

A060546


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


33



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
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OFFSET

0,2


COMMENTS

a(n) is also the number of medianreflective (palindrome) symmetric patterns in a topdown 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 topdown triangle of three neighboring cells in the arrangement contains either one or three white cells.
The number of possibilities for an ngame (sub)set of tennis with neither player gaining a 2game advantage. (Motivated by the marathon IsnerMahut 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), 138.
Index entries for sequences related to cellular automata
Index to divisibility sequences
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(n1) for n >= 1.
G.f.: (1+2*x) / (12*x^2).  Ralf Stephan, Jul 15 2013 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]
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) { for (n=0, 500, write("b060546.txt", 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.
Cf. A016116, A008619.
Cf. A000079 (oriented), A005418(n+1) (unoriented), A122746(n2) (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
Sequence in context: A152166 A320770 A016116 * A163403 A231208 A306663
Adjacent sequences: A060543 A060544 A060545 * A060547 A060548 A060549


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



