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%I #36 Apr 04 2023 07:46:09
%S 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,
%T 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,
%U 32,32,32,32,32,32,33,38,48,58,63,64,64,64,64,64,64,65,71,86,106,121,127
%N k=11 case of family of sequences beyond Fibonacci and Padovan.
%C 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.
%C 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.
%D Zanten, A. J. van, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.
%H J.-P. Allouche and T. Johnson, <a href="http://www.math.jussieu.fr/~allouche/johnson2.pdf">Narayana's Cows and Delayed Morphisms</a>
%H Richard Padovan, <a href="http://www.nexusjournal.com/conferences/N2002-Padovan.html">Dom Hans van der Laan and the Plastic Number</a>.
%H Kevin I. Piterman and Leandro Vendramin, <a href="https://crossroads-2023.github.io/vendramin/gap.pdf">Computer algebra with GAP</a>, 2023. See p. 39.
%H E. S. Selmer, <a href="http://www.mscand.dk/article/view/10478/8499">On the irreducibility of certain trinomials</a>, Math. Scand., 4 (1956) 287-302.
%H J. Shallit, <a href="http://dx.doi.org/10.1016/0304-3975(88)90103-X">A generalization of automatic sequences</a>, Theoretical Computer Science, 61 (1988), 1-16.
%H Leandro Vendramin, <a href="https://www.mun.ca/aac/AACMiniCourses/GAP/problems.pdf">Mini-couse on GAP - Exercises</a>, Universidad de Buenos Aires (Argentina, 2020).
%H <a href="/index/Rec/order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,1,1).
%F 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.
%F G.f.: x*(1-x^11) / ((1-x)*(1-x^11-x^12)). - _Colin Barker_, Mar 26 2013
%p 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
%t 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_ *)
%t 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 *)
%Y Cf. A000931, A079398, A103372-A103378, A103380, A103389, A103399.
%K easy,nonn
%O 1,13
%A _Jonathan Vos Post_, Feb 15 2005
%E Corrected from a(11) on by _R. J. Mathar_, Aug 30 2008
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