A157469 Positive numbers y such that y^2 is of the form x^2 + (x+97)^2 with integer x.

85, 97, 113, 397, 485, 593, 2297, 2813, 3445, 13385, 16393, 20077, 78013, 95545, 117017, 454693, 556877, 682025, 2650145, 3245717, 3975133, 15446177, 18917425, 23168773, 90026917, 110258833, 135037505, 524715325, 642635573, 787056257

Offset: 1

Author

Klaus Brockhaus, Mar 12 2009

Keywords

• nonn
• easy

Comments

(-13,a(1)) and (A129836(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2 + (x+97)^2 = y^2.

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 = {0, 2}.

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

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

Examples

			(-13, a(1)) = (-13, 85) is a solution: (-13)^2+(-13+97)^2 = 169+7056 = 7225 = 85^2.
(A129836(1), a(2)) = (0, 97) is a solution: 0^2+(0+97)^2 = 9409 = 97^2.
(A129836(3), a(4)) = (228, 397) is a solution: 228^2+(228+97)^2 = 51984+105625 = 157609 = 397^2.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Crossrefs

Cf. A129836, A001653, 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).

Programs

Magma

I:=[85,97,113,397,485,593]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..50]]; // G. C. Greubel, Mar 31 2018

Mathematica

LinearRecurrence[{0,0,6,0,0,-1},{85,97,113,397,485,593},30] (* Harvey P. Dale, Apr 04 2013 *)

PARI

{forstep(n=-20, 800000000, [3, 1], if(issquare(2*n^2+194*n+9409, &k), print1(k, ",")))};

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=85, a(2)=97, a(3)=113, a(4)=397, a(5)=485, a(6)=593.

G.f.: (1-x)*(85 + 182*x + 295*x^2 + 182*x^3 + 85*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 97*A001653(k) for k >= 1.