1,12

k=10 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 and k=9 case is A103377. 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=10 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^11 - x - 1 = 0. This is the real constant (to 111 digits accuracy): 1.06829718892084127636942958832387828209363101692083344450761194664700697029324883112914781008997954899432302064. Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/11))^(1/11)))^(1/11))))^(1/11)))))^(1/11))))). The sequence of prime values in this k=10 case is A103388; The sequence of semiprime values in this k=10 case is A103398.

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 DelayedMorphisms

Richard Padovan, Dom Hans van der Laanand the Plastic Number.

G.f.: -x*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)/(-1+x^10+x^11) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 22 2007

a(52)=17 because a(52)=a(52-10)+a(52-11) = a(42)+a(41) = 9 + 8. The sequence has as elements 5, 17 and 257, which are all Fermat Primes.

A103378 := proc(n) option remember; if n <= 11 then 1 ; else A103378(n-10)+A103378(n-11) ; fi ; end: seq(A103378(n), n=1..78) ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 22 2007

SemiprimeQ[n_]:=Plus@@FactorInteger[n][[All, 2]]?2; Clear[a]; k=10; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[n-k]+a[n-k-1]; A103377=Array[a, 100] A103387=Union[Select[Array[a, 1000], PrimeQ]] A103397=Union[Select[Array[a, 300], SemiprimeQ]] N[Solve[x^10 - x - 1 == 0, x], 111][[2]]

Cf. A000045, A000931, A079398, A103372-103381, A103388, A103398.

Sequence in context: A102684 A156821 A025856 this_sequence A103663 A083311 A037968

Adjacent sequences: A103375 A103376 A103377 this_sequence A103379 A103380 A103381

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 15 2005

Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 22 2007