1,7

k=5 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1), k=3 case is A079398 (offset so as to begin 1,1,1,1) and k=4 case is A103372.

The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).

For this k=5 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^6 - x - 1 = 0. This is the real constant (to 100 digits accuracy): 1.134724138401519492605446054506472840279667226382801485925149551668236893999842671279689011614820249

The sequence of prime values in this k=5 case is A103383; The sequence of semiprime values in this k=5 case is A103393.

Selmer, E.S., "On the irreducibility of certain trinomials", Math. Scand., 4 (1956) 287-302

Shallit, J., "A generalization of automatic sequences", Theoretical Computer Science, 61(1988)1-16.

Zanten, A. J. van, "The golden ratio in the arts of painting, building and mathematics", Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.

Richard Padovan, Dom Hans van der Laan and the Plastic Number.

J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms

a(22) = 9 because a(22) = a(22-5) + a(22-6) = a(17) + a(16) = 5 + 4 = 9.

k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 65]

Cf. A000045, A000931, A079398, A103372-A103381, A103383, A103393.

Sequence in context: A025783 A025780 A109697 this_sequence A038539 A109368 A046774

Adjacent sequences: A103370 A103371 A103372 this_sequence A103374 A103375 A103376

nonn,easy

Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 03 2005

Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net) and Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 06 2005