A157257 -id:A157257 - cp's OEIS frontend

A129288 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 41)^2 = y^2.

0, 36, 39, 123, 319, 336, 820, 1960, 2059, 4879, 11523, 12100, 28536, 67260, 70623, 166419, 392119, 411720, 970060, 2285536, 2399779, 5654023, 13321179, 13987036, 32954160, 77641620, 81522519, 192071019, 452528623, 475148160, 1119472036

Offset: 1

Mohamed Bouhamida, May 26 2007

Also values x of Pythagorean triples (x, x+41, y).
Corresponding values y of solutions (x, y) are in A157257.
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 = {1, 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 = 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. A157257, A001652, 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))).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(36+3*x+84*x^2-20*x^3-x^4-20*x^5)/((1-x)*(1-6*x^3+ x^6)))); // G. C. Greubel, May 07 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,36,39,123,319,336,820},40] (* Harvey P. Dale, Jan 18 2015 *)

forstep(n=0, 1200000000, [3 ,1], if(issquare(2*n^2+82*n+1681), print1(n, ",")))

a(n) = 6*a(n-3) - a(n-6) + 82 for n > 6; a(1)=0, a(2)=36, a(3)=39, a(4)=123, a(5)=319, a(6)=336.
G.f.: x*(36 + 3*x + 84*x^2 - 20*x^3 - x^4 - 20*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 41*A001652(k) for k >= 0.

Edited and extended by Klaus Brockhaus, Feb 26 2009

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

Original entry on oeis.org

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

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

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

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.

cp's OEIS Frontend

A129288 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 41)^2 = y^2.

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

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

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

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

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