A060546 a(n) = 2^ceiling(n/2).

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

Offset: 0

Author

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Keywords

• easy
• nonn

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

Also the number of walks of length n on the graph x--y--z starting at y. - Sean A. Irvine, May 30 2025

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 sequences related to cellular automata
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (0,2).

Crossrefs

Column k=2 of A321391.

Cf. A016116, A008619.

Cf. A000079 (oriented), A005418(n+1) (unoriented), A122746(n-2) (chiral).

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^Ceiling(n/2): n in [0..50]]; // G. C. Greubel, Nov 07 2018

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 *)

PARI

a(n) = { 2^ceil(n/2) } \\ Harry J. Smith, Jul 06 2009

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

Extensions

More terms from James Sellers, Apr 04 2001

a(0)=1 prepended by Robert A. Russell, Nov 07 2018

Edited by N. J. A. Sloane, Nov 10 2018