalized Wythoff arrays, shuffles and interspersions</a>, Discr. Math. 126 (1-3) (1994) 137-149.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-2).

%F 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

%F 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

%F 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]

%F a(n) = a(n-1) + a(n-2) * A000034(n-1). [_Reinhard Zumkeller_, Jan 21 2012]

%t 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 *)

%o (Haskell)

%o a007068 n = a007068_list !! (n-1)

%o a007068_list = 1 : 3 : zipWith (+)

%o    (tail a007068_list) (zipWith (*) a000034_list a007068_list)

%o -- _Reinhard Zumkeller_, Jan 21 2012

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

%K nonn,easy,nice

%O 1,2

%A _N. J. A. Sloane_, _Mira Bernstein_

%E Better description and more terms from _Olivier Gérard_, Jun 05 2001