A160205 -id:A160205 - cp's OEIS frontend

A123654 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.

0, 264, 1491, 2427, 3811, 10764, 16180, 24220, 64711, 96271, 143127, 379120, 563064, 836160, 2211627, 3283731, 4875451, 12892260, 19140940, 28418164, 75143551, 111563527, 165635151, 437970664, 650241840, 965394360, 2552682051

Offset: 1

Mohamed Bouhamida, Jun 03 2007

Also values x of Pythagorean triples (x, x+809, y).
Corresponding values y of solutions (x, y) are in A160203.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (873+232*sqrt(2))/809 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (989043+524338*sqrt(2))/809^2 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. A160203, A001652, A115135, A156035 (decimal expansion of 3+2*sqrt(2)), A160204 (decimal expansion of (873+232*sqrt(2))/809), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).

I:=[0,264,1491,2427,3811,10764,16180]; [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}, {0,264,1491,2427,3811,10764,16180}, 50] (* G. C. Greubel, May 04 2018 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481), print1(n, ",")))}

x='x+O('x^30); concat([0], Vec(x*(264+1227*x+936*x^2-200*x^3 -409*x^4 -200*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)+1618 for n > 6; a(1)=0, a(2)=264, a(3)=1491, a(4)=2427, a(5)=3811, a(6)=10764.
G.f.: x*(264+1227*x+936*x^2-200*x^3-409*x^4-200*x^5) / ((1-x)*(1-6*x^3 +x^6)).
a(3*k+1) = 809*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, May 18 2009

A160204 Decimal expansion of (873+232*sqrt(2))/809.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 18 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A123654.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160203.

			(873+232*sqrt(2))/809 = 1.48466940231218547753...

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

Cf. A123654, A160203, A002193 (decimal expansion of sqrt(2)), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).

(873+232*Sqrt(2))/809; // G. C. Greubel, Apr 25 2018

RealDigits[(873+232Sqrt[2])/809,10,120][[1]] (* Harvey P. Dale, Mar 22 2018 *)

(873+232*sqrt(2))/809 \\ G. C. Greubel, Apr 25 2018

Equals (29+4*sqrt(2))/(29-4*sqrt(2)).

A160203 Positive numbers y such that y^2 is of the form x^2+(x+809)^2 with integer x.

Original entry on oeis.org

641, 809, 1105, 2741, 4045, 5989, 15805, 23461, 34829, 92089, 136721, 202985, 536729, 796865, 1183081, 3128285, 4644469, 6895501, 18232981, 27069949, 40189925, 106269601, 157775225, 234244049, 619384625, 919581401, 1365274369

Offset: 1

Klaus Brockhaus, May 18 2009

nonn

(-200, a(1)) and (A123654(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (873+232*sqrt(2))/809 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (989043+524338*sqrt(2))/809^2 for n mod 3 = 1.

			(-200, a(1)) = (-200, 641) is a solution: (-200)^2+(-200+809)^2 = 40000+370881 = 410881 = 641^2.
(A123654(1), a(2)) = (0, 809) is a solution: 0^2+(0+809)^2 = 654481 = 809^2.
(A123654(3), a(4)) = (1491, 2741) is a solution: 1491^2+(1491+809)^2 = 2223081+5290000 = 7513081 = 2741^2.

Cf. A123654, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160204 (decimal expansion of (873+232*sqrt(2))/809), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).

{forstep(n=-200, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=641, a(2)=809, a(3)=1105, a(4)=2741, a(5)=4045, a(6)=5989.
G.f.: (1-x)*(641+1450*x+2555*x^2+1450*x^3+641*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 809*A001653(k) for k >= 1.

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A123654 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.

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A160204 Decimal expansion of (873+232*sqrt(2))/809.

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A160203 Positive numbers y such that y^2 is of the form x^2+(x+809)^2 with integer x.

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