A186830 Keith sequence for the number 197.

1, 9, 7, 17, 33, 57, 107, 197, 361, 665, 1223, 2249, 4137, 7609, 13995, 25741, 47345, 87081, 160167, 294593, 541841, 996601, 1833035, 3371477, 6201113, 11405625, 20978215, 38584953, 70968793, 130531961

Offset: 1

Author

N. J. A. Sloane, Feb 27 2011

Keywords

• nonn
• base
• easy

Comments

This sequence illustrates why 197 is a Keith number (cf. A007629).

Other multiples of 197 in the sequence are 4137 and 992207243244533. - Alonso del Arte, Mar 14 2011

Links

• M. Klazar and F. Luca, Counting Keith numbers, Journal of Integer Sequences, Vol. 10 (2007), #07.2.2.
• Eric Weisstein's World of Mathematics, Keith Number
• Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Crossrefs

Cf. A007629.

Programs

Mathematica

keithSeq[n_Integer, b_:10, goBeyondN_:0] := Module[{seq = IntegerDigits[n, b], ord, max = n + goBeyondN, curr}, ord = Length[seq]; curr = seq[[-1]]; While[curr < max, curr = Plus@@Take[seq, -ord]; seq = Append[seq, curr]]; Return[seq]]; keithSeq[197, 10, 10^8] (* Alonso del Arte, Mar 14 2011 *)

PARI

Vec((1+8*x-3*x^2)/(1-x-x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Feb 04 2013

Formula

a(1)=1, a(2)=9, a(3)=7; thereafter a(n) = sum of previous three terms. Note that 197 appears in the sequence, which is why 197 is a Keith number.

G.f.: x*(1+8*x-3*x^2)/(1-x-x^2-x^3). [Colin Barker, Jun 19 2012]