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%I #89 Dec 19 2023 17:41:06
%S 5,10,20,40,80,160,320,640,1280,2560,5120,10240,20480,40960,81920,
%T 163840,327680,655360,1310720,2621440,5242880,10485760,20971520,
%U 41943040,83886080,167772160,335544320,671088640,1342177280,2684354560,5368709120,10737418240
%N a(n) = 5 * 2^n.
%C Same as Pisot sequences E(5,10), L(5,10), P(5,10), T(5,10). See A008776 for definitions of Pisot sequences.
%C The first differences are the sequence itself. - _Alexandre Wajnberg_ & _Eric Angelini_, Sep 07 2005
%C 5 times powers of 2. - _Omar E. Pol_, Dec 16 2008
%C Subsequence of A051916. - _Reinhard Zumkeller_, Mar 20 2010
%C With the addition of "2, 3," at the beginning, this sequence gives terms (n + 3) through the first term greater than 2^n, for n odd, of the negabinary Keith sequence for 2^n, thus proving that with the exception of 2 itself, no odd-indexed power of 2 is a negabinary Keith number (see A188381). - _Alonso del Arte_, Feb 02 2012
%C Let b(0) = 5 and b(n+1) = the smallest number not in the sequence such that b(n+1) - Product_{i=0..n} b(i) divides b(n+1) - Sum_{i=0..n} b(i). Then b(n+2) = a(n) for n > 0. - _Derek Orr_, Jan 15 2015
%H Vincenzo Librandi, <a href="/A020714/b020714.txt">Table of n, a(n) for n = 0..238</a>
%H John Elias, <a href="/A020714/a020714.png">Illustration: Alternating Tetrahedrons of Tetrahedrons</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H Petro Kosobutskyy, Anastasiia Yedyharova, and Taras Slobodzyan, <a href="https://doi.org/10.23939/cds2023.01.121">From Newton's binomial and Pascal's triangle to Collatz's problem</a>, Comp. Des. Sys., Theor. Practice (2023) Vol. 5, No. 1, 121-127.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1003">Encyclopedia of Combinatorial Structures 1003</a>.
%H Everett Sullivan, <a href="https://arxiv.org/abs/1611.02771">Linear chord diagrams with long chords</a>, arXiv preprint arXiv:1611.02771 [math.CO], 2016. See Table 1.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (2).
%F a(n) = 5*2^n. a(n) = 2*a(n-1).
%F G.f.: 5/(1-2*x).
%F If m is a term greater than 5 of this sequence then m = 5*phi(phi(m)). - _Farideh Firoozbakht_, Aug 16 2005
%F a(n) = A118416(n+1,3) for n>2. - _Reinhard Zumkeller_, Apr 27 2006
%F a(n) = A000079(n)*5. - _Omar E. Pol_, Dec 16 2008
%F a(n) = A173786(n+2,n) for n > 1. - _Reinhard Zumkeller_, Feb 28 2010
%F a(n) = A001045(n+4) - A001045(n). - _Paul Curtz_, Nov 08 2012
%F Sum_{n>=1} 1/a(n) = 2/5. - _Amiram Eldar_, Oct 28 2020
%F E.g.f.: 5*exp(2*x). - _Stefano Spezia_, May 15 2021
%t Table[5*2^n, {n, 0, 31}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 16 2008 *)
%t NestList[2#&,5,40] (* _Harvey P. Dale_, Mar 13 2022 *)
%o (Magma) [5*2^n: n in [0..40]]; // _Vincenzo Librandi_, Apr 28 2011
%o (PARI) a(n)=5<<n \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Row sums of (4, 1)-Pascal triangle A093561.
%Y Row sums of (9, 1)-Pascal triangle A093644.
%Y Row sums of (1, 4)-Pascal triangle A095666 (with leading 4).
%Y Cf. A000079, A001045, A008776, A051916, A118416, A173786, A188381.
%K nonn,easy
%O 0,1
%A _David W. Wilson_
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