A055271 a(n) = 5*a(n-1) - a(n-2) with a(0)=1, a(1)=7.

1, 7, 34, 163, 781, 3742, 17929, 85903, 411586, 1972027, 9448549, 45270718, 216905041, 1039254487, 4979367394, 23857582483, 114308545021, 547685142622, 2624117168089, 12572900697823, 60240386321026, 288629030907307, 1382904768215509, 6625894810170238, 31746569282635681, 152106951603008167

Offset: 0

Author

Barry E. Williams, May 10 2000

Keywords

• easy
• nonn

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), pp. 181-193.
• E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (5,-1).

Crossrefs

Cf. A030221.

Programs

Magma

I:=[1,7]; [n le 2 select I[n] else 5*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Mar 16 2020

Maple

A055271:= n-> simplify(ChebyshevU(n, 5/2) + 2*ChebyshevU(n-1, 5/2)); seq(A055271(n), n=0..30); # G. C. Greubel, Mar 16 2020

Mathematica

LinearRecurrence[{5,-1}, {1,7}, 30] (* G. C. Greubel, Mar 16 2020 *)

Sage

[chebyshev_U(n, 5/2) + 2*chebyshev_U(n-1, 5/2) for n in (0..30)] # G. C. Greubel, Mar 16 2020

Formula

a(n) = (7*(((5+sqrt(21))/2)^n - ((5-sqrt(21))/2)^n) - (((5+sqrt(21))/2)^(n-1) - ((5-sqrt(21))/2)^(n-1)))/sqrt(21).

G.f.: (1+2*x)/(1-5*x+x^2).

a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-8)^k. - Philippe Deléham, Mar 05 2014

a(n) = ChebyshevT(n, 5/2) + (9/2)*ChebyshevU(n-1,5/2) = ChebyshevU(n, 5/2) + 2*ChebyshevU(n-1, 5/2). - G. C. Greubel, Mar 16 2020

Extensions

Terms a(22) onward added by G. C. Greubel, Mar 16 2020