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A103378
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a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = a(11) = 1 and for n>11: a(n) = a(n-10) + a(n-11).
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9
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 64, 64, 65, 71, 86, 106, 121, 127, 128
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OFFSET
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1,12
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COMMENTS
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k=10 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 and k=9 case is A103377. 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=10 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^11 - x - 1 = 0. This is the real constant (to 111 digits accuracy): 1.06829718892084127636942958832387828209363101692083344450761194664700697029324883112914781008997954899432302064. Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/11))^(1/11)))^(1/11))))^(1/11)))))^(1/11))))). The sequence of prime values in this k=10 case is A103388; The sequence of semiprime values in this k=10 case is A103398.
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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..78.
J.-P. Allouche and T. Johnson, Narayana's Cows and DelayedMorphisms
Richard Padovan, Dom Hans van der Laanand the Plastic Number.
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FORMULA
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G.f.: -x*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)/(-1+x^10+x^11) . - R. J. Mathar, Nov 22 2007
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EXAMPLE
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a(52)=17 because a(52)=a(52-10)+a(52-11) = a(42)+a(41) = 9 + 8. The sequence has as elements 5, 17 and 257, which are all Fermat Primes.
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MAPLE
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A103378 := proc(n) option remember; if n <= 11 then 1 ; else A103378(n-10)+A103378(n-11) ; fi ; end: seq(A103378(n), n=1..78) ; - R. J. Mathar, Nov 22 2007
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MATHEMATICA
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SemiprimeQ[n_]:=Plus@@FactorInteger[n][[All, 2]]?2; Clear[a]; k=10; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[n-k]+a[n-k-1]; A103377=Array[a, 100] A103387=Union[Select[Array[a, 1000], PrimeQ]] A103397=Union[Select[Array[a, 300], SemiprimeQ]] N[Solve[x^10 - x - 1 == 0, x], 111][[2]]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 80] (* From Harvey P. Dale, May 11 2011 *)
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CROSSREFS
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Cf. A000045, A000931, A079398, A103372-103381, A103388, A103398.
Sequence in context: A102684 A156821 A025856 * A103663 A083311 A037968
Adjacent sequences: A103375 A103376 A103377 * A103379 A103380 A103381
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post, Feb 15 2005
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EXTENSIONS
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Corrected and extended by R. J. Mathar, Nov 22 2007
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STATUS
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approved
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