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, 240085707, 441586461, 812204129, 1493876297, 2747666887, 5053747313

Offset

1,2

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

Amiram Eldar, Table of n, a(n) for n = 1..1000

Martin Klazar and Florian Luca, Counting Keith numbers, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.2.

Eric Weisstein's World of Mathematics, Keith Number.

Index entries for linear recurrences with constant coefficients, signature (1,1,1).

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

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 *)
LinearRecurrence[{1, 1, 1}, {1, 9, 7}, 36] (* Amiram Eldar, Jan 20 2026 *)

Prog

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

Crossrefs

Cf. A007629.

Sequence in context: A131724 A393103 A190995 * A124050 A107663 A298780

Adjacent sequences: A186827 A186828 A186829 * A186831 A186832 A186833

Keyword

nonn,base,easy

Author

N. J. A. Sloane, Feb 27 2011

Extensions

a(31)-a(36) from Amiram Eldar, Jan 20 2026

Status

approved