A151685 a(n) = Sum_{k >= 0} bin2(wt(n+k),k+1), where bin2(i,j) = A013609(i,j), wt(i) = A000120(i).

3, 7, 5, 7, 17, 17, 7, 7, 17, 17, 19, 41, 51, 31, 9, 7, 17, 17, 19, 41, 51, 31, 21, 41, 51, 55, 101, 143, 113, 49, 11, 7, 17, 17, 19, 41, 51, 31, 21, 41, 51, 55, 101, 143, 113, 49, 23, 41, 51, 55, 101, 143, 113, 73, 103, 143, 161, 257, 387, 369, 211, 71, 13, 7, 17, 17, 19, 41, 51

Offset

0,1

Comments

Or, a(n) = Sum_{k >= 0} 2^wt(k) * binomial(wt(n+k),k).

Links

Table of n, a(n) for n=0..68.

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

G.f.: Product_{ k >= 0 } (1 + 2*x^(2^k-1) + x^(2^k)).

Example

Contribution from Omar E. Pol, Jun 09 2009: (Start)
Triangle begins:
.3;
.7,5;
.7,17,17,7;
.7,17,17,19,41,51,31,9;
.7,17,17,19,41,51,31,21,41,51,55,101,143,113,49,11;
.7,17,17,19,41,51,31,21,41,51,55,101,143,113,49,23,41,51,55,101,143,113,...
(End)

Maple

bin2:=proc(n, k) option remember; if k<0 or k>n then 0
elif k=0 then 1 else 2*bin2(n-1, k-1)+bin2(n-1, k); fi; end;
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:=n->add( bin2(wt(n+k), k), k=0..120 );
# or:
f := n->add( 2^k*binomial(wt(n+k), k), k=0..20 );

Mathematica

max = 70; (* number of terms *)
CoefficientList[Product[1 + 2*x^(2^k-1) + x^(2^k), {k, 0, Log2[max+1] // Ceiling}] + O[x]^max, x] (* Jean-François Alcover, Aug 03 2022 *)

Crossrefs

For generating functions of the form Product_{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. A151689, A151691.

Cf. A000079. [From Omar E. Pol, Jun 09 2009]

Sequence in context: A287660 A122001 A161327 * A019809 A305741 A021270

Adjacent sequences: A151682 A151683 A151684 * A151686 A151687 A151688

Keyword

nonn

Author

N. J. A. Sloane, Jun 01 2009

Status

approved