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A254758
Part of the positive proper solutions x of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (1, 5).
3
1, 23, 137, 799, 4657, 27143, 158201, 922063, 5374177, 31322999, 182563817, 1064059903, 6201795601, 36146713703, 210678486617, 1227924205999, 7156866749377, 41713276290263, 243122790992201, 1417023469662943
OFFSET
0,2
COMMENTS
The corresponding y solutions are given in A254759.
The other part of the proper (sometimes called primitive) solutions are given in (A254757(n), A220414(n)) for n >= 1.
The improper positive solutions come from 7*(x(n), y(n)) with the positive proper solutions of the Pell equation x^2 - 2*y^2 = -1 given in (A001653(n-1), A002315(n)), for n >= 1.
REFERENCES
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198.
FORMULA
a(n) = rational part of z(n), where z(n) = (1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 0.
G.f.: (1+17*x)/(1-6*x+x^2).
a(n) = 6*a(n-1) - a(n-2), n >= 1, with a(-1) = -17 and a(0) = 1.
a(n) = S(n, 6) + 17*S(n-1, 6), n >= 0, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310).
EXAMPLE
The first pairs of positive solutions of this part of the Pell equation x^2 - 2*y^2 = - 7^2 are: [1, 5], [23, 17], [137, 97], [799, 565], [4657, 3293], [27143, 19193], [158201, 111865], [922063, 651997], [5374177, 3800117], ...
MAPLE
with(orthopoly): a := n -> `if`(n=0, 1, U(n, 3)+17*U(n-1, 3)):
seq(a(n), n=0..19); # Peter Luschny, Feb 07 2015
MATHEMATICA
LinearRecurrence[{6, -1}, {1, 23}, 20] (* Jean-François Alcover, Jun 28 2019 *)
PROG
(PARI) Vec((1+17*x)/(1-6*x+x^2) + O(x^30)) \\ Michel Marcus, Feb 08 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Feb 07 2015
STATUS
approved