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A129288 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 41)^2 = y^2. 12
0, 36, 39, 123, 319, 336, 820, 1960, 2059, 4879, 11523, 12100, 28536, 67260, 70623, 166419, 392119, 411720, 970060, 2285536, 2399779, 5654023, 13321179, 13987036, 32954160, 77641620, 81522519, 192071019, 452528623, 475148160, 1119472036 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also values x of Pythagorean triples (x, x+41, y).

Corresponding values y of solutions (x, y) are in A157257.

lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).

lim_{n -> infinity} a(n)/a(n-1) = (7 + 2*sqrt(2))/(7 - 2*sqrt(2)) for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (3 + 2*sqrt(2))*(7 - 2*sqrt(2))^2/(7 + 2*sqrt(2))^2 for n mod 3 = 0.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Mohammad K. Azarian, Diophantine Pair, Problem B-881, Fibonacci Quarterly, Vol. 37, No. 3, August 1999, pp. 277-278; Solution to Problem B-881, Fibonacci Quarterly, Vol. 38, No. 2, May 2000, pp. 183-184.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

FORMULA

a(n) = 6*a(n-3) - a(n-6) + 82 for n > 6; a(1)=0, a(2)=36, a(3)=39, a(4)=123, a(5)=319, a(6)=336.

G.f.: x*(36 + 3*x + 84*x^2 - 20*x^3 - x^4 - 20*x^5)/((1-x)*(1 - 6*x^3 + x^6)).

a(3*k + 1) = 41*A001652(k) for k >= 0.

MATHEMATICA

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 36, 39, 123, 319, 336, 820}, 40] (* Harvey P. Dale, Jan 18 2015 *)

PROG

(PARI) forstep(n=0, 1200000000, [3 , 1], if(issquare(2*n^2+82*n+1681), print1(n, ", ")))

(MAGMA) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(36+3*x+84*x^2-20*x^3-x^4-20*x^5)/((1-x)*(1-6*x^3+ x^6)))); // G. C. Greubel, May 07 2018

CROSSREFS

Cf. A157257, A001652, A156035 (decimal expansion of 3 + 2*sqrt(2)), A157258 (decimal expansion of 7 + 2*sqrt(2)), A157259 (decimal expansion of 7 - 2*sqrt(2)), A157260 (decimal expansion of (7 + 2*sqrt(2))/(7 - 2*sqrt(2))).

Sequence in context: A261380 A261373 A248372 * A083248 A077090 A254836

Adjacent sequences:  A129285 A129286 A129287 * A129289 A129290 A129291

KEYWORD

nonn,easy

AUTHOR

Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 26 2007

EXTENSIONS

Edited and extended by Klaus Brockhaus, Feb 26 2009

STATUS

approved

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)