A054489 Expansion of (1+4*x)/(1-6*x+x^2).

1, 10, 59, 344, 2005, 11686, 68111, 396980, 2313769, 13485634, 78600035, 458114576, 2670087421, 15562409950, 90704372279, 528663823724, 3081278570065, 17959007596666, 104672767009931, 610077594462920

Offset: 0

Author

Barry E. Williams, May 04 2000

Keywords

• easy
• nonn

Comments

Numbers n such that 8*n^2 + 41 is a square.

(x, y) = (a(n), a(n+1)) are solutions to x^2 + y^2 - 6*x*y = 41. - John O. Oladokun, Mar 17 2021

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 (6,-1).

Crossrefs

Cf. A000129, A002203, A038761, A054488.

Programs

GAP

a:=[1,10];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020

Magma

I:=[1,10]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2020

Maple

a[0]:=1: a[1]:=10: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..19); # Zerinvary Lajos, Jul 26 2006

Mathematica

Table[(LucasL[2*n+1, 2] + 3*Fibonacci[2*n, 2])/2, {n,0,30}] (* G. C. Greubel, Jan 19 2020 *)
LinearRecurrence[{6,-1},{1,10},20] (* Harvey P. Dale, Jun 11 2024 *)

PARI

vector(31, n, polchebyshev(n-1,2,3) +4*polchebyshev(n-2,2,3) ) \\ G. C. Greubel, Jan 19 2020

Sage

[chebyshev_U(n,3) +4*chebyshev_U(n-1,3) for n in (0..30)] # G. C. Greubel, Jan 19 2020

Formula

a(n) = 6*a(n-1) - a(n-2), a(0)=1, a(1)=10.

a(n) = (10*((3+2*sqrt(2))^n - (3-2*sqrt(2))^n) - ((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1)))/(4*sqrt(2)).

From G. C. Greubel, Jan 19 2020: (Start)

a(n) = ChebyshevU(n,3) + 4*ChebyshevU(n-1,3).

a(n) = (Pell(2*n+2) + 4*Pell(2*n))/2 = (Pell-Lucas(2*n+1) + 3*Pell(2*n))/2.

E.g.f.: exp(3*x)*( cosh(2*sqrt(2)*x) + 7*sinh(2*sqrt(2)*x)/(2*sqrt(2)) ). (End)

Extensions

More terms from James Sellers, May 05 2000