A103375 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 1 and for n>8: a(n) = a(n-7) + a(n-8).

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 65, 71, 86, 106, 121, 127, 128, 129, 136, 157, 192, 227, 248, 255, 257, 265, 293, 349, 419, 475, 503, 512

Offset: 1

Author

Jonathan Vos Post, Feb 03 2005

Keywords

• nonn
• easy

Comments

k=7 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), k=4 case is A103372, k=5 case is A103373 and k=6 case is A103374.

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=7 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^8 - x - 1 = 0. This is the real constant 1.09698155779855981790827896716753708959253010821278671381232885124855898059....

The sequence of prime values in this k=7 case is A103385; the sequence of semiprime values in this k=7 case is A103395.

Examples

			a(30) = 12 because a(30) = a(30-7) + a(30-8) = a(24) + a(23) = 7 + 5 = 12.
		

References

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

Links

• J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - _N. J. A. Sloane_, Feb 19 2025]
• J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from _N. J. A. Sloane_, Feb 19 2025]
• Richard Padovan, Dom Hans van der Laan and the Plastic Number.
• E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.
• J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988) 1-16.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1,1).

Crossrefs

Cf. A000045, A000931, A079398, A103372-A103374, A103376-A103380, A103385, A103395.

Programs

Mathematica

k = 7; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 73]
LinearRecurrence[{0,0,0,0,0,0,1,1},{1,1,1,1,1,1,1,1},80]

PARI

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; 1,1,0,0,0,0,0,0]^(n-1)*[1;1;1;1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Formula

G.f.: -x*(1+x+x^2+x^3+x^4+x^5+x^6)/(-1+x^7+x^8). - R. J. Mathar, Dec 14 2009

Extensions

Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005

Corrected (one more 8 inserted) by R. J. Mathar, Dec 14 2009