cp's OEIS Frontend

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

%I A151666 #26 Aug 07 2021 01:44:45
%S A151666 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,
%T A151666 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,
%U A151666 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
%N A151666 Number of partitions of n into distinct powers of 4.
%H A151666 Reinhard Zumkeller, <a href="/A151666/b151666.txt">Table of n, a(n) for n = 0..10000</a>
%H A151666 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H A151666 Lukasz Merta, <a href="https://arxiv.org/abs/1803.00292">Composition inverses of the variations of the Baum-Sweet sequence</a>, arXiv:1803.00292 [math.NT], 2018. See q(n) (with different offset) p. 9.
%H A151666 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A151666 G.f.: Prod_{k >= 0 } (1+x^(4^k)). Exponents give A000695.
%F A151666 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^4). - _Ilya Gutkovskiy_, Aug 12 2019
%t A151666 terms = 105;
%t A151666 kmax = Log[4, terms] // Ceiling;
%t A151666 CoefficientList[Product[1+x^(4^k), {k, 0, kmax}] + O[x]^(kmax terms), x][[1 ;; terms]] (* _Jean-François Alcover_, Jul 31 2018 *)
%o A151666 (Haskell)
%o A151666 a151666 n = fromEnum (n < 2 || m < 2 && a151666 n' == 1)
%o A151666    where (n', m) = divMod n 4
%o A151666 -- _Reinhard Zumkeller_, Dec 03 2011
%Y A151666 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.
%Y A151666 Cf. A039966, A151667, A000695, A269707.
%K A151666 nonn
%O A151666 0,1
%A A151666 _N. J. A. Sloane_, May 30 2009