A131572 a(0) = 0 and a(1) = 1, continued such that absolute values of 2nd differences equal the original sequence.

0, 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

Offset: 0

Author

Paul Curtz, Aug 28 2007

Keywords

• nonn
• easy

Comments

This is the main sequence of a family of sequences starting at a(0) = A and a(1) = B, continuing a(3, ...) = 2B, 2B, 4B, 4B, 8B, 8B, 16B, 16B, 32B, 32B, ... such that the absolute values of the 2nd differences, abs(a(n+2) - 2*a(n+1) + a(n)), equal the original sequence. Alternatively starting at a(0) = a(1) = 1 gives A016116.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (0,2).

Crossrefs

Cf. A000007, A131575.

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

Magma

[2^Floor(n/2)-0^n: n in [0..50]]; // Vincenzo Librandi, Aug 18 2011

Mathematica

LinearRecurrence[{0,2},{0,1,2},50] (* Harvey P. Dale, Jul 10 2018 *)

SageMath

[0]+[2^(n//2) for n in range(1,51)] # G. C. Greubel, Apr 22 2023

Formula

a(n) = 2*a(n-2), n>2.

O.g.f.: x*(1+2*x)/(1-2*x^2). - R. J. Mathar, Jul 16 2008

a(n) = A016116(n) - A000007(n), that is, a(0)=0, a(n) = A016116(n) for n>=1. - Bruno Berselli, Apr 13 2011

First differences: a(n+1) - a(n) = A131575(n).

Second differences: A131575(n+1) - A131575(n) = (-1)^n*a(n).

E.g.f.: -1 + cosh(sqrt(2)*x) + (1/sqrt(2))*sinh(sqrt(2)*x). - G. C. Greubel, Apr 22 2023

Extensions

Edited by R. J. Mathar, Jul 16 2008

More terms from Vincenzo Librandi, Aug 18 2011