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 A103375 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 1 and for n>8: a(n) = a(n-7) + a(n-8). 15
 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 65, 71, 86, 106, 121, 127, 128, 129, 136, 157, 192, 227, 248, 255, 257, 265, 293, 349, 419, 475, 503, 512 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS k=7 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 and k=6 case is A103374. 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=7 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^8 - x - 1 = 0. This is the real constant 1.09698155779855981790827896716753708959253010821278671381232885124855898059.... The sequence of prime values in this k=7 case is A103385; the sequence of semiprime values in this k=7 case is A103395. 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..72. Richard Padovan, Dom Hans van der Laan and the Plastic Number. J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms 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 (0,0,0,0,0,0,1,1). FORMULA G.f.: -x*(1+x+x^2+x^3+x^4+x^5+x^6)/(-1+x^7+x^8). - R. J. Mathar, Dec 14 2009 EXAMPLE a(30) = 12 because a(30) = a(30-7) + a(30-8) = a(24) + a(23) = 7 + 5 = 12. MATHEMATICA k = 7; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 73] LinearRecurrence[{0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}, 80] PROG (PARI) a(n)=([0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 1; 1, 1, 0, 0, 0, 0, 0, 0]^(n-1)*[1; 1; 1; 1; 1; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016 CROSSREFS Cf. A000045, A000931, A079398, A103372-A103374, A103376-A103380, A103385, A103395. Sequence in context: A303904 A173021 A109703 * A285758 A340959 A246869 Adjacent sequences: A103372 A103373 A103374 * A103376 A103377 A103378 KEYWORD nonn,easy AUTHOR Jonathan Vos Post, Feb 03 2005 EXTENSIONS Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005 Corrected (one more 8 inserted) by R. J. Mathar, Dec 14 2009 STATUS approved

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