

A151550


Expansion of g.f. Product_{n >= 1} (1 + x^(2^n1) + 2*x^(2^n)).


16



1, 1, 2, 1, 3, 4, 4, 1, 3, 4, 5, 5, 10, 12, 8, 1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 16, 1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 17, 5, 10, 13, 15, 20, 32, 33, 23, 20, 33, 41, 50, 72, 96, 80, 32, 1, 3, 4, 5, 5, 10, 12, 9, 5, 10, 13, 15, 20, 32, 32, 17, 5, 10, 13
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OFFSET

0,3


COMMENTS

When convolved with [1, 2, 2, 2, ...] gives the toothpick sequence A153006: (1, 3, 6, 9, ...).  Gary W. Adamson, May 25 2009
This sequence and the Adamson's comment both are mentioned in the ApplegatePolSloane article, see chapter 8 "generating functions".  Omar E. Pol, Sep 20 2011


REFERENCES

D. Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157191


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..16383
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.], which is also available at arXiv:1004.3036v2, [math.CO], 2010.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS


FORMULA

To get a nice recurrence, change the offset to 0 and multiply the g.f. by x as in the triangle in the example lines. Then we have: a(0)=0; a(2^i)=1; a(2^i1)=2^(i1) for i >= 1; otherwise write n = 2^i+j with 1 <= j <= 2^i2, then a(n) = a(2^i+j) = 2*a(j) + a(j+1).


EXAMPLE

From Omar E. Pol, Jun 09 2009, edited by N. J. A. Sloane, Jun 17 2009:
May be written as a triangle:
0;
1;
1,2;
1,3,4,4;
1,3,4,5,5,10,12,8;
1,3,4,5,5,10,12,9,5,10,13,15,20,32,32,16;
1,3,4,5,5,10,12,9,5,10,13,15,20,32,32,17,5,10,13,15,20,32,33,23,20,33,41,...
The rows of the triangle converge to A151555.


MATHEMATICA

terms = 100;
CoefficientList[Product[(1+x^(2^n1) + 2 x^(2^n)), {n, 1, Log[2, terms] // Ceiling}] + O[x]^terms, x] (* JeanFrançois Alcover, Aug 05 2018 *)


CROSSREFS

For generating functions of the form Product_{k>=c} (1+a*x^(2^k1)+b*x^2^k)) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
Cf. A139250, A151551, A151552, A151553, A151554, A151555, A152980, A153006, A151688.
Cf. A000079.  Omar E. Pol, Jun 09 2009
Sequence in context: A097564 A345233 A128270 * A097003 A336926 A193788
Adjacent sequences: A151547 A151548 A151549 * A151551 A151552 A151553


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, May 19 2009, Jun 17 2009


STATUS

approved



