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A103376
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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).
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13
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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
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OFFSET
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1,10
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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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.
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FORMULA
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G.f.: x*(1+x)*(1+x^2)*(1+x^4)/(1-x^8-x^9). [From R. J. Mathar, Dec 14 2009]
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EXAMPLE
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a(93) = 1200 because a(93) = a(93-8) + a(93-9) = a(85) + a(84) = 642 + 558.
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MATHEMATICA
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k = 8; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 76]
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CROSSREFS
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Cf. A000045, A000931, A079398, A103372-A103381, A103386, A103396.
Sequence in context: A013941 A061798 A029241 * A189819 A045818 A064128
Adjacent sequences: A103373 A103374 A103375 * A103377 A103378 A103379
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KEYWORD
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nonn,easy
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AUTHOR
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Jonathan Vos Post, Feb 05 2005
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EXTENSIONS
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Edited by Ray Chandler, Feb 10 2005
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STATUS
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approved
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