A151666 Number of partitions of n into distinct powers of 4.

1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,1

Links

Reinhard Zumkeller, 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.]

Lukasz Merta, Composition inverses of the variations of the Baum-Sweet sequence, arXiv:1803.00292 [math.NT], 2018. See q(n) (with different offset) p. 9.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Formula

G.f.: Prod_{k >= 0 } (1+x^(4^k)). Exponents give A000695.

G.f. A(x) satisfies: A(x) = (1 + x) * A(x^4). - Ilya Gutkovskiy, Aug 12 2019

Mathematica

terms = 105;
kmax = Log[4, terms] // Ceiling;
CoefficientList[Product[1+x^(4^k), {k, 0, kmax}] + O[x]^(kmax terms), x][[1 ;; terms]] (* Jean-François Alcover, Jul 31 2018 *)

Prog

(Haskell)
a151666 n = fromEnum (n < 2 || m < 2 && a151666 n' == 1)
   where (n', m) = divMod n 4
-- Reinhard Zumkeller, Dec 03 2011

Crossrefs

For generating functions Prod_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.

Cf. A039966, A151667, A000695, A269707.

Sequence in context: A157686 A181115 A284527 * A355681 A214284 A361466

Adjacent sequences: A151663 A151664 A151665 * A151667 A151668 A151669

Keyword

nonn

Author

N. J. A. Sloane, May 30 2009

Status

approved