A161487 -id:A161487 - cp's OEIS frontend

A161486 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+191)^2 = y^2.

0, 69, 336, 573, 936, 2449, 3820, 5929, 14740, 22729, 35020, 86373, 132936, 204573, 503880, 775269, 1192800, 2937289, 4519060, 6952609, 17120236, 26339473, 40523236, 99784509, 153518160, 236187189, 581587200, 894769869, 1376600280

Offset: 1

Klaus Brockhaus, Jun 13 2009

nonn

Corresponding values y of solutions (x, y) are in A161487.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (209+60*sqrt(2))/191 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (52323+26522*sqrt(2))/191^2 for n mod 3 = 0.

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

Cf. A161487, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161488 (decimal expansion of (209+60*sqrt(2))/191), A161489 (decimal expansion of (52323+26522*sqrt(2))/191^2).

Transpose[NestList[Flatten[{Rest[#],6#[[4]]-First[#]+382}]&,{0,69, 336, 573, 936,2449},40]][[1]]  (* Harvey P. Dale, Apr 01 2011 *)
LinearRecurrence[{1,0,6,-6,0,-1,1},{0,69,336,573,936,2449,3820},40] (* Harvey P. Dale, Mar 29 2016 *)

{forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+382*n+36481), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+382 for n > 6; a(1)=0, a(2)=69, a(3)=336, a(4)=573, a(5)=936, a(6)=2449.
G.f.: x*(69+267*x+237*x^2-51*x^3-89*x^4-51*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 191*A001652(k) for k >= 0.

A161488 Decimal expansion of (209+60*sqrt(2))/191.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Jun 13 2009

lim_{n -> infinity} b(n)/b(n-1) = (209+60*sqrt(2))/191 for n mod 3 = {1, 2}, b = A161486.

lim_{n -> infinity} b(n)/b(n-1) = (209+60*sqrt(2))/191 for n mod 3 = {0, 2}, b = A161487.

			(209+60*sqrt(2))/191 = 1.53849640702819739752...

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

Cf. A161486, A161487, A002193 (decimal expansion of sqrt(2)), A161489 (decimal expansion of (52323+26522*sqrt(2))/191^2).

(209 + 60*Sqrt(2))/191; // G. C. Greubel, Apr 06 2018

RealDigits[(209+60*Sqrt[2])/191, 10, 100][[1]] (* G. C. Greubel, Apr 06 2018 *)

(209+60*sqrt(2))/191 \\ G. C. Greubel, Apr 06 2018

Equals (20+3*sqrt(2))/(20-3*sqrt(2)).

A161489 Decimal expansion of (52323 + 26522*sqrt(2))/191^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Jun 13 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A161486.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A161487.

			(52323 + 26522*sqrt(2))/191^2 = 2.46239884052682840065...

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

Cf. A161486, A161487, A002193 (decimal expansion of sqrt(2)), A161488 (decimal expansion of (209+60*sqrt(2))/191).

(52323 + 26522*Sqrt(2))/191^2; // G. C. Greubel, Apr 06 2018

RealDigits[(52323+26522*Sqrt[2])/191^2, 10, 100][[1]] (* G. C. Greubel, Apr 06 2018 *)

(52323 + 26522*sqrt(2))/191^2 \\ G. C. Greubel, Apr 06 2018

Equals (298 + 89*sqrt(2))/(298 - 89*sqrt(2)).

Equals (3 + 2*sqrt(2))*(20 - 3*sqrt(2))^2/(20 + 3*sqrt(2))^2.

cp's OEIS Frontend

A161486 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+191)^2 = y^2.

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A161488 Decimal expansion of (209+60*sqrt(2))/191.

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A161489 Decimal expansion of (52323 + 26522*sqrt(2))/191^2.

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