A103374 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n-6) + a(n-7).

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 17, 21, 27, 31, 32, 32, 33, 38, 48, 58, 63, 64, 65, 71, 86, 106, 121, 127, 129, 136, 157, 192, 227, 248, 256, 265, 293, 349, 419, 475, 504, 521, 558, 642, 768, 894, 979, 1025, 1079

Offset: 1

Author

Jonathan Vos Post, Feb 03 2005

Keywords

• nonn
• easy

Comments

k=6 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 and k=5 case is A103373.

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=6 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^7 - x - 1 = 0. This is the real constant 1.1127756842787... (see A230160).

The sequence of prime values in this k=6 case is A103384; the sequence of semiprime values in this k=6 case is A103394.

Examples

			a(32) = 17 because a(32) = a(32-6) + a(32-7) = a(26) + a(25) = 9 + 8 = 17.
		

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

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• 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 (0,0,0,0,0,1,1).

Crossrefs

Cf. A000045, A000931, A079398, A103384, A103394, A230160.

Cf. A103372, A103373, A103375, A103376, A103377, A103378, A103379, A103380

Programs

Mathematica

k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 70]
RecurrenceTable[{a[n] == a[n - 6] + a[n - 7], a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == a[7] == 1}, a, {n, 70}] (* or *)
Rest@ CoefficientList[Series[-x (1 + x) (1 + x + x^2) (x^2 - x + 1)/(-1 + x^6 + x^7), {x, 0, 70}], x] (* Michael De Vlieger, Oct 03 2016 *)
LinearRecurrence[{0,0,0,0,0,1,1},{1,1,1,1,1,1,1},80] (* Harvey P. Dale, Sep 02 2024 *)

PARI

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

PARI

x='x+O('x^50); Vec(x*(1+x)*(1+x+x^2)*(x^2-x+1)/(1-x^6-x^7)) \\ G. C. Greubel, May 01 2017

Formula

G.f.: x*(1+x)*(1+x+x^2)*(x^2-x+1) / ( 1-x^6-x^7 ). - R. J. Mathar, Aug 26 2011

Extensions

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