A159894 -id:A159894 - cp's OEIS frontend

A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

0, 56, 1925, 2181, 2465, 13056, 14540, 16188, 77865, 86513, 96117, 455588, 505992, 561968, 2657117, 2950893, 3277145, 15488568, 17200820, 19102356, 90275745, 100255481, 111338445, 526167356, 584333520, 648929768, 3066729845

Offset: 1

Mohamed Bouhamida, Jun 20 2007

Also values x of Pythagorean triples (x, x+727, y).
Corresponding values y of solutions (x, y) are in A159893.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (731+54*sqrt(2))/727 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159893, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,56,1925,2181,2465,13056,14540},40] (* or *) RecurrenceTable[{a[1]==0,a[2]==56,a[3]==1925,a[4]==2181,a[5] == 2465, a[6] == 13056, a[n] ==6a[n-3]-a[n-6]+1454},a,{n,40}] (* Harvey P. Dale, Jan 16 2013 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1454*n+528529), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1454 for n > 6; a(1)=0, a(2)=56, a(3)=1925, a(4)=2181, a(5)=2465, a(6)=13056.
G.f.: x*(56+1869*x+256*x^2-52*x^3-623*x^4-52*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 727*A001652(k) for k >= 0.

Edited and one term added by Klaus Brockhaus, Apr 30 2009

A159893 Positive numbers y such that y^2 is of the form x^2+(x+727)^2 with integer x.

Original entry on oeis.org

677, 727, 785, 3277, 3635, 4033, 18985, 21083, 23413, 110633, 122863, 136445, 644813, 716095, 795257, 3758245, 4173707, 4635097, 21904657, 24326147, 27015325, 127669697, 141783175, 157456853, 744113525, 826372903, 917725793

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-52, a(1)) and (A130646(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (731+54*sqrt(2))/727 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 1.
For the generic case x^2+(x+p)^2=y^2 with p= m^2 -2 a prime number in A028871, m>=5, the x values are given by the sequence defined by: a(n)= 6*a(n-3) -a(n-6) +2*p with a(1)= 0, a(2)= 2*m +2, a(3)= 3*m^2 -10*m +8, a(4)= 3*p, a(5)= 3*m^2 +10*m +8, a(6)= 20*m^2 -58*m +42. Y values are given by the sequence defined by: b(n)= 6*b(n-3) -b(n-6) with b(1)= p, b(2)= m^2 +2*m +2, b(3)= 5*m^2 -14*m +10, b(4)= 5*p, b(5)= 5*m^2 +14*m +10, b(6)= 29*m^2 -82*m +58. - Mohamed Bouhamida, Sep 09 2009

			(-52, a(1)) = (-52, 677) is a solution: (-52)^2+(-52+727)^2 = 2704+455625 = 458329 = 677^2.
(A130646(1), a(2)) = (0, 727) is a solution: 0^2+(0+727)^2 = 528529 = 727^2.
(A130646(3), a(4)) = (1925, 3277) is a solution: 1925^2+(1925+727)^2 = 3705625+7033104 = 10738729 = 3277^2.

G. C. Greubel, Table of n, a(n) for n = 1..3895
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A130646, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).

I:=[677,727,785,3277,3635,4033]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 17 2018

LinearRecurrence[{0,0,6,0,0,-1}, {677,727,785,3277,3635,4033}, 50] (* G. C. Greubel, May 17 2018 *)

{forstep(n=-52, 10000000, [1, 3], if(issquare(2*n^2+1454*n+528529, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=677, a(2)=727, a(3)=785, a(4)=3277, a(5)=3635, a(6)=4033.
G.f.: (1-x)*(677+1404*x+2189*x^2+1404*x^3+677*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 727*A001653(k) for k >= 1.

A159895 Decimal expansion of (1304787+843542*sqrt(2))/727^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2009

lim_{n -> infinity} b(n)/b(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 0, b = A130646.

lim_{n -> infinity} b(n)/b(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 1, b = A159893.

			(1304787+843542*sqrt(2))/727^2 = 4.72582495346769116482...

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

Cf. A130646, A159893, A002193 (decimal expansion of sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727).

(1304787 +843542*Sqrt(2))/727^2; // G. C. Greubel, May 17 2018

RealDigits[(1304787+843542*Sqrt[2])/727^2, 10, 100][[1]] (* G. C. Greubel, May 17 2018 *)

(1304787 +843542*sqrt(2))/727^2 \\ G. C. Greubel, May 17 2018

Equals (1354 +623*sqrt(2))/(1354 -623*sqrt(2)).

Equals (3 +2*sqrt(2))*(27 -sqrt(2))^2/(27 +sqrt(2))^2.

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A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

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A159893 Positive numbers y such that y^2 is of the form x^2+(x+727)^2 with integer x.

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A159895 Decimal expansion of (1304787+843542*sqrt(2))/727^2.

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