OFFSET
0,2
COMMENTS
a(n) is also the number of median-reflective (palindrome) symmetric patterns in a top-down equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.
The number of possibilities for an n-game (sub)set of tennis with neither player gaining a 2-game advantage. (Motivated by the marathon Isner-Mahut match at Wimbledon, 2010.) - Barry Cipra, Jun 28 2010
Number of achiral rows of n colors using up to two colors. For a(3)=4, the rows are AAA, ABA, BAB, and BBB. - Robert A. Russell, Nov 07 2018
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for linear recurrences with constant coefficients, signature (0,2).
FORMULA
a(n) = 2^ceiling(n/2).
a(n) = A016116(n+1) for n >= 1.
a(n) = 2^A008619(n-1) for n >= 1.
G.f.: (1 + 2*x) / (1 - 2*x^2). - Ralf Stephan, Jul 15 2013
E.g.f.: cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Feb 02 2023
MAPLE
for n from 0 to 100 do printf(`%d, `, 2^ceil(n/2)) od:
MATHEMATICA
2^Ceiling[Range[0, 50]/2] (* or *) Riffle[2^Range[0, 25], 2^Range[25]] (* Harvey P. Dale, Mar 05 2013 *)
LinearRecurrence[{0, 2}, {1, 2}, 40] (* Robert A. Russell, Nov 07 2018 *)
PROG
(PARI) a(n) = { 2^ceil(n/2) } \\ Harry J. Smith, Jul 06 2009
(Magma) [2^Ceiling(n/2): n in [0..50]]; // G. C. Greubel, Nov 07 2018
CROSSREFS
Column k=2 of A321391.
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
easy,nonn
AUTHOR
André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
EXTENSIONS
More terms from James A. Sellers, Apr 04 2001
a(0)=1 prepended by Robert A. Russell, Nov 07 2018
Edited by N. J. A. Sloane, Nov 10 2018
STATUS
approved