A054488 Expansion of (1+2*x)/(1-6*x+x^2).

1, 8, 47, 274, 1597, 9308, 54251, 316198, 1842937, 10741424, 62605607, 364892218, 2126747701, 12395593988, 72246816227, 421085303374, 2454265004017, 14304504720728, 83372763320351, 485932075201378, 2832219687887917

Offset

0,2

Comments

Bisection (even part) of Chebyshev sequence with Diophantine property.

b(n)^2 - 8*a(n)^2 = 17, with the companion sequence b(n)= A077240(n).

The odd part is A077413(n) with Diophantine companion A077239(n).

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 sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (6,-1).

Formula

a(n) = 6*a(n-1) - a(n-2), a(0)=1, a(1)=8.

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

a(n) = S(n, 6) + 2*S(n-1, 6), with S(n, x) Chebyshev's polynomials of the second kind, A049310. S(n, 6) = A001109(n+1).

a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-9)^k. - Philippe Deléham, Mar 05 2014

a(n) = (Pell(2*n+2) + 2*Pell(2*n))/2 = (Pell-Lucas(2*n+1) + Pell(2*n))/2. - G. C. Greubel, Jan 19 2020

E.g.f.: (1/4)*exp(3*x)*(4*cosh(2*sqrt(2)*x) + 5*sqrt(2)*sinh(2*sqrt(2)*x)). - Stefano Spezia, Jan 27 2020

Example

8 = a(1) = sqrt((A077240(1)^2 - 17)/8) = sqrt((23^2 - 17)/8)= sqrt(64) = 8.

Maple

a[0]:=1: a[1]:=8: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); # Zerinvary Lajos, Jul 26 2006

Mathematica

LinearRecurrence[{6, -1}, {1, 8}, 30] (* Harvey P. Dale, Oct 09 2017 *)
Table[(LucasL[2*n+1, 2] + Fibonacci[2*n, 2])/2, {n, 0, 30}] (* G. C. Greubel, Jan 19 2020 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1+2*x)/(1-6*x+x^2)) \\ G. C. Greubel, Jan 19 2020
(PARI) apply( {A054488(n)=[1, 8]*([0, -1; 1, 6]^n)[, 1]}, [0..30]) \\ M. F. Hasler, Feb 27 2020
(Magma) I:=[1, 8]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2020
(Magma) R<x>:=PowerSeriesRing(Integers(), 21); Coefficients(R!( (1+2*x)/(1-6*x+x^2))); // Marius A. Burtea, Jan 20 2020
(Sage) [(lucas_number2(2*n+1, 2, -1) + lucas_number1(2*n, 2, -1))/2 for n in (0..30)] # G. C. Greubel, Jan 19 2020
(GAP) a:=[1, 8];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020

Crossrefs

Cf. A000129, A002203, A002315, A038761, A077239, A077240, A077413.

Cf. A077241 (even and odd parts).

Sequence in context: A255720 A014524 A098891 * A034349 A296797 A024108

Adjacent sequences: A054485 A054486 A054487 * A054489 A054490 A054491

Keyword

easy,nonn

Author

Barry E. Williams, May 04 2000

Extensions

More terms from James A. Sellers, May 05 2000

Chebyshev comments from Wolfdieter Lang, Nov 08 2002

Status

approved