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%I #82 Nov 12 2022 07:39:00
%S 0,1,5,6,25,26,30,31,125,126,130,131,150,151,155,156,625,626,630,631,
%T 650,651,655,656,750,751,755,756,775,776,780,781,3125,3126,3130,3131,
%U 3150,3151,3155,3156,3250,3251,3255,3256,3275,3276,3280,3281,3750,3751
%N Sums of distinct powers of 5.
%C Numbers without any base-5 digits larger than 1.
%C a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. - _Philippe Deléham_, Oct 17 2011
%C Values of k where A008977(k) does not end with 0. - _Henry Bottomley_, Nov 09 2022
%H T. D. Noe, <a href="/A033042/b033042.txt">Table of n, a(n) for n = 0..1023</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 K. Dilcher and L. Ericksen, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p24">Hyperbinary expansions and Stern polynomials</a>, Elec. J. Combin, 22, 2015, #P2.24.
%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 45.
%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 a(n) = Sum_{i=0..m} d(i)*5^i, where Sum_{i=0..m} d(i)*2^i is the base-2 representation of n.
%F Numbers j such that the coefficient of x^j is > 0 in Product_{k>=0} (1 + x^(5^k)). - _Benoit Cloitre_, Jul 29 2003
%F a(n) = A097251(n)/4.
%F a(2n) = 5*a(n), a(2n+1) = a(2n)+1.
%F a(n) = Sum_{k>=0} A030308(n,k)*5^k. - _Philippe Deléham_, Oct 17 2011
%F liminf a(n)/n^(log(5)/log(2)) = 1/4 and limsup a(n)/n^(log(5)/log(2)) = 1. - _Gheorghe Coserea_, Sep 15 2015
%F G.f.: (1/(1 - x))*Sum_{k>=0} 5^k*x^(2^k)/(1 + x^(2^k)). - _Ilya Gutkovskiy_, Jun 04 2017
%p a:= proc(n) local m, r, b; m, r, b:= n, 0, 1;
%p while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*5 od; r
%p end:
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Mar 16 2013
%t t = Table[FromDigits[RealDigits[n, 2], 5], {n, 1, 100}]
%t (* _Clark Kimberling_, Aug 02 2012 *)
%t FromDigits[#,5]&/@Tuples[{0,1},7] (* _Harvey P. Dale_, May 22 2018 *)
%o (PARI) a(n) = subst(Pol(binary(n)), 'x, 5);
%o vector(50, i, a(i-1)) \\ _Gheorghe Coserea_, Sep 15 2015
%o (PARI) a(n)=fromdigits(binary(n),5) \\ _Charles R Greathouse IV_, Jan 11 2017
%o (Julia)
%o function a(n)
%o m, r, b = n, 0, 1
%o while m > 0
%o m, q = divrem(m, 2)
%o r += b * q
%o b *= 5
%o end
%o r end; [a(n) for n in 0:49] |> println # _Peter Luschny_, Jan 03 2021
%Y For generating functions Product_{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. A000695, A005836, A008977, A010060, A033043-A033052.
%Y Row 5 of array A104257.
%K nonn,base,easy
%O 0,3
%A _Clark Kimberling_
%E Extended by _Ray Chandler_, Aug 03 2004
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