OFFSET
0,2
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
Recurrence: a(0)=1, a(1) = a(2) = 2; a(2^m-1)=2 for m >= 2; a(2^m) = 3 for m >= 2; a(2^m-2) = m for m >= 3; otherwise, for m >= 5, if m=2^i+j (0 <= j < 2^i - 1), a(m) = a(j) + a(j+1).
a(n) = Sum_{k>=0, n+k odd} binomial(A000120(n+k),k); the sum may be restricted further to k <= 2*A000523(n+1)+1 [based on Hagen von Eitzen's formula for A151552]. [corrected by Amiram Eldar, Jul 29 2023]
EXAMPLE
If formatted as a triangle:
.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,
.3,4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26,25,16,6,2,
.3,4,4,5,7,7,5,5,7,8,9,12,14,11,6,5,7,8,9,12,14,12,10,12,15,17,21,26,25,16,7
... 5,7,8,9,12,14,12,10,12,15,17,21,26,25,17,11,12,15,17,21,26,26,22,22,27,32,38,47,51,41,22,7,2,
.3,4,4,5,7,7,4,2, ...
MATHEMATICA
CoefficientList[Series[(1+x)Product[1+x^(2^n-1)+x^2^n, {n, 10}], {x, 0, 100}], x] (* Harvey P. Dale, Jul 13 2019 *)
a[n_] := Sum[If[OddQ[n + k], Binomial[DigitCount[n + k, 2, 1], k], 0], {k, 0, 2*Floor[Log2[n + 1]] + 1}]; Array[a, 92, 0] (* Amiram Eldar, Jul 29 2023 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, May 20 2009
STATUS
approved