A118674 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2.

0, 9, 60, 93, 140, 429, 620, 893, 2576, 3689, 5280, 15089, 21576, 30849, 88020, 125829, 179876, 513093, 733460, 1048469, 2990600, 4274993, 6111000, 17430569, 24916560, 35617593, 101592876, 145224429, 207594620, 592126749, 846430076

Offset: 1

Author

Mohamed Bouhamida, May 19 2006

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+31, y).

Corresponding values y of solutions (x, y) are in A157646.

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) = (33 + 8*sqrt(2))/31 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1539 + 850*sqrt(2))/31^2 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).

Crossrefs

cf. A157646, A066436 (primes of the form 2*n^2-1), A118673, A129836, A001652, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3 + 2*sqrt(2)), A157647 (decimal expansion of (33 + 8*sqrt(2))/31), A157648 (decimal expansion of (1539 + 850*sqrt(2))/31^2).

Programs

Magma

I:=[0,9,60,93,140,429,620]; [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

Mathematica

ClearAll[a]; Evaluate[Array[a, 6]] = {0, 9, 60, 93, 140, 429}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 62; Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Dec 27 2011, after given formula *)
LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,9,60,93,140,429,620}, 50] (* G. C. Greubel, Mar 31 2018 *)

PARI

{forstep(n=0, 850000000, [1, 3], if(issquare(2*n^2+62*n+961), print1(n, ",")))};

Formula

a(n) = 6*a(n-3) - a(n-6) + 62 for n > 6; a(1)=0, a(2)=9, a(3)=60, a(4)=93, a(5)=140, a(6)=429.

G.f.: x*(9 + 51*x + 33*x^2 - 7*x^3 - 17*x^4 - 7*x^5)/((1-x)*(1 - 6*x^3 + x^6)).

a(3*k + 1) = 31*A001652(k) for k >= 0.

Extensions

Edited by Klaus Brockhaus, Mar 11 2009