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%I A060546 #49 Feb 03 2023 01:37:33
%S A060546 1,2,2,4,4,8,8,16,16,32,32,64,64,128,128,256,256,512,512,1024,1024,
%T A060546 2048,2048,4096,4096,8192,8192,16384,16384,32768,32768,65536,65536,
%U A060546 131072,131072,262144,262144,524288,524288,1048576,1048576,2097152,2097152
%N A060546 a(n) = 2^ceiling(n/2).
%C A060546 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.
%C A060546 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
%C A060546 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
%H A060546 Harry J. Smith, <a href="/A060546/b060546.txt">Table of n, a(n) for n = 0..500</a>
%H A060546 A. Barbé, <a href="http://dx.doi.org/10.1016/S0166-218X(00)00211-0">Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2</a>, Discr. Appl. Math. 105(2000), 1-38.
%H A060546 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A060546 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A060546 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,2).
%F A060546 a(n) = 2^ceiling(n/2).
%F A060546 a(n) = A016116(n+1) for n >= 1.
%F A060546 a(n) = 2^A008619(n-1) for n >= 1.
%F A060546 G.f.: (1+2*x) / (1-2*x^2). - _Ralf Stephan_, Jul 15 2013 [Adapted to offset 0 by _Robert A. Russell_, Nov 07 2018]
%F A060546 E.g.f.: cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x). - _Stefano Spezia_, Feb 02 2023
%p A060546 for n from 0 to 100 do printf(`%d,`,2^ceil(n/2)) od:
%t A060546 2^Ceiling[Range[0,50]/2] (* or *) Riffle[2^Range[0, 25], 2^Range[25]] (* _Harvey P. Dale_, Mar 05 2013 *)
%t A060546 LinearRecurrence[{0, 2}, {1, 2}, 40] (* _Robert A. Russell_, Nov 07 2018 *)
%o A060546 (PARI) { for (n=0, 500, write("b060546.txt", n, " ", 2^ceil(n/2)); ) } \\ _Harry J. Smith_, Jul 06 2009
%o A060546 (Magma) [2^Ceiling(n/2): n in [0..50]]; // _G. C. Greubel_, Nov 07 2018
%Y A060546 Column k=2 of A321391.
%Y A060546 Cf. A016116, A008619.
%Y A060546 Cf. A000079 (oriented), A005418(n+1) (unoriented), A122746(n-2) (chiral).
%Y A060546 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
%K A060546 easy,nonn
%O A060546 0,2
%A A060546 André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
%E A060546 More terms from _James A. Sellers_, Apr 04 2001
%E A060546 a(0)=1 prepended by _Robert A. Russell_, Nov 07 2018
%E A060546 Edited by _N. J. A. Sloane_, Nov 10 2018

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