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 A033042 Sums of distinct powers of 5. 36

%I

%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

%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="http://neilsloane.com/doc/tooth.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 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 n such that the coefficient of x^n 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

%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. A000695, A005836, 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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Last modified January 21 08:28 EST 2019. Contains 319351 sequences. (Running on oeis4.)