A103379 k=11 case of family of sequences beyond Fibonacci and Padovan.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 64, 64, 64, 65, 71, 86, 106, 121, 127

Offset

1,13

Comments

k=11 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, k=6 case is A103374, k=7 case is A103375, k=8 case is A103376, k=9 case is A103377 and k=10 case is A103378.

The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1)= 1 and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]). For this k=11 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^12 - x - 1 = 0. This is the real constant 1.062169167864255148458944... Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/12))^(1/12)))^(1/12))))^(1/12)))))^(1/12))))). The sequence of prime values in this k=11 case is A103389; the sequence of semiprime values in this k=11 case is A103399.

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

Table of n, a(n) for n=1..84.

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

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

Kevin I. Piterman and Leandro Vendramin, Computer algebra with GAP, 2023. See p. 39.

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.

Leandro Vendramin, Mini-couse on GAP - Exercises, Universidad de Buenos Aires (Argentina, 2020).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,1,1).

Formula

For n>12: a(n) = a(n-11) + a(n-12). a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = a(11) = a(12) = 1.

G.f.: x*(1-x^11) / ((1-x)*(1-x^11-x^12)). - Colin Barker, Mar 26 2013

Maple

A103379 := proc(n) option remember ; if n <= 12 then 1; else procname(n-11)+procname(n-12) ; fi; end: for n from 1 to 120 do printf("%d, ", A103379(n)) ; od: # R. J. Mathar, Aug 30 2008

Mathematica

SemiprimeQ[n_]:=Plus@@FactorInteger[n][[All, 2]]?2; Clear[a]; k11; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[n-k]+a[n-k-1]; A103379=Array[a, 100] A103389=Union[Select[Array[a, 1000], PrimeQ]] A103399=Union[Select[Array[a, 300], SemiprimeQ]] N[Solve[x^12 - x - 1 == 0, x], 111][[2]] (* Program, edit and extension by Ray Chandler and Robert G. Wilson v *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 100] (* Harvey P. Dale, Jan 31 2015 *)

Crossrefs

Cf. A000931, A079398, A103372-A103378, A103380, A103389, A103399.

Sequence in context: A125891 A153675 A111895 * A032550 A339172 A036450

Adjacent sequences: A103376 A103377 A103378 * A103380 A103381 A103382

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 15 2005

Extensions

Corrected from a(11) on by R. J. Mathar, Aug 30 2008

Status

approved