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A022096
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Fibonacci sequence beginning 1 6.
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11
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1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364, 589, 953, 1542, 2495, 4037, 6532, 10569, 17101, 27670, 44771, 72441, 117212, 189653, 306865, 496518, 803383, 1299901, 2103284, 3403185, 5506469, 8909654, 14416123, 23325777, 37741900, 61067677, 98809577, 159877254
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OFFSET
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0,2
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COMMENTS
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a(n-1)=sum(P(6;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=5. These are the sums of the SW-NE diagonals in P(6;n,k), the (6,1) Pascal triangle A093563. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also sums of SW-NE diagonals in (1,5)-Pascal triangle A096940.
Subsequence of primes: 7, 13, 53, 139, 953, 44771, 189653, 1494692464747, ... - R. J. Mathar, Aug 09 2012
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LINKS
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Table of n, a(n) for n=0..37.
Tanya Khovanova, Recursive Sequences
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368, 2012
Index to sequences with linear recurrences with constant coefficients, signature (1,1)
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FORMULA
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a(n) = a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=6.
a(-1)=5.
G.f.: (1+5*x)/(1-x-x^2).
Row sums of triangle A131777. - Gary W. Adamson, Jul 14 2007
a(n) = 5*Fibonacci(n+2)-4*Fibonacci(n+1) [From Gary Detlefs Dec 21 2010]
a(n) = (2^(-1-n) ((1-Sqrt[5])^n (-11+Sqrt[5])+(1+Sqrt[5])^n (11+Sqrt[5])))/Sqrt[5] - Herbert Kociemba
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MATHEMATICA
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a={}; b=1; c=6; AppendTo[a, b]; AppendTo[a, c]; Do[b=b+c; AppendTo[a, b]; c=b+c; AppendTo[a, c], {n, 1, 9, 1}]; a (* Vladimir Orlovsky, Jul 22 2008 *)
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PROG
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(MAGMA) a0:=1; a1:=6; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013
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CROSSREFS
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a(n) = A101220(5, 0, n+1).
a(n) = A109754(5, n+1).
Cf. A131777.
Sequence in context: A127020 A154662 A070398 * A041175 A041074 A041749
Adjacent sequences: A022093 A022094 A022095 * A022097 A022098 A022099
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Spelling correction by Jason G. Wurtzel, Aug 22 2010
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STATUS
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approved
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