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A056854
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a(n)=7a(n-1)-a(n-2), a(0)=2, a(1)=7.
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12
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2, 7, 47, 322, 2207, 15127, 103682, 710647, 4870847, 33385282, 228826127, 1568397607, 10749957122, 73681302247, 505019158607, 3461452808002, 23725150497407, 162614600673847, 1114577054219522, 7639424778862807, 52361396397820127, 358890350005878082, 2459871053643326447
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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REFERENCES
| E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pps. 231-242.
D. Gerdemann, Combinatorial proofs of Zeckendorf family identities, Fib. Q., 46/47 (2008/2009), 249-261. [From N. J. A. Sloane, Dec 05 2009]
A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., 5 (1967), pps. 424-434.
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LINKS
| Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| a(n) = 7*S(n-1, 7) - 2*S(n-2, 7) = S(n, 7) - S(n-2, 7) = 2*T(n, 7/2), with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. U(n, x), resp. T(n, x), are Chebyshev's polynomials of the second, resp. first, kind. S(n-1, 7) = A004187(n), n>=0. See A049310 and A053120.
a(n)=((7+sqrt(45))/2)^n + ((7-sqrt(45))/2)^n.
G.f.: (2-7x)/(1-7x+x^2).
a(n)=A005248(2*n). Bisection of A005248.
a(n)=Fibonacci(8*n)/Fibonacci(4*n), n>0, [From Gary Detlefs (gdetlefs(AT)aol.com) Dec 26 2010]
a(n) = Lucas(4*n), n>=0. // Vincenzo Librandi, Apr 14 2011
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MATHEMATICA
| a[0] = 2; a[1] = 7; a[n_] := 7a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 19}] (from Robert G. Wilson v Jan 30 2004)
LinearRecurrence[{7, -1}, {2, 7}, 25] (* or *) LucasL[4*Range[0, 25]] (* From Harvey P. Dale, Aug 08 2011 *)
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PROG
| (PARI) a(n)=if(n<0, 0, polsym(1-7*x+x^2, n)[n+1])
(PARI) a(n)=if(n<0, 0, 2*subst(poltchebi(n), x, 7/2))
sage: [lucas_number2(n, 7, 1) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(MAGMA) [ Lucas(4*n) : n in [0..100]]; // Vincenzo Librandi, Apr 14 2011
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CROSSREFS
| a(n)=sqrt{[45*(A004187(n))^2]+4}.
a(n) = A000032(4n) = Lucas numbers L(4n).
Sequence in context: A054555 A072287 A091117 * A117141 A125813 A106159
Adjacent sequences: A056851 A056852 A056853 * A056855 A056856 A056857
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KEYWORD
| easy,nonn
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AUTHOR
| Barry E. Williams, Aug 29 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Aug 31 2000
Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 31 2002
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