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%I #26 Aug 07 2021 01:44:45
%S 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 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 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 Number of partitions of n into distinct powers of 4.
%H Reinhard Zumkeller, <a href="/A151666/b151666.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 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 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 G.f.: Prod_{k >= 0 } (1+x^(4^k)). Exponents give A000695.
%F G.f. A(x) satisfies: A(x) = (1 + x) * A(x^4). - _Ilya Gutkovskiy_, Aug 12 2019
%t terms = 105;
%t kmax = Log[4, terms] // Ceiling;
%t CoefficientList[Product[1+x^(4^k), {k, 0, kmax}] + O[x]^(kmax terms), x][[1 ;; terms]] (* _Jean-François Alcover_, Jul 31 2018 *)
%o (Haskell)
%o a151666 n = fromEnum (n < 2 || m < 2 && a151666 n' == 1)
%o where (n', m) = divMod n 4
%o -- _Reinhard Zumkeller_, Dec 03 2011
%Y 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 Cf. A039966, A151667, A000695, A269707.
%K nonn
%O 0,1
%A _N. J. A. Sloane_, May 30 2009
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