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 A103379 k=11 case of family of sequences beyond Fibonacci and Padovan. 11
 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, 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, 32, 32, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 64, 64, 64, 65, 71, 86, 106, 121, 127 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,13 COMMENTS 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. 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 Table of n, a(n) for n=1..84. J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms Richard Padovan, Dom Hans van der Laan and the Plastic Number. Kevin I. Piterman and Leandro Vendramin, Computer algebra with GAP, 2023. See p. 39. 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. Leandro Vendramin, Mini-couse on GAP - Exercises, Universidad de Buenos Aires (Argentina, 2020). Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,1,1). FORMULA 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. G.f.: x*(1-x^11) / ((1-x)*(1-x^11-x^12)). - Colin Barker, Mar 26 2013 MAPLE 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 MATHEMATICA 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 *) 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 *) CROSSREFS Cf. A000931, A079398, A103372-A103378, A103380, A103389, A103399. Sequence in context: A125891 A153675 A111895 * A032550 A339172 A036450 Adjacent sequences: A103376 A103377 A103378 * A103380 A103381 A103382 KEYWORD easy,nonn AUTHOR Jonathan Vos Post, Feb 15 2005 EXTENSIONS Corrected from a(11) on by R. J. Mathar, Aug 30 2008 STATUS approved

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Last modified April 21 14:30 EDT 2024. Contains 371874 sequences. (Running on oeis4.)