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A151552 G.f.: Prod_{ k >= 1} (1 + x^(2^k-1) + x^(2^k)). 18
1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 4, 3, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 5, 1, 1, 2, 2, 2, 3, 4, 3, 2, 3, 4, 4, 5, 7, 7, 4, 2, 3, 4, 4, 5, 7, 7, 5, 5, 7, 8, 9, 12, 14, 11, 5, 2, 3, 4, 4, 5, 7, 7, 5, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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), 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

FORMULA

a(n) = 1 for 0 <= n <= 3; thereafter write n = 2^i + j, with 0 <= j < 2^i, then a(n) = a(j) + a(j+1), except that a(2^(i+1)-2) = a(2^(i+1)-1) = 1.

a(n) = sum_{k >= 1, n+k even} binomial(A000120(n+k),k); the sum may be restricted further to k <= A000523(n+1). [From Hagen von Eitzen, May 20 2009]

EXAMPLE

Written as a triangle:

1,

1,

1,1,

2,2,1,1,

2,2,2,3,4,3,1,1,

2,2,2,3,4,3,2,3,4,4,5,7,7,4,1,1,

2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,1,1,

2,2,2,3,4,3,2,3,4,4,5,7,7,4,2,3,4,4,5,7,7,5,5,7,8,9,12,14,11,5,2,3,4,4,5,7,7,5,5,...

The rows converge to A151714.

MAPLE

G := mul( 1 + x^(2^n-1) + x^(2^n), n=1..20);

wt := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:

f:=proc(n) local t1, k; global wt; t1:=0; for k from 0 to 20 do if n+k mod 2 = 0 then t1:=t1+binomial(wt(n+k), k); fi; od; t1; end;

CROSSREFS

For generating functions of the form Prod_{k>=c} (1+a*x^(2^k-1)+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, A151550, A151551, A160573, A151702, A151714.

Sequence in context: A156268 A053257 A151702 * A160418 A168115 A269982

Adjacent sequences:  A151549 A151550 A151551 * A151553 A151554 A151555

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, May 19 2009, Dec 26 2009

STATUS

approved

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Last modified January 19 23:25 EST 2019. Contains 319319 sequences. (Running on oeis4.)