A129288 -id:A129288 - cp's OEIS frontend

A157259 Decimal expansion of 7 - 2*sqrt(2).

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

Offset: 1

Klaus Brockhaus, Feb 26 2009

lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {1, 2}, b = A129288.

lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}, b = A157257.

			7 - 2*sqrt(2) = 4.17157287525380990239...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Pierre-Antoine Guihéneuf, Rotations Discrètes, Images des Mathématiques, CNRS, 2018. See 3-2*sqrt(2), the fractional part of this constant, about the loss of information when rotating an image.
Index entries for algebraic numbers, degree 2

Cf. A129288, A157257, A157258 (decimal expansion of 7+2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).

[7 - 2*Sqrt(2)]; // G. C. Greubel, Nov 28 2017

RealDigits[7-2*Sqrt[2],10,120][[1]] (* Harvey P. Dale, May 01 2012 *)

7 - 2*sqrt(2) \\ G. C. Greubel, Nov 28 2017

Equals 3 + Sum_{k>=0} binomial(2*k,k)/((k+1) * 8^k). - Amiram Eldar, Aug 03 2020
Equals 4 + exp(-arccosh(3)). - Amiram Eldar, Jul 06 2023
Equals 4 + (2-sqrt(2))/(2+sqrt(2)). - Davide Rotondo, Jun 08 2024

A129289 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.

Original entry on oeis.org

0, 44, 95, 219, 455, 744, 1460, 2832, 4515, 8687, 16683, 26492, 50808, 97412, 154583, 296307, 567935, 901152, 1727180, 3310344, 5252475, 10066919, 19294275, 30613844, 58674480, 112455452, 178430735, 341980107, 655438583, 1039970712, 1993206308, 3820176192

Offset: 1

Mohamed Bouhamida, May 26 2007

nonn

Also values x of Pythagorean triples (x, x+73, y).
Corresponding values y of solutions (x, y) are in A160041.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (89+36*sqrt(2))/73 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 for n mod 3 = 0.

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

Cf. A160041, A129288, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160042 (decimal expansion of (89+36*sqrt(2))/73), A160043 (decimal expansion of (5907+1802*sqrt(2))/73^2).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)))); // G. C. Greubel, May 07 2018

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+73)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,44,95,219,455,744,1460},70] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

{forstep(n=0, 100000000, [3 ,1], if(issquare(2*n^2+146*n+5329), print1(n, ",")))}

a(n) = 6*a(n-3) -a(n-6) +146 for n > 6; a(1)=0, a(2)=44, a(3)=95, a(4)=219, a(5)=455, a(6)=744.
G.f.: x*(44+51*x+124*x^2-28*x^3-17*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 73*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, May 04 2009

A157258 Decimal expansion of 7 + 2*sqrt(2).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 26 2009

lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {1, 2}, b = A129288.

lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}, b = A157257.

			7 + 2*sqrt(2) = 9.82842712474619009760...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for algebraic numbers, degree 2.

Cf. A129288, A157257, A157259 (decimal expansion of 7-2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).

[7+2*Sqrt(2)]; // G. C. Greubel, Nov 28 2017

RealDigits[7+2Sqrt[2],10,120][[1]]  (* Harvey P. Dale, Mar 20 2011 *)

7+2*sqrt(2) \\ G. C. Greubel, Nov 28 2017

Equals 4 + A156035. - R. J. Mathar, Feb 27 2009

A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

Original entry on oeis.org

0, 115, 136, 411, 1036, 1155, 2740, 6375, 7068, 16303, 37488, 41527, 95352, 218827, 242368, 556083, 1275748, 1412955, 3241420, 7435935, 8235636, 18892711, 43340136, 48001135, 110115120, 252605155, 279771448, 641798283, 1472291068, 1630627827, 3740674852

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+137, y).
Corresponding values y of solutions (x, y) are in A157213.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 0.

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

Cf. A157213, A001652, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))), A129288, A129289, A129298.

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,115,136,411,1036,1155,2740},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 1500000000, [3, 1], if(issquare(2*n^2+274*n+18769), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+274 for n > 6; a(1)=0, a(2)=115, a(3)=136, a(4)=411, a(5)=1036, a(6)=1155.
G.f.: x*(115+21*x+275*x^2-65*x^3-7*x^4-65*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 137*A001652(k) for k >= 0.

Edited and extended by Klaus Brockhaus, Feb 25 2009

A157257 Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.

Original entry on oeis.org

29, 41, 85, 89, 205, 481, 505, 1189, 2801, 2941, 6929, 16325, 17141, 40385, 95149, 99905, 235381, 554569, 582289, 1371901, 3232265, 3393829, 7996025, 18839021, 19780685, 46604249, 109801861, 115290281, 271629469, 639972145, 671961001

Offset: 1

Klaus Brockhaus, Feb 26 2009

(-20, a(1)) and (A129288(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+41)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(7-2*sqrt(2))^2/(7+2*sqrt(2))^2 for n mod 3 = 1.

			(-20, a(1)) = (-20, 29) is a solution: (-20)^2+(-20+41)^2 = 400+441 = 841 = 29^2.
(A129288(1), a(2)) = (0, 41) is a solution: 0^2+(0+41)^2 = 1681 = 41^2.
(A129288(3), a(4)) = (39, 89) is a solution: 39^2+(39+41)^2 = 1521+6400 = 7921 = 89^2.

G. C. Greubel, Table of n, a(n) for n = 1..1001
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A129288, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157258 (decimal expansion of 7+2*sqrt(2)), A157259 (decimal expansion of 7-2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6))) // G. C. Greubel, Feb 04 2018

CoefficientList[Series[(1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6), {x,0,50}], x] (* G. C. Greubel, Feb 04 2018 *)

{forstep(n=-20, 500000000, [3 ,1], if(issquare(n^2+(n+41)^2, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6)) \\ G. C. Greubel, Feb 04 2018

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=29, a(2)=41, a(3)=85, a(4)=89, a(5)=205, a(6)=481.
G.f.: (1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 41*A001653(k) for k >= 1.

A157260 Decimal expansion of (7 + 2sqrt(2))/(7 - 2sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 26 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129288.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A157257.

			(7 +2*sqrt(2))/(7 -2*sqrt(2)) = 2.35604828649869905771...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for algebraic numbers, degree 2.

Cf. A129288, A157257, A157258 (decimal expansion of 7+2*sqrt(2)), A157259 (decimal expansion of 7-2*sqrt(2)).

Cf. A157300 (decimal expansion of (1683+58*sqrt(2))/41^2). - Klaus Brockhaus, May 01 2009

[(7+2*Sqrt(2))/(7-2*Sqrt(2))]; // G. C. Greubel, Nov 28 2017

RealDigits[(7 + 2*Sqrt[2])/(7 - 2*Sqrt[2]), 10, 50][[1]] (* G. C. Greubel, Nov 28 2017 *)

(7+2*sqrt(2))/(7-2*sqrt(2)) \\ G. C. Greubel, Nov 28 2017

Equals (57 + 28*sqrt(2))/41. - Klaus Brockhaus, May 01 2009

A129625 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

Original entry on oeis.org

0, 75, 432, 699, 1092, 3115, 4660, 6943, 18724, 27727, 41032, 109695, 162168, 239715, 639912, 945747, 1397724, 3730243, 5512780, 8147095, 21742012, 32131399, 47485312, 126722295, 187276080, 276765243, 738592224, 1091525547, 1613106612

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+233, y).
Corresponding values y of solutions (x, y) are in A157297.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 0.

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

Cf. A157297, A001652, A129288, A129289, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).

I:=[0,75,432,699,1092,3115,4660]; [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, Mar 29 2018

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,75,432,699,1092,3115,4660}, 50] (* G. C. Greubel, Mar 29 2018 *)

{forstep(n=0, 1700000000, [3, 1], if(issquare(2*n^2+466*n+54289), print1(n, ",")))};

a(n) = 6*a(n-3) -a(n-6) +466 for n > 6; a(1)=0, a(2)=75, a(3)=432, a(4)=699, a(5)=1092, a(6)=3115.
G.f.: x*(75 +357*x +267*x^2 -57*x^3 -119*x^4 -57*x^5)/((1-x)*(1 -6*x^3 +x^6)).
a(3*k+1) = 233*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 11 2009

A129626 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

Original entry on oeis.org

0, 76, 559, 843, 1239, 3976, 5620, 7920, 23859, 33439, 46843, 139740, 195576, 273700, 815143, 1140579, 1595919, 4751680, 6648460, 9302376, 27695499, 38750743, 54218899, 161421876, 225856560, 316011580, 940836319, 1316389179, 1841851143, 5483596600

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+281, y).
Corresponding values y of solutions (x, y) are in A157348.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 0.

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

Cf. A157348, A001652, A129288, A129289, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 76, 559, 843, 1239, 3976, 5620}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 1000000000, [3, 1], if(issquare(2*n^2+562*n+78961), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+562 for n > 6; a(1)=0, a(2)=76, a(3)=559, a(4)=843, a(5)=1239, a(6)=3976.
G.f.: x*(76+483*x+284*x^2-60*x^3-161*x^4-60*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 281*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Apr 12 2009

A157300 Decimal expansion of (1683 + 58*sqrt(2))/41^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 01 2009

Limit_{n -> oo} b(n)/b(n-1) = (1683+58*sqrt(2))/41^2 for n mod 3 = 0, b = A129288.

Limit_{n -> oo} b(n)/b(n-1) = (1683+58*sqrt(2))/41^2 for n mod 3 = 1, b = A157257.

			(1683+58*sqrt(2))/41^2 = 1.04998476300870881191...

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

Cf. A129288, A157257, A002193 (decimal expansion of sqrt(2)), A157260 (decimal expansion of (57+28*sqrt(2))/41).

SetDefaultRealField(RealField(100)); (1683+58*Sqrt(2))/41^2; // G. C. Greubel, Sep 28 2018

evalf((1683+58*sqrt(2))/41^2,120); # Muniru A Asiru, Sep 28 2018

RealDigits[(1683+58Sqrt[2])/41^2,10,120][[1]] (* Harvey P. Dale, May 05 2014 *)

default(realprecision, 100); (1683+58*sqrt(2))/41^2 \\ G. C. Greubel, Sep 28 2018

Equals (58+sqrt(2))/(58-sqrt(2)).

Equals (3+2*sqrt(2))*(7-2*sqrt(2))^2/(7+2*sqrt(2))^2.

A160054 Primes prime(k) such that prime(k)^2 + prime(k+1)^2 - 1 is a perfect square.

Original entry on oeis.org

7, 11, 23, 109, 211, 307, 1021, 4583, 42967, 297779, 1022443, 1459811, 10781809, 125211211, 11673806759, 3019843939831, 40047392632801, 88212019638251209, 444190204424015227, 57852556614292865039, 9801250757169593701501, 64747502900142088755541, 619216322498658374863033

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 01 2009

nonn

An infinite number of solutions exists for a^2 + b^2 - 1 = c^2 over the set of natural numbers a, b, c.
If we constrain these to b=a+2, i.e., 2a^2 + 4a + 3 = c^2, the solutions are with a = 1, 11, 69, 407, 2377, ... (The twin prime 11 is also in this sequence here. The solutions can be generated recursively from a(0)=1, m(0)=3 and a(k+1) = 3*a(k) + 2*m(k) + 2, m(k+1) = 4*a(k) + 3*m(k) + 4.)
Filtering these solutions for prime pairs a(k) and b(k) would generate the subset of lower twin primes in the sequence.
The equivalent procedure can be carried out for other prime gaps 2*d such that prime(k)=a, prime(k+1)=a+2*d, 2*a^2 + 4*a*d + 4*d^2 - 1 = m^2. This decomposes the sequence into classes according to the gap 2*d.
a(17) > 5*10^12. - Donovan Johnson, May 17 2010

			7^2 + 11^2 - 1 = 13^2.
11^2 + 13^2 - 1 = 17^2.
23^2 + 29^2 - 1 = 37^2.
109^2 + 113^2 - 1 = 157^2.
211^2 + 223^2 - 1 = 307^2.
307^2 + 311^2 - 1 = 19^2*23^2.
1021^2 + 1031^2 - 1 = 1451^2.
4583^2 + 4591^2 - 1 = 13^2*499^2.

Cf. A050791, A129288, A271050.

[n: n in [0..2*10^7] | IsSquare(n^2+NextPrime(n+1)^2-1) and IsPrime(n)]; // Vincenzo Librandi, Aug 02 2015

lst = {}; p = q = 2; While[p < 4000000000, q = NextPrime@ p; If[ IntegerQ[ Sqrt[p^2 + q^2 - 1]], AppendTo[lst, p]; Print@ p]; p = q]; lst (* Robert G. Wilson v, May 31 2009 *)

p=2;forprime(q=3,1e6,if(issquare(q^2+p^2-1),print1(p", "));p=q) \\ Charles R Greathouse IV, Nov 06 2014

is(n)=issquare(n^2+nextprime(n+1)^2-1)&&isprime(n) \\ Charles R Greathouse IV, Nov 29 2014

{A000040(k): A069484(k)-1 in A000290}.

Edited and 4 more terms from R. J. Mathar, May 08 2009
a(13) from Robert G. Wilson v, May 31 2009
a(15)-a(16) from Donovan Johnson, May 17 2010
More terms from Jinyuan Wang, Jan 09 2021

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A157259 Decimal expansion of 7 - 2*sqrt(2).

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A129289 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.

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A157258 Decimal expansion of 7 + 2*sqrt(2).

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A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

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A157257 Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.

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A157260 Decimal expansion of (7 + 2*sqrt(2))/(7 - 2*sqrt(2)).

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A129625 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

A157260 Decimal expansion of (7 + 2sqrt(2))/(7 - 2sqrt(2)).