%I #52 Jul 11 2020 17:12:17
%S 1,2,10,50,250,1250,6250,31250,156250,781250,3906250,19531250,
%T 97656250,488281250,2441406250,12207031250,61035156250,305175781250,
%U 1525878906250,7629394531250,38146972656250,190734863281250,953674316406250
%N Expansion of (1-3*x)/(1-5*x).
%C Partial sums are A034478.
%C Except for the first two terms 1 and 2, these are the integers that satisfy phi(n) = 2*n/5. - _Michel Marcus_, Jul 14 2015
%C For n>=1, period of powers of 4 mod 10^n. See A000302. - _Martin Renner_, Jun 12 2020
%H Nathaniel Johnston, <a href="/A020699/b020699.txt">Table of n, a(n) for n = 0..250</a>
%H D Bevan, D Levin, P Nugent, J Pantone, L Pudwell, <a href="http://arxiv.org/abs/1510.08036">Pattern avoidance in forests of binary shrubs</a>, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1037">Encyclopedia of Combinatorial Structures 1037</a>
%H INRIA Algorithms Project, <a href="http://tinyurl.com/A020699">Encyclopedia of Combinatorial Structures 1037</a> (archived version of page)
%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (5).
%F a(n) = 2*5^(n-1) for n>0.
%F E.g.f.: (2*exp(5*x)+3)/5; a(n)=(2*5^n+3*0^n)/5. - _Paul Barry_, Sep 03 2003
%F a(n) = sum{k=0..n, C(n-1, k)*(Jac(2n-2k)+Jac(2n-2k-1))}+0^n/2, where Jac(n)=A001045(n). - _Paul Barry_, Jun 07 2005
%F a(0)=1, a(1)=2, a(n) = 5*a(n-1) for n>=2. [_Vincenzo Librandi_, Jan 01 2011]
%F a(n) = A020729(n-1), n>0. - _R. J. Mathar_, Sep 16 2016
%p seq(`if`(n=0,1,2*5^(n-1)), n=0..22); # _Nathaniel Johnston_, Jun 26 2011
%t CoefficientList[Series[(1 - 3 x)/(1 - 5 x), {x, 0, 22}], x] (* _Michael De Vlieger_, Jul 14 2015 *)
%o (PARI) Vec((1-3*x)/(1-5*x) + O(x^30)) \\ _Michel Marcus_, Jul 14 2015
%Y Cf. A034478, A001045, A020729, A000302.
%K nonn,easy
%O 0,2
%A _David W. Wilson_