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A054413
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a(n)=7*a(n-1)+a(n-2).
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28
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1, 7, 50, 357, 2549, 18200, 129949, 927843, 6624850, 47301793, 337737401, 2411463600, 17217982601, 122937341807, 877779375250, 6267392968557, 44749530155149, 319514104054600, 2281348258537349, 16288951913816043, 116304011655249650, 830417033500563593
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| In general sequences with recurrence a(n)= k*a(n-1)+a(n-2) and a(0)=1 [and a(-1)=0] have generating function 1/(1-k*x-x^2). If k is odd (k>=3) they satisfy a(3n)=b(5n), a(3n+1)=b(5n+3), a(3n+2)=2*b(5n+4) where b(n) is the sequence of denominators of continued fraction convergents to sqrt(k^2+4). [If k is even then a(n) is the sequence of denominators of continued fraction convergents to sqrt(k^2/4+1).]
a(p) == 53^((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 the sequence given above k=7 which implies that it is associated with A041091.
For a similar statement about sequences with recurrence a(n) = k*a(n-1)+a(n-2) but with a(0) = 2 [and a(-1) = 0] see A086902; a sequence that is associated with A041090.
For more information follow the Khovanova link and see A087130, A140455 and A178765.
(End)
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 7'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 to sequences with linear recurrences with constant coefficients, signature (7,1).
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(3n)=A041091(5n), a(3n+1)=A041091(5n+3), a(3n+2)=2*A041091(5n+4) G.f.: 1/(1-7x-x^2)
a(n)= U(n, 7*I/2)*(-I)^n with I^2=-1 and Chebyshev's U(n, x/2)=S(n, x) polynomials. See A049310.
a(n)=F(n, 7), the n-th Fibonacci polynomial evaluated at x=7. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
a(n) = (sigma^n-(-sigma)^{-n})/(Sqrt[53]) with sigma=(7+Sqrt[53])/2; a(n) = Sum_0^{Floor[(n-1)/2]} Binomial[n-1-i,i]*7^{n-1-2i} - Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Sep 24 2007
a(n)=-(7/106)*sqrt(53)*[7/2-(1/2)*sqrt(53)]^n+(1/2)*[7/2+(1/2)*sqrt(53)]^n+(1/2)*[7/2-(1/2) *sqrt(53)]^n+(7/106)*[7/2+(1/2)*sqrt(53)]^n*sqrt(53), with n>=0 - Paolo P. Lava (paoloplava(AT)gmail.com), Jun 25 2008
a(n)=((7+sqrt53)^n-(7-sqrt53)^n)/(2^n*sqrt53). Offset 1. a(3)=50. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 17 2009]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 12 2010: (Start)
a(2n+1) = 7*A097836(n), a(2n) = A097838(n).
Limit(a(n+k)/a(k), k=infinity) = (A086902(n) + A054413(n-1)*sqrt(53))/2.
Limit(A086902(n)/A054413(n-1), n=infinity) = sqrt(53).
(End)
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MATHEMATICA
| a=0; lst={a}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*7, {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]
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PROG
| (Other) sage: [lucas_number1(n, 7, -1) for n in xrange(1, 19)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009]
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CROSSREFS
| Cf. A099367, A000045, A000129, A006190, A001076, A052918, A005668.
Sequence in context: A096882 A033125 A022037 * A163458 A081571 A081189
Adjacent sequences: A054410 A054411 A054412 * A054414 A054415 A054416
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KEYWORD
| nonn,easy
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), May 10 2000
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EXTENSIONS
| Formula corrected by Johannes W. Meijer (meijgia(AT)hotmail.com), May 30 2010, Jun 02 2010
Extended by T. D. Noe (noe(AT)sspectra.com), May 23 2011
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