A129836 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 97)^2 = y^2.

0, 15, 228, 291, 368, 1575, 1940, 2387, 9416, 11543, 14148, 55115, 67512, 82695, 321468, 393723, 482216, 1873887, 2295020, 2810795, 10922048, 13376591, 16382748, 63658595, 77964720, 95485887, 371029716, 454411923, 556532768

Offset

1,2

Comments

Also values x of Pythagorean triples (x, x + 97, y).

Corresponding values y of solutions (x, y) are in A157469.

For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436, the x values are given by the sequence defined by a(n) = 6*a(n-3) - a(n-6) + 2p with a(1)=0, a(2) = 2m + 1, a(3) = 6m^2 - 10m + 4, a(4) = 3p, a(5) = 6m^2 + 10m + 4, a(6) = 40m^2 - 58m + 21 (cf. A118673).

Pairs (p, m) are (7, 2), (17, 3), (31, 4), (71, 6), (97, 7), (127, 8), (199, 10), (241, 11), (337, 13), (449, 15), (577, 17), (647, 18), (881, 21), (967, 22), ...

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

lim_{n -> infinity} a(n)/a(n-1) = (99 + 14*sqrt(2))/97 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (19491 + 12070*sqrt(2))/97^2 for n mod 3 = 0.

For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a prime number in A066436, m>=2, Y values are given by the sequence defined by b(n) = 6*b(n-3) - b(n-6) with b(1) = p, b(2) = 2m^2 + 2m + 1, b(3) = 10m^2 - 14m + 5, b(4) = 5p, b(5) = 10m^2 + 14m + 5, b(6) = 58m^2 - 82m + 29. - Mohamed Bouhamida, Sep 09 2009

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) + 194 for n > 6; a(1)=0, a(2)=15, a(3)=228, a(4)=291, a(5)=368, a(6)=1575.

G.f.: x*(15 + 213*x + 63*x^2 - 13*x^3 - 71*x^4 - 13*x^5)/((1-x)*(1 - 6*x^3 + x^6)).

a(3*k + 1) = 97*A001652(k) for k >= 0.

Mathematica

ClearAll[a]; Evaluate[Array[a, 6]] = {0, 15, 228, 291, 368, 1575}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 194; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Dec 27 2011, after given formula *)
LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 15, 228, 291, 368, 1575, 1940}, 50] (* G. C. Greubel, May 07 2018 *)

Prog

(PARI) forstep(n=0, 600000000, [3, 1], if(issquare(2*n^2+194*n+9409), print1(n, ", ")))
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(15+213*x+63*x^2-13*x^3-71*x^4-13*x^5)/((1-x)*(1-6*x^3 + x^6)))); // G. C. Greubel, May 07 2018

Crossrefs

Cf. A157469, A066436 (primes of the form 2*n^2 - 1), A001652, A118673, A118674, A156035 (decimal expansion of 3 + 2*sqrt(2)), A157470 (decimal expansion of (99 + 14*sqrt(2))/97), A157471 (decimal expansion of (19491 + 12070*sqrt(2))/97^2).

Sequence in context: A067222 A154597 A041422 * A075262 A286722 A250418

Adjacent sequences: A129833 A129834 A129835 * A129837 A129838 A129839

Keyword

nonn,easy

Author

Mohamed Bouhamida, May 21 2007

Extensions

Edited and two terms added by Klaus Brockhaus, Mar 12 2009

Status

approved