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A022113
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Fibonacci sequence beginning 2, 7.
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14
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2, 7, 9, 16, 25, 41, 66, 107, 173, 280, 453, 733, 1186, 1919, 3105, 5024, 8129, 13153, 21282, 34435, 55717, 90152, 145869, 236021, 381890, 617911, 999801, 1617712, 2617513, 4235225, 6852738, 11087963, 17940701, 29028664, 46969365, 75998029, 122967394
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OFFSET
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0,1
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REFERENCES
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H. S. M. Coxeter, Introduction to Geometry, Second Edition, Wiley Classics Library Edition Published 1989, p. 172.
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LINKS
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Ivan Panchenko, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).
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FORMULA
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From Colin Barker, Oct 18 2013: (Start)
G.f.: -(5*x + 2)/(x^2 + x - 1).
a(n) = a(n-1) + a(n-2). (End)
a(n) = ((5+6*sqrt(5))/5)*((1+sqrt(5))/2)^n + ((5-6*sqrt(5))/5)*((1-sqrt(5))/2)^n starting at n=0. - Bogart B. Strauss, Oct 27 2013
a(n) = h*Fibonacci(n+k) + Fibonacci(n+k-h) with h=5, k=1. - Bruno Berselli, Feb 20 2017
a(n) = 8*F(n) + F(n-3) for F = A000045. - J. M. Bergot, Jul 14 2017
a(n) = Fibonacci(n+4) + Lucas(n-1). - Greg Dresden and Henry Sauer, Mar 04 2022
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 6*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Jul 18 2022
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MATHEMATICA
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RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == a[n - 1] + a[n - 2]}, a, {n, 0, 40}] (* Bruno Berselli, Mar 12 2015 *)
LinearRecurrence[{1, 1}, {2, 7}, 37] (* or *)
CoefficientList[Series[-(5 x + 2)/(x^2 + x - 1), {x, 0, 36}], x] (* Michael De Vlieger, Jul 14 2017 *)
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PROG
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(Magma) a0:=2; a1:=7; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013
(PARI) a(n)=8*fibonacci(n)+fibonacci(n-3) \\ Charles R Greathouse IV, Jul 14 2017
(PARI) a(n)=([0, 1; 1, 1]^n*[2; 7])[1, 1] \\ Charles R Greathouse IV, Jul 14 2017
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CROSSREFS
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Cf. A000032. A000045.
Sequence in context: A165995 A287575 A267212 * A041643 A041395 A042345
Adjacent sequences: A022110 A022111 A022112 * A022114 A022115 A022116
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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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STATUS
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approved
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