A297182 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.

1, 2, 3, 4, 5, 6, 7, 20, 33, 46, 59, 72, 85, 254, 423, 592, 761, 930, 1099, 3296, 5493, 7690, 9887, 12084, 14281, 42842, 71403, 99964, 128525, 157086, 185647, 556940, 928233, 1299526, 1670819, 2042112, 2413405, 7240214, 12067023, 16893832, 21720641, 26547450

Offset: 1

Author

N. J. A. Sloane, Dec 28 2017

Keywords

• nonn

Links

• Lars Blomberg, 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 (1,0,0,0,0,13,-13).

Programs

Mathematica

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

Formula

Conjectures from Colin Barker, Dec 28 2017: (Start)

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

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

(End)

The second conjecture by Colin Barker is true up to n=1000. - Lars Blomberg, Dec 29 2017

Extensions

Terms a(21) and beyond from Lars Blomberg, Dec 29 2017