1,6

k=4 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) and k=3 case is A079398 (offset so as to begin 1,1,1,1).

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=4 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the irreducible characteristic polynomial: x^5 - x - 1 = 0. This is the real constant (to 100 digits accuracy): 1.167303978261418684256045899854842180720560371525489039140082449275651903429527053180685205049728672

The sequence of prime values in this k=4 case is A103382; The sequence of semiprime values in this k=4 case is A103392.

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(14) = 5 because a(14) = a(14-4) + a(14-5) = a(10) + a(9) = 3 + 2 = 5.

k = 4; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 61]

Cf. A000045, A000931, A079398, A103373-A103381, A103382, A103392.

Sequence in context: A035451 A124746 A124789 this_sequence A029082 A035450 A029126

Adjacent sequences: A103369 A103370 A103371 this_sequence A103373 A103374 A103375

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