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A103374
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a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n-6) + a(n-7).
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15
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1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 17, 21, 27, 31, 32, 32, 33, 38, 48, 58, 63, 64, 65, 71, 86, 106, 121, 127, 129, 136, 157, 192, 227, 248, 256, 265, 293, 349, 419, 475, 504, 521, 558, 642, 768, 894, 979, 1025, 1079
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| k=6 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 and k=5 case is A103373.
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=6 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^7 - x - 1 = 0. This is the real constant (to 100 digits accuracy): 1.112775684278705470629704020571092935606859271855283681485701628007166332579528443459272836948847449
The sequence of prime values in this k=6 case is A103384; The sequence of semiprime values in this k=6 case is A103394.
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REFERENCES
| 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
| Richard Padovan, Dom Hans van der Laan and the Plastic Number.
J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms
Index to sequences with linear recurrences with constant coefficients, signature (0,0,0,0,0,1,1).
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FORMULA
| G.f. -x*(1+x)*(1+x+x^2)*(x^2-x+1) / ( -1+x^6+x^7 ). - R. J. Mathar, Aug 26 2011
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EXAMPLE
| a(32) = 17 because a(32) = a(32-6) + a(32-7) = a(26) + a(25) = 9 + 8 = 17.
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MATHEMATICA
| k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 70]
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CROSSREFS
| Cf. A000045, A000931, A079398, A103372-A103381, A103384, A103394.
Sequence in context: A093354 A109701 A124751 * A137722 A081305 A035665
Adjacent sequences: A103371 A103372 A103373 * A103375 A103376 A103377
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KEYWORD
| nonn,easy
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 03 2005
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EXTENSIONS
| Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net) and Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 06 2005
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