1,13

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.

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.

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

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

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.

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: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 30 2008]

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.

Cf. A000931, A079398, A103372-103381, A103389, A103399.

Adjacent sequences: A103376 A103377 A103378 this_sequence A103380 A103381 A103382

Sequence in context: A064458 A125891 A111895 this_sequence A032550 A109399 A036450

easy,nonn

Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 15 2005

Corrected from a(11) on by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 30 2008