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A099371
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Generalized Fibonacci sequence.
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19
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0, 1, 9, 82, 747, 6805, 61992, 564733, 5144589, 46866034, 426938895, 3889316089, 35430783696, 322766369353, 2940328107873, 26785719340210, 244011802169763, 2222891938868077, 20250039251982456, 184473245206710181
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(p) == 85^((p-1)/2)) mod p for odd primes p [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 22 2009]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765.
(End)
For n >=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 9's along the main diagonal and 1's along the superdiagonal and the subdiagonal. [From John M. Campbell, Jul 08 2011]
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REFERENCES
| J. H. Han and M. D. Hirschhorn, "Another Look at an Amazing Identity of Ramanujan", Mathematics Magazine, Vol. 79 (2006), pp. 302-304. See equation 6 on page 303.
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| G.f.: x/(1 - 9*x - x^2).
a(n) = 9*a(n-1)+a(n-2), n>=2, a(0)=0, a(1)=1.
a(n) = ((-i)^(n-1))*S(n-1, 9*i) with S(n, x) Chebyshev's polynomials of the second kind (see A049310) and i^2=-1.
a(n)= (ap^n - am^p)/(ap-am) with ap:= (9+sqrt(85))/2 and am:= (9-sqrt(85))/2=-1/ap (Binet form).
a(n)= sum(binomial(n-1-k, k)*9^(n-1-2*k), k=0..floor((n-1)/2)), n>=1.
a(n)=F(n, 9), the n-th Fibonacci polynomial evaluated at x=9. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
a(n)=((9+sqrt85)^n-(9-sqrt85)^n)/(2^n*sqrt85). Offset 1. a(3)=82. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
a(2n+2) = 9*A097839(n), a(2n+1) = A097841(n).
a(3n+1) = A041151(5n), a(3n+2) = A041151(5n+3), a(3n+3) = 2*A041151(5n+4).
Limit(a(n+k)/a(k), k=infinity) = (A087798(n) + A099371(n)*sqrt(85))/2.
Limit(A087798(n)/A099371(n), n=infinity) = sqrt(85).
(End)
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MATHEMATICA
| a=0; lst={a}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*9, {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]
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PROG
| sage: from sage.combinat.sloane_functions import recur_gen3 sage: it = recur_gen3(0, 1, 9, 9, 1, 0) sage: [it.next() for i in xrange(1, 22)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008
(Other) sage: [lucas_number1(n, 9, -1) for n in xrange(0, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]
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CROSSREFS
| Cf. A099372 (squares).
Cf. A099371.
Sequence in context: A067506 A033119 A033127 * A068109 A163460 A081191
Adjacent sequences: A099368 A099369 A099370 * A099372 A099373 A099374
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 18 2004
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