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 A103376 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = 1 and for n>9: a(n) = a(n-8) + a(n-9). 15
 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 129, 136, 157, 192, 227, 248, 255, 256, 257, 265, 293 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,10 COMMENTS k=8 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 and k=7 case is A103375. The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]). For this k=8 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^9 - x - 1 = 0. This is the real constant (to 50 digits accuracy): 1.0850702454914508283368958640973142340506536310308. Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/9))^(1/9)))^(1/9))))^(1/9)))))^(1/9))))). The sequence of prime values in this k=8 case is A103386; The sequence of semiprime values in this k=8 case is A103396. 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..76. J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms Richard Padovan, Dom Hans van der Laan and the Plastic Number. 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. Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1,1). FORMULA G.f.: x*(1+x)*(1+x^2)*(1+x^4)/(1-x^8-x^9). - R. J. Mathar, Dec 14 2009 a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1, a(7)=1, a(8)=1, a(9)=1, a(n)=a(n-8)+a(n-9). - Harvey P. Dale, May 07 2015 EXAMPLE a(93) = 1200 because a(93) = a(93-8) + a(93-9) = a(85) + a(84) = 642 + 558. MATHEMATICA k = 8; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 76] LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1}, 80] (* Harvey P. Dale, May 07 2015 *) PROG (PARI) a(n)=([0, 1, 0, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0, 1; 1, 1, 0, 0, 0, 0, 0, 0, 0]^(n-1)*[1; 1; 1; 1; 1; 1; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016 CROSSREFS Cf. A000045, A000931, A079398, A103372-A103375, A103377-A103380, A103386, A103396. Sequence in context: A029241 A226749 A277090 * A189819 A145992 A045818 Adjacent sequences: A103373 A103374 A103375 * A103377 A103378 A103379 KEYWORD nonn,easy AUTHOR Jonathan Vos Post, Feb 05 2005 EXTENSIONS Edited by Ray Chandler, Feb 10 2005 STATUS approved

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Last modified March 4 07:29 EST 2024. Contains 370522 sequences. (Running on oeis4.)