A137171 -id:A137171 - cp's OEIS frontend

A137172 Numbers in the array A137171.

0, 1, 2, 3, 4, 6, 10, 12, 16, 20, 28, 36, 56, 64, 72, 120, 136, 240, 256, 272, 496, 528, 992, 1024, 1056, 2016, 2080, 4032, 4096, 4160, 8128, 8256, 16256, 16384, 16512, 32640, 32896, 65280, 65536, 65792, 130816, 131328, 261632, 262144, 262656, 523776, 524800, 1047552

Offset: 1

Paul Curtz, May 11 2008

nonn

Union of the first three rows of the array in A137171, sorted.

Definition corrected and more terms added by R. J. Mathar, Jul 29 2009

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

Original entry on oeis.org

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

Paul Curtz, Mar 26 2009

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2).

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

[0,1] cat [2^Floor((n-2)/2): n in [2..50]]; // G. C. Greubel, Apr 19 2023

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 *)

a(n)=if(n>3,([0,1; 2,0]^n*[1;1])[1,1]/2,n>0) \\ Charles R Greathouse IV, Oct 18 2022

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

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

Edited by R. J. Mathar, Apr 08 2009

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A137172 Numbers in the array A137171.

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

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