A158780 a(2n) = A131577(n), a(2n+1) = A011782(n).

0, 1, 1, 1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152, 4194304

Offset

0,5

Comments

This construction combines the 2 basic sequences which equal their first differences in the same way as A138635 does for sequences which equal their 3rd differences and A137171 does for sequences which equal their fourth differences.

Essentially the same as A016116, A060546, and A131572. - R. J. Mathar, Apr 08 2009

Dropping a(0), this is the inverse binomial transform of A024537. - R. J. Mathar, Apr 08 2009

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

a(2n) + a(2n+1) = A000079(n).

G.f.: x*(1+x-x^2)/(1-2*x^2). - R. J. Mathar, Apr 08 2009

a(n) = (1/2)*(2^floor(n/2) + [n=1] - [n=0]). - G. C. Greubel, Apr 19 2023

E.g.f.: (2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) + 2*x - 2)/4. - Stefano Spezia, May 13 2023

Mathematica

Table[(2^Floor[n/2] +Boole[n==1] -Boole[n==0])/2, {n, 0, 50}] (* or *) LinearRecurrence[{0, 2}, {0, 1, 1, 1}, 51] (* G. C. Greubel, Apr 19 2023 *)

Prog

(PARI) a(n)=if(n>3, ([0, 1; 2, 0]^n*[1; 1])[1, 1]/2, n>0) \\ Charles R Greathouse IV, Oct 18 2022
(Magma) [0, 1] cat [2^Floor((n-2)/2): n in [2..50]]; // G. C. Greubel, Apr 19 2023
(SageMath)
def A158780(n): return (2^(n//2) + int(n==1) - int(n==0))/2
[A158780(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

Crossrefs

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

Sequence in context: A016116 A060546 A163403 * A231208 A306663 A222955

Adjacent sequences: A158777 A158778 A158779 * A158781 A158782 A158783

Keyword

nonn,easy

Author

Paul Curtz, Mar 26 2009

Extensions

Edited by R. J. Mathar, Apr 08 2009

Status

approved