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 A294566 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. 2
 1, 2, 3, 8, 13, 38, 63, 188, 313, 938, 1563, 4688, 7813, 23438, 39063, 117188, 195313, 585938, 976563, 2929688, 4882813, 14648438, 24414063, 73242188, 122070313, 366210938, 610351563, 1831054688, 3051757813, 9155273438, 15258789063, 45776367188, 76293945313 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Colin Barker, Table of n, a(n) for n = 1..1000 G. Bell, A. Lawson, N. Pritchard, and D. Yasaki, Locally infinite Cayley graphs of the integers, arXiv:1711.00809 [math.GT], 2017. Index entries for linear recurrences with constant coefficients, signature (1,5,-5). FORMULA 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) From Colin Barker, Nov 02 2017: (Start) G.f.: x*(1 + x - 4*x^2) / ((1 - x)*(1 - 5*x^2)). a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) for n > 3. a(n) = (3*5^(n/2) + 5)/10 for n even. a(n) = (5^((n-1)/2) + 1)/2 for n odd. (End) EXAMPLE 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. 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). 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. MATHEMATICA LinearRecurrence[{1, 5, -5}, Range@ 3, 30] (* or *) 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 *) PROG (PARI) Vec(x*(1 + x - 4*x^2) / ((1 - x)*(1 - 5*x^2)) + O(x^40)) \\ Colin Barker, Nov 02 2017 CROSSREFS Sequence in context: A317911 A151889 A294151 * A129873 A025575 A077055 Adjacent sequences:  A294563 A294564 A294565 * A294567 A294568 A294569 KEYWORD nonn,easy AUTHOR Austin Lawson, Nov 02 2017 STATUS approved

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Last modified January 20 23:20 EST 2020. Contains 331104 sequences. (Running on oeis4.)