A057090 Scaled Chebyshev U-polynomials evaluated at i*sqrt(7)/2. Generalized Fibonacci sequence.
1, 7, 56, 441, 3479, 27440, 216433, 1707111, 13464808, 106203433, 837677687, 6607167840, 52113918689, 411047605703, 3242130670744, 25572247935129, 201700650241111, 1590910287233680, 12548276562323537, 98974307946900519, 780658091564568392
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
- A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=7, q=7.
- Tanya Khovanova, Recursive Sequences
- Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=7.
- Index entries for sequences related to Chebyshev polynomials.
- Index entries for linear recurrences with constant coefficients, signature (7,7).
Crossrefs
Cf. A000045.
Programs
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Magma
I:=[1,7]; [n le 2 select I[n] else 7*Self(n-1) + 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018
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Maple
a:= n-> (<<0|1>, <7|7>>^n. <<1, 7>>)[1, 1]: seq(a(n), n=0..30);
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Mathematica
Join[{a=0,b=1},Table[c=7*b+7*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *) LinearRecurrence[{7,7},{1,7},30] (* Harvey P. Dale, Nov 30 2012 *) -
PARI
Vec(1/(1-7*x-7*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015
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Sage
[lucas_number1(n,7,-7) for n in range(1, 21)] # Zerinvary Lajos, Apr 24 2009
Formula
a(n) = 7*(a(n-1) + a(n-2)), a(0)=1, a(1)=7.
a(n) = S(n, i*sqrt(7))*(-i*sqrt(7))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1 - 7*x - 7*x^2).
a(n) = Sum_{k=0..n} 6^k*A063967(n,k). - Philippe Deléham, Nov 03 2006
Comments