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%I #33 Nov 10 2017 19:19:43
%S 1,2,3,8,13,38,63,188,313,938,1563,4688,7813,23438,39063,117188,
%T 195313,585938,976563,2929688,4882813,14648438,24414063,73242188,
%U 122070313,366210938,610351563,1831054688,3051757813,9155273438,15258789063,45776367188,76293945313
%N a(n) is the smallest positive integer of length (distance from origin) n in the Cayley graph of the integers generated by all powers of 5.
%H Colin Barker, <a href="/A294566/b294566.txt">Table of n, a(n) for n = 1..1000</a>
%H G. Bell, A. Lawson, N. Pritchard, and D. Yasaki, <a href="https://arxiv.org/abs/1711.00809">Locally infinite Cayley graphs of the integers</a>, arXiv:1711.00809 [math.GT], 2017.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-5).
%F Let r,q satisfy the division algorithm so that n = q*2 + r. If r= 0, then a(n) = (5^q - 2*5^(q-1) + 1)/2. Otherwise, a(n) = ((2*r-1)*5^q + 1)/2. (Proved)
%F From _Colin Barker_, Nov 02 2017: (Start)
%F G.f.: x*(1 + x - 4*x^2) / ((1 - x)*(1 - 5*x^2)).
%F a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n > 3.
%F a(n) = (3*5^(n/2) + 5)/10 for n even.
%F a(n) = (5^((n-1)/2) + 1)/2 for n odd.
%F (End)
%e The Cayley graph of the integers generated by the powers of 5 is a graph whose vertices are integers and an edge between integers whenever they differ by a power of 5. The length of an integer in this graph is its edge distance from 0.
%e For example, 1 = 5^0 and thus has length 1. 2 = 5^0 + 5^0 and thus has length 2. The same pattern holds for 3. But 4 = 5 - 5^0 and thus has length 2. It does not appear in the sequence because there is a smaller positive integer of length 2 (namely 2).
%e We can see the smallest integer of length 4 is 8 = 5^1 + 5^0 + 5^0 + 5^0. 8 cannot be written as a sum of 3 or fewer powers of 5.
%t LinearRecurrence[{1, 5, -5}, Range@ 3, 30] (* or *)
%t Rest@ CoefficientList[Series[x (1 + x - 4 x^2)/((1 - x) (1 - 5 x^2)), {x, 0, 30}], x] (* _Michael De Vlieger_, Nov 03 2017 *)
%o (PARI) Vec(x*(1 + x - 4*x^2) / ((1 - x)*(1 - 5*x^2)) + O(x^40)) \\ _Colin Barker_, Nov 02 2017
%K nonn,easy
%O 1,2
%A _Austin Lawson_, Nov 02 2017
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