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
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (0,2).
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
Programs
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Magma
[0,1] cat [2^Floor((n-2)/2): n in [2..50]]; // G. C. Greubel, Apr 19 2023
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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 *) -
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
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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
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
Extensions
Edited by R. J. Mathar, Apr 08 2009
Comments