A129993 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+199)^2 = y^2.

0, 21, 504, 597, 704, 3441, 3980, 4601, 20540, 23681, 27300, 120197, 138504, 159597, 701040, 807741, 930680, 4086441, 4708340, 5424881, 23818004, 27442697, 31619004, 138821981, 159948240, 184289541, 809114280, 932247141, 1074118640

Offset

1,2

Comments

Also values x of Pythagorean triples (x, x+199, y).

Corresponding values y of solutions (x, y) are in A159548.

For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.

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

Lim_{n -> infinity} a(n)/a(n-1) = (201+20*sqrt(2))/199 for n mod 3 = {1, 2}.

Lim_{n -> infinity} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 0.

Links

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

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) + 398 for n > 6; a(1)=0, a(2)=21, a(3)=504, a(4)=597, a(5)=704, a(6)=3441.

G.f.: x*(21+483*x+93*x^2-19*x^3-161*x^4-19*x^5) / ((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 199*A001652(k) for k >= 0.

a(1)=0, a(2)=21, a(3)=504, a(4)=597, a(5)=704, a(6)=3441, a(7)=3980, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Jun 03 2012

Mathematica

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 21, 504, 597, 704, 3441, 3980}, 30] (* Harvey P. Dale, Jun 03 2012 *)

Prog

(PARI) {forstep(n=0, 500000000, [1, 3], if(issquare(2*n^2+398*n+39601), print1(n, ", ")))};
(Magma) I:=[0, 21, 504, 597, 704, 3441, 3980]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // G. C. Greubel, Mar 31 2018

Crossrefs

Cf. A159548, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

Sequence in context: A006299 A231542 A065921 * A359852 A209348 A095655

Adjacent sequences: A129990 A129991 A129992 * A129994 A129995 A129996

Keyword

nonn,easy

Author

Mohamed Bouhamida, Jun 14 2007

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 14 2009

Status

approved