A151548 When A160552 is regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ..., this is what the rows converge to.

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, 39, 53, 59, 81, 127, 129, 63, 5, 11, 17, 19, 21, 39, 49, 35, 21, 39, 53, 59, 81, 127, 129, 67, 21, 39, 53, 59, 81, 127, 133, 91, 81, 131, 165, 199, 289, 383, 321, 127, 5, 11, 17, 19, 21, 39

Offset: 0

Author

David Applegate, May 18 2009

Keywords

• nonn

Comments

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

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

Equals (1, 0, -2, 2, -2, 2, ...) convolved with the Toothpick sequence, A139250. - Gary W. Adamson, Mar 06 2012

It appears that the sums of two successive terms give A147646. - Omar E. Pol, Feb 18 2015

Examples

			From _Omar E. Pol_, Jul 24 2009: (Start)
When written as a triangle:
1;
3;
5,7;
5,11,17,15;
5,11,17,19,21,39,49,31;
5,11,17,19,21,39,49,35,21,39,53,59,81,127,129,63;
5,11,17,19,21,39,49,35,21,39,53,59,81,127,129,67,21,39,53,59,81,127,133,91,...
(End)
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000
• David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, 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.]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015

Crossrefs

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

Programs

Maple

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
S2:=proc(n) option remember; local i,j;
if n <= 1 then RETURN(2*n+1); fi;
i:=floor(log(n)/log(2));
j:=n-2^i;
if j=0 then 5 elif j=2^i-1 then 2*n+1
else 2*S2(j)+S2(j+1); fi;
end; # - N. J. A. Sloane, May 22 2009

Mathematica

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 *)

Formula

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

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

a(n) = A147646(n) - a(n-1), n >= 1. - Omar E. Pol, Feb 19 2015