A151548 - OEIS

%I #41 Feb 24 2021 02:48:18

%S 1,3,5,7,5,11,17,15,5,11,17,19,21,39,49,31,5,11,17,19,21,39,49,35,21,

%T 39,53,59,81,127,129,63,5,11,17,19,21,39,49,35,21,39,53,59,81,127,129,

%U 67,21,39,53,59,81,127,133,91,81,131,165,199,289,383,321,127,5,11,17,19,21,39

%N When A160552 is regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., this is what the rows converge to.

%C When convolved with A151575: (1, 0, 2, -2, 6, -10, 22, -42, 86, -170, 342, ...) equals the toothpick sequence A139250: (1, 3, 7, 11, 15, 23, 35, 43, ...). - _Gary W. Adamson_, May 25 2009

%C Equals A160552: [1, 1, 3, 1, 3, 5, ...] convolved with [1, 2, 0, 0, 0, ...], equivalent to a(n) = 2*A160552(n) + A160552(n+1). - _Gary W. Adamson_, Jun 04 2009

%C Equals (1, 0, -2, 2, -2, 2, ...) convolved with the Toothpick sequence, A139250. - _Gary W. Adamson_, Mar 06 2012

%C It appears that the sums of two successive terms give A147646. - _Omar E. Pol_, Feb 18 2015

%H N. J. A. Sloane, <a href="/A151548/b151548.txt">Table of n, a(n) for n = 0..10000</a>

%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168, 2015

%F a(2^k-1) = 2^(k+1)-1 for k >= 0; otherwise a(2^k) = 5 for k >= 1; otherwise a(2^i+j) = 2a(j)+a(j+1) for i >= 2, 1 <= j <= 2^i-2. - _N. J. A. Sloane_, May 22 2009

%F G.f.: 1/(1+x) + 4*x*mul(1+x^(2^k-1)+2*x^(2^k),k=1..oo). - _N. J. A. Sloane_, May 23 2009

%F a(n) = A147646(n) - a(n-1), n >= 1. - _Omar E. Pol_, Feb 19 2015

%e From _Omar E. Pol_, Jul 24 2009: (Start)

%e When written as a triangle:

%e 1;

%e 3;

%e 5,7;

%e 5,11,17,15;

%e 5,11,17,19,21,39,49,31;

%e 5,11,17,19,21,39,49,35,21,39,53,59,81,127,129,63;

%e 5,11,17,19,21,39,49,35,21,39,53,59,81,127,129,67,21,39,53,59,81,127,133,91,...

%e (End)

%p G := 1/(1+x) + 4*x*mul(1+x^(2^k-1)+2*x^(2^k),k=1..20); # _N. J. A. Sloane_, May 23 2009

%p S2:=proc(n) option remember; local i,j;

%p if n <= 1 then RETURN(2*n+1); fi;

%p i:=floor(log(n)/log(2));

%p j:=n-2^i;

%p if j=0 then 5 elif j=2^i-1 then 2*n+1

%p else 2*S2(j)+S2(j+1); fi;

%p end; # - _N. J. A. Sloane_, May 22 2009

%t terms = 70; CoefficientList[1/(1 + x) + 4*x*Product[1 + x^(2^k - 1) + 2*x^(2^k), {k, 1, Log[2, terms] // Ceiling}] + O[x]^terms, x] (* _Jean-François Alcover_, Nov 14 2017, after _N. J. A. Sloane_ *)

%Y Cf. A139250, A160552, A151549, A078008, A246336, A147646, A151575.

%K nonn

%O 0,2

%A _David Applegate_, May 18 2009