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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 13.
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%I #15 May 31 2023 12:40:07

%S 1,2,3,4,5,6,7,20,33,46,59,72,85,254,423,592,761,930,1099,3296,5493,

%T 7690,9887,12084,14281,42842,71403,99964,128525,157086,185647,556940,

%U 928233,1299526,1670819,2042112,2413405,7240214,12067023,16893832,21720641,26547450

%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 13.

%H Lars Blomberg, <a href="/A297182/b297182.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_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,13,-13).

%F Conjectures from _Colin Barker_, Dec 28 2017: (Start)

%F G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 - 12*x^6) / ((1 - x)*(1 - 13*x^6)).

%F a(n) = a(n-1) + 13*a(n-6) - 13*a(n-7) for n>7.

%F (End)

%F The second conjecture by Colin Barker is true up to n=1000. - _Lars Blomberg_, Dec 29 2017

%t LinearRecurrence[{1,0,0,0,0,13,-13},{1,2,3,4,5,6,7},50] (* _Harvey P. Dale_, May 31 2023 *)

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Dec 28 2017

%E Terms a(21) and beyond from _Lars Blomberg_, Dec 29 2017