

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
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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^(q1) + 1)/2. Otherwise, a(n) = ((2*r1)*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(n1) + 5*a(n2)  5*a(n3) for n > 3.
a(n) = (3*5^(n/2) + 5)/10 for n even.
a(n) = (5^((n1)/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



