A103380 k=12 case of family of sequences beyond Fibonacci and Padovan: a(n) = a(n-12) + a(n-13).

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

Offset

1,14

Comments

k=12 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, k=10 case is A103378 and k=11 case is A103379.

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^13 - x - 1 = 0. This is the real constant 1.0570505752... . Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/13))^(1/13)))^(1/13))))^(1/13)))))^(1/13))))).

The sequence of prime values in this k=12 case is A103390.

The sequence of semiprime values in this k=12 case is A103400.

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

T. D. Noe, Table of n, a(n) for n = 1..1000

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

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 (0,0,0,0,0,0,0,0,0,0,0,1,1).

Formula

For n>13: a(n) = a(n-12) + a(n-13). a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = a(11) = a(12) a(13) = 1.

G.f.: x*(1-x^12) / ((1-x)*(1-x^12-x^13)). - Colin Barker, Mar 27 2013

Maple

A103380 := proc(n) option remember ; if n <= 13 then 1; else procname(n-12)+procname(n-13) ; fi; end: for n from 1 to 120 do printf("%d, ", A103380(n)) ; od: # R. J. Mathar, Aug 30 2008

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 90] (* Harvey P. Dale, Jul 16 2012 *)

Crossrefs

Cf. A000931, A079398, A103372-A103379, A103390, A103400.

Sequence in context: A094382 A146167 A346622 * A309120 A098708 A067394

Adjacent sequences: A103377 A103378 A103379 * A103381 A103382 A103383

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 16 2005

Extensions

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

Status

approved