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

0,3

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).

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

Mathematica

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

Prog

(Magma) [2^Floor(n/2)-0^n: n in [0..50]]; // Vincenzo Librandi, Aug 18 2011
(SageMath) [0]+[2^(n//2) for n in range(1, 51)] # G. C. Greubel, Apr 22 2023

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

Sequence in context: A338739 A076939 A117575 * A152166 A320770 A016116

Adjacent sequences: A131569 A131570 A131571 * A131573 A131574 A131575

Keyword

nonn,easy

Author

Paul Curtz, Aug 28 2007

Extensions

Edited by R. J. Mathar, Jul 16 2008

More terms from Vincenzo Librandi, Aug 18 2011

Status

approved