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A103389
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Primes in A103379 (= 11-delayed Fibonacci b(n) = b(n-11)+b(n-12) or = 1 for n<12).
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4
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2, 3, 5, 7, 17, 31, 71, 127, 157, 227, 257, 293, 349, 419, 503, 8179, 65657, 68053, 72421, 80429, 258949, 493109, 16399511, 33609887, 34225183, 1387603957, 5575987679, 15932884421, 35689079297, 693128029907, 957136790429, 1129233918343, 10363690074667, 41632551979939, 10815125582078291
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OFFSET
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1,1
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COMMENTS
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These primes are elements of the k=11 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, k=9 case is A103377 and k=10 case is A103378. 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=11 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^12 - x - 1 = 0. This is the real constant 1.062169167864255148458944276143126923146557407121804298167945495796... . Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/12))^(1/12)))^(1/12))))^(1/12)))))^(1/12))))). This sequence of prime values in this k=11 case is this sequence. The sequence of semiprime values in this k=11 case is A103399.
N.B.: a(n) in the above does not refer to the present sequence but to the delayed Fibonacci sequence itself, here A103379. - M. F. Hasler, Sep 19 2015
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REFERENCES
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A. J. van Zanten, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, vol 17 no 2 (1999) 229-245.
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LINKS
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FORMULA
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Intersection of A103379 and A000040, where A103379 is: for n>12: a(n) = a(n-11) + a(n-12). a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = a(11) = a(12) = 1.
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EXAMPLE
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A103379(20) = 3, which is prime, hence 3 is in this sequence.
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MAPLE
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A103379 := proc(n) option remember ; if n <= 12 then 1; else procname(n-11)+procname(n-12) ; fi; end: isA103379 := proc(n) option remember ; local i ; for i from 1 do if A103379(i) = n then RETURN(true) ; elif A103379(i) > n then RETURN(false) ; fi; od: end: A103389 := proc(n) option remember ; local a; if n = 1 then 2; else for a from procname(n-1)+1 do if isprime(a) then if isA103379(a) then RETURN(a) ; fi; fi; od: fi; end: for n from 1 to 37 do printf("%d, ", A103389(n)) ; od: # R. J. Mathar, Aug 30 2008
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MATHEMATICA
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Clear[a]; k11; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[n-k]+a[n-k-1]; A103389=Union[Select[Array[a, 1000], PrimeQ]] N[Solve[x^12 - x - 1 == 0, x], 111][[2]] (* Program, edit and extension by Ray Chandler and Robert G. Wilson v, irrelevant code deleted by M. F. Hasler, Sep 19 2015 *)
Select[LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 700], PrimeQ]//Union (* Harvey P. Dale, Apr 22 2016 *)
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PROG
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(PARI) {a=vector(m=12, n, 1); L=0; for(n=m, 10^5, isprime(a[i=n%m+1]+=a[(n+1)%m+1])&&L<a[i]&&print1(L=a[i], ", "))} \\ M. F. Hasler, Sep 19 2015
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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