A157122 -id:A157122 - cp's OEIS frontend

A090734 Decimal expansion of 4th Madelung constant (negated).

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

Offset: 1

Benoit Cloitre, Jan 18 2004

			-1.83939908404504706624730547956723047642278359481773...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 77.

Cf. A059750, A088537, A085469.

RealDigits[8*(5 - 3*Sqrt[2])*Zeta[1/2]*Zeta[-1/2], 10, 120][[1]] (* Amiram Eldar, May 26 2023 *)

8*(5-3*sqrt(2))*zeta(1/2)*zeta(-1/2) \\ Charles R Greathouse IV, Jun 07 2016

M_4 = -8*(5-3*sqrt(2))*zeta(1/2)*zeta(-1/2) = -8 *(A157122-6) * A059750 * A211113.

A157119 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

Original entry on oeis.org

0, 84, 105, 309, 765, 884, 2060, 4712, 5405, 12257, 27713, 31752, 71688, 161772, 185313, 418077, 943125, 1080332, 2436980, 5497184, 6296885, 14204009, 32040185, 36701184, 82787280, 186744132, 213910425, 482519877, 1088424813, 1246761572

Offset: 1

Klaus Brockhaus, Feb 25 2009

Corresponding values y of solutions (x, y) are in A157120.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(11-3*sqrt(2))^2/(11+3*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. A157120, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

{forstep(n=0, 1300000000, [1, 3], if(issquare(2*n^2+206*n+10609), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+206 for n > 6; a(1) = 0, a(2) = 84, a(3) = 105, a(4) = 309, a(5) = 765, a(6) = 884.
G.f.: x*(84+21*x+204*x^2-48*x^3-7*x^4-48*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 103*A001652(k) for k >= 0.

A157121 Decimal expansion of 11+3*sqrt(2).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Feb 25 2009

lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {1, 2}, b = A157119.

lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}, b = A157120.

			11+3*sqrt(2) = 15.24264068711928514640...

Muniru A Asiru, Table of n, a(n) for n = 2..2000
Index entries for algebraic numbers, degree 2.

Cf. A157119, A157120, A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

evalf[120](11+3*sqrt(2)); # Muniru A Asiru, Feb 12 2019

RealDigits[11+3*Sqrt[2],10,120][[1]] (* Harvey P. Dale, Sep 12 2012 *)

Equals 7+A083729 = 11+A010474. [R. J. Mathar, Feb 27 2009]

A157123 Decimal expansion of (11 + 3sqrt(2))/(11 - 3sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 25 2009

Lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {1, 2}, b = A157119.

Lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}, b = A157120.

			(11 + 3*sqrt(2))/(11 - 3*sqrt(2)) = 2.25570966132644925457...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2.

Cf. A157119, A157120, A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)).

(11+3*Sqrt(2))/(11-3*Sqrt(2)) // G. C. Greubel, Jan 27 2018

RealDigits[(11 + 3*Sqrt[2])/(11 - 3*Sqrt[2]), 10, 100][[1]] (* G. C. Greubel, Jan 27 2018 *)

(11+3*sqrt(2))/(11-3*sqrt(2)) \\ G. C. Greubel, Jan 27 2018

A157120 Positive numbers y such that y^2 is of the form x^2+(x+103)^2 with integer x.

Original entry on oeis.org

73, 103, 205, 233, 515, 1157, 1325, 2987, 6737, 7717, 17407, 39265, 44977, 101455, 228853, 262145, 591323, 1333853, 1527893, 3446483, 7774265, 8905213, 20087575, 45311737, 51903385, 117078967, 264096157, 302515097, 682386227, 1539265205

Offset: 1

Klaus Brockhaus, Feb 25 2009

(-48, a(1)) and (A157119(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

			(-48, a(1)) = (-48, 73) is a solution: (-48)^2+(-48+103)^2 = 2304+3025 = 5329 = 73^2.
(A157119(1), a(2)) = (0, 103) is a solution: 0^2+(0+103)^2 = 10609 = 103^2,
(A157119(3), a(4)) = (105, 233) is a solution: 105^2+(105+103)^2 = 11025+43264 = 54289 = 233^2.

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

Cf. A157119, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

Select[Table[Sqrt[x^2+(x+103)^2],{x,-50,3*10^6}],IntegerQ] (* THe program generates the first 20 terms of the sequence. *) (* or *) LinearRecurrence[ {0,0,6,0,0,-1},{73,103,205,233,515,1157},50](* Harvey P. Dale, Aug 19 2020 *)

{forstep(n=-48, 1100000000, [1, 3], if(issquare(2*n^2+206*n+10609, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1) = 73, a(2) = 103, a(3) = 205, a(4) = 233, a(5) = 515, a(6) = 1157.
G.f.: x*(1-x)*(73+176*x+381*x^2+176*x^3+73*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 103*A001653(k) for k >= 1.
Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))*(11-3*sqrt(2))^2/(11+3*sqrt(2))^2 for n mod 3 = 1.
Limit_{n -> oo} a(n)/a(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}.

Typo corrected by Klaus Brockhaus, Mar 01 2009

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A090734 Decimal expansion of 4th Madelung constant (negated).

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A157119 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

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A157121 Decimal expansion of 11+3*sqrt(2).

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A157123 Decimal expansion of (11 + 3*sqrt(2))/(11 - 3*sqrt(2)).

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A157120 Positive numbers y such that y^2 is of the form x^2+(x+103)^2 with integer x.

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A157123 Decimal expansion of (11 + 3sqrt(2))/(11 - 3sqrt(2)).