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).
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
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
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