1,14

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 (to 111 digits accuracy): 1.05705057522122838488168672780475393004610754746179195044509883164897227401211724373116252402900494800729490222. 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.

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>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.

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

SemiprimeQ[n_]:=Plus@@FactorInteger[n][[All, 2]]?2; Clear[a]; k12; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[n-k]+a[n-k-1]; A103380=Array[a, 100] A103390=Union[Select[Array[a, 1000], PrimeQ]] A103400=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, A103390, A103400.

Sequence in context: A082996 A094382 A146167 this_sequence A098708 A067394 A076925

Adjacent sequences: A103377 A103378 A103379 this_sequence A103381 A103382 A103383

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 16 2005

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