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A041041
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Denominators of continued fraction convergents to sqrt(26).
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9
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1, 10, 101, 1020, 10301, 104030, 1050601, 10610040, 107151001, 1082120050, 10928351501, 110365635060, 1114584702101, 11256212656070, 113676711262801, 1148023325284080, 11593909964103601, 117087122966320090, 1182465139627304501, 11941738519239365100
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Generalized Fibonacci sequence.
Sqrt(26) = 10/2 + 10/101 + 10/(101*10301) + 10/(10301*1050601) + ... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2008
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 10'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
| S. Falcon & A. Plaza: The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals, 33 (2007)
S. Falcon & A. Plaza: On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007)
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n) = 10*a(n-1) + a(n-2), n>=1; a(-1):=0, a(0)=1.
G.f.: 1/(1-10*x-x^2).
a(n) = S(n, 10*I)*(-I)^n with I^2:=-1 and S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind. See A049310.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap:= 5 + sqrt(26), am:= 5 - sqrt(26)=-1/ap.
a(n)=F(n, 10), the n-th Fibonacci polynomial evaluated at x=10. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
a(n) = ((5+Sqrt[26])^n-(5-Sqrt[26])^n)/(2Sqrt[26]); a(n) = Sum[Binomial[n-1-i,i]*10^{n-1-2i}, {i,0,Floor[(n-1)/2]}] - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Sep 24 2007
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MATHEMATICA
| a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*10, {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]
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PROG
| (Other) sage: [lucas_number1(n, 10, -1) for n in xrange(1, 19)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]
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CROSSREFS
| Cf. A041040.
Cf. squares A099374.
Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371.
Sequence in context: A190480 A175541 A180175 * A163461 A081192 A108892
Adjacent sequences: A041038 A041039 A041040 * A041042 A041043 A041044
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KEYWORD
| nonn,cofr,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Extended by T. D. Noe (noe(AT)sspectra.com), May 23 2011
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