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A165900
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Values of Fibonacci polynomial n^2 - n - 1.
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21
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-1, -1, 1, 5, 11, 19, 29, 41, 55, 71, 89, 109, 131, 155, 181, 209, 239, 271, 305, 341, 379, 419, 461, 505, 551, 599, 649, 701, 755, 811, 869, 929, 991, 1055, 1121, 1189, 1259, 1331, 1405, 1481, 1559, 1639, 1721, 1805, 1891, 1979, 2069, 2161, 2255
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OFFSET
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0,4
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COMMENTS
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a(n) = - A188652(2*n) for n > 0. - Reinhard Zumkeller, Apr 13 2011
For n > 0: a(n) = A214803(A015614(n+1)). - Reinhard Zumkeller, Jul 29 2012
Shifted version of the array denoted rB(0,2) in A132382, whose e.g.f. is exp(x)(1-x)^2. Taking the derivative gives the e.g.f. of A165900. - Tom Copeland, Dec 02 2013
The Fibonacci numbers are generated by the series x/(1 - x - x^2). - T. D. Noe, Dec 04 2013
Absolute value of expression f(k)*f(k+1)-f(k-1)*f(k+2) where f(1)=1, f(2)=n. Sign is alternately +1 and -1. - Carmine Suriano, Jan 28 2014 [Can anybody clarify what is meant here? - Joerg Arndt, Nov 24 2014]
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
J. J. Heed and L Kelly, An interesting sequence of Fibonacci sequence generators, Fibonacci Quarterly, 13 (1975), pp. 29-30.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n+2) = (n+1)*a(n+1) - (n+2)*a(n).
G.f.: (x^2+2*x-1)/(1-x)^3.
E.g.f.: exp(x)*(x^2-1).
G.f.: -1-x+x^2*G(0) where G(k) = 1 + x*(k+1)*(k+4)/(1 - 1/(1 + (k+1)*(k+4)/G(k+1)); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 16 2012
E.g.f.: G(0) where G(k)= -1 - x^2/(1 - 1/(1 + x*(k+1)/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 17 2012
a(n+1) = a(n) + A005843(n) = A002378(n) - 1. - Ivan N. Ianakiev, Feb 18 2013
a(n+2) = A028387(n). - Michael B. Porter, Sep 26 2018
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MATHEMATICA
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Table[n^2 - n - 1, {n, 0, 50}] (* Ron Knott, Oct 27 2010 *)
LinearRecurrence[{3, -3, 1}, {-1, -1, 1}, 60] (* Harvey P. Dale, Jul 05 2021 *)
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PROG
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(PARI) a(n)=n^2-n-1 \\ Charles R Greathouse IV, Jan 12 2012
(Haskell)
a165900 n = n * (n - 1) - 1 -- Reinhard Zumkeller, Jul 29 2012
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CROSSREFS
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A028387 and A110331 are very similar sequences.
Sequence in context: A209188 A215886 A088059 * A110331 A028387 A106071
Adjacent sequences: A165897 A165898 A165899 * A165901 A165902 A165903
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KEYWORD
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sign,easy
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AUTHOR
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Philippe Deléham, Sep 29 2009
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EXTENSIONS
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a(22) corrected by Reinhard Zumkeller, Apr 13 2011
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STATUS
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approved
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