

A054488


Expansion of (1+2x)/(16x+x^2).


9



1, 8, 47, 274, 1597, 9308, 54251, 316198, 1842937, 10741424, 62605607, 364892218, 2126747701, 12395593988, 72246816227, 421085303374, 2454265004017, 14304504720728, 83372763320351, 485932075201378, 2832219687887917
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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. 122125, 194196.


LINKS

Table of n, a(n) for n=0..20.
I. Adler, Three Diophantine equations  Part II, Fib. Quart., 7 (1969), pp. 181193.
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231242.
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(n1)a(n2), 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(n1, 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


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[n1]a[n2] 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 *)


CROSSREFS

Cf. A002315 and A038761.
A077241 (even and odd parts).
Sequence in context: A296631 A255720 A014524 * 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



