A160201 -id:A160201 - cp's OEIS frontend

A122694 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+761)^2 = y^2.

0, 583, 820, 2283, 5440, 6783, 15220, 33579, 41400, 90559, 197556, 243139, 529656, 1153279, 1418956, 3088899, 6723640, 8272119, 18005260, 39190083, 48215280, 104944183, 228418380, 281021083, 611661360, 1331321719, 1637912740

Offset: 1

Mohamed Bouhamida, Jun 03 2007

Also values x of Pythagorean triples (x, x+761, y).
Corresponding values y of solutions (x, y) are in A160200.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2) = A156035.
lim_{n -> infinity} a(n)/a(n-1) = (1003+462*sqrt(2))/761 = A160201 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (591603+85478*sqrt(2))/761^2 = A160202 for n mod 3 = 0.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A160200, A001652, A115135.

I:=[0,583,820,2283,5440,6783,15220]; [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..30]]; // G. C. Greubel, May 04 2018

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 583, 820, 2283, 5440, 6783, 15220}, 27] (* Jean-François Alcover, Nov 13 2017 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1522*n+579121), print1(n, ",")))}

x='x+O('x^30); concat([0], Vec(x*(583 +237*x +1463*x^2 -341*x^3 -79*x^4 -341*x^5)/((1-x)*(1 -6*x^3 +x^6)))) \\ G. C. Greubel, May 04 2018

a(n) = 6*a(n-3) -a(n-6) +1522 for n > 6; a(1)=0, a(2)=583, a(3)=820, a(4)=2283, a(5)=5440, a(6)=6783.
G.f.: x*(583 +237*x +1463*x^2 -341*x^3 -79*x^4 -341*x^5)/((1-x)*(1 -6*x^3 +x^6)).
a(3*k+1) = 761*A001652(k) for k >= 0.

Edited and one term added by Klaus Brockhaus, May 18 2009

A160202 Decimal expansion of (591603+85478*sqrt(2))/761^2.

Original entry on oeis.org

1, 2, 3, 0, 2, 9, 0, 6, 4, 1, 9, 9, 8, 0, 0, 6, 3, 2, 0, 9, 2, 6, 8, 3, 6, 9, 6, 2, 8, 5, 9, 2, 2, 3, 8, 4, 5, 0, 3, 4, 0, 7, 4, 7, 6, 6, 0, 0, 4, 8, 0, 8, 6, 1, 0, 6, 3, 4, 0, 4, 0, 6, 0, 7, 4, 7, 9, 2, 1, 7, 3, 4, 9, 7, 3, 1, 7, 2, 0, 9, 7, 9, 6, 6, 8, 9, 3, 5, 2, 0, 6, 3, 8, 9, 0, 9, 3, 3, 7, 8, 3, 3, 4, 1, 3

Offset: 1

Klaus Brockhaus, May 18 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A122694.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160200.

			(591603+85478*sqrt(2))/761^2 = 1.23029064199800632092...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A122694, A160200, A002193 (decimal expansion of sqrt(2)), A160201 (decimal expansion of (1003+462*sqrt(2))/761).

(591603+85478*Sqrt(2))/761^2; // G. C. Greubel, Apr 25 2018

RealDigits[(591603+85478Sqrt[2])/761^2,10,120][[1]] (* Harvey P. Dale, Oct 25 2011 *)

(591603+85478*sqrt(2))/761^2 \\ G. C. Greubel, Apr 25 2018

Equals (1082 +79*sqrt(2))/(1082 -79*sqrt(2)).

Eausl (3+2*sqrt(2))*(42-11*sqrt(2))^2/(42+11*sqrt(2))^2.

A160200 Positive numbers y such that y^2 is of the form x^2+(x+761)^2 with integer x.

Original entry on oeis.org

541, 761, 1465, 1781, 3805, 8249, 10145, 22069, 48029, 59089, 128609, 279925, 344389, 749585, 1631521, 2007245, 4368901, 9509201, 11699081, 25463821, 55423685, 68187241, 148414025, 323032909, 397424365, 865020329, 1882773769

Offset: 1

Klaus Brockhaus, May 18 2009

nonn

(-341, a(1)) and (A122694(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+761)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (1003+462*sqrt(2))/761 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (591603+85478*sqrt(2))/761^2 for n mod 3 = 1.

			(-341, a(1)) = (-341, 541) is a solution: (-341)^2+(-341+761)^2 = 116281+176400 = 292681 = 541^2.
(A122694(1), a(2)) = (0, 761) is a solution: 0^2+(0+761)^2 = 579121 = 761^2.
(A122694(3), a(4)) = (820, 1781) is a solution: 820^2+(820+761)^2 = 672400+2499561 = 3171961 = 1781^2.

Cf. A122694, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160201 (decimal expansion of (1003+462*sqrt(2))/761), A160202 (decimal expansion of (591603+85478*sqrt(2))/761^2).

{forstep(n=-344, 10000000, [3, 1], if(issquare(2*n^2+1522*n+579121, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=541, a(2)=761, a(3)=1465, a(4)=1781, a(5)=3805, a(6)=8249.
G.f.: (1-x)*(541+1302*x+2767*x^2+1302*x^3+541*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 761*A001653(k) for k >= 1.

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A122694 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+761)^2 = y^2.

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A160202 Decimal expansion of (591603+85478*sqrt(2))/761^2.

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A160200 Positive numbers y such that y^2 is of the form x^2+(x+761)^2 with integer x.

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