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

Author

N. J. A. Sloane, May 30 2009

Keywords

• nonn

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

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.

Programs

Haskell

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

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 *)

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