A010465 -id:A010465 - cp's OEIS frontend

A154246 a(n) = ( (5 + sqrt(7))^n - (5 - sqrt(7))^n )/(2*sqrt(7)).

1, 10, 82, 640, 4924, 37720, 288568, 2206720, 16872976, 129008800, 986374432, 7541585920, 57661119424, 440862647680, 3370726327168, 25771735613440, 197044282245376, 1506551581411840, 11518718733701632, 88069258871603200

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

Second binomial transform of A086901 without initial term 1.

Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(7) = 7.6457513110....

G. C. Greubel, Table of n, a(n) for n = 1..750
Index entries for linear recurrences with constant coefficients, signature (10,-18).

Cf. A010465 (decimal expansion of square root of 7), A086901.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[((5+r)^n-(5-r)^n)/(2*r): n in [1..25]]; [Integers()!S[j]: j in [1..#S]]; // Klaus Brockhaus, Jan 07 2009

I:=[1,10]; [n le 2 select I[n] else 10*Self(n-1)-18*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Sep 08 2016

Table[Simplify[((5+Sqrt[7])^n -(5-Sqrt[7])^n)/(2*Sqrt[7])], {n,1,25}] (* or *) LinearRecurrence[{10, -18}, {1, 10}, 25] (* G. C. Greubel, Sep 07 2016 *)

my(x='x+O('x^25)); Vec(x/(1-10*x+18*x^2)) \\ G. C. Greubel, May 31 2019

[lucas_number1(n,10,18) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 10*a(n-1) - 18*a(n-2) for n > 1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 10*x + 18*x^2). (End)
E.g.f.: (1/sqrt(7))*exp(5*x)*sinh(sqrt(7)*x). - G. C. Greubel, Sep 07 2016

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

A154247 a(n) = ( (6 + sqrt(7))^n - (6 - sqrt(7))^n )/(2*sqrt(7)).

Original entry on oeis.org

1, 12, 115, 1032, 9049, 78660, 681499, 5896848, 50998705, 440975868, 3812747971, 32964675480, 285006414601, 2464101386292, 21304030612075, 184189427142432, 1592456237959009, 13767981468377580, 119034546719719699

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

nonn

Lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(7) = 8.6457513110....

G. C. Greubel, Table of n, a(n) for n = 1..750
Index entries for linear recurrences with constant coefficients, signature (12, -29).

Cf. A010465 (decimal expansion of square root of 7).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

I:=[1,12]; [n le 2 select I[n] else 12*Self(n-1)-29*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 08 2016

Join[{a=1,b=12},Table[c=12*b-29*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
With[{c=Sqrt[7]},Simplify/@Table[((6+c)^n-(6-c)^n)/(2c),{n,20}]] (* or *) LinearRecurrence[{12,-29},{1,12},20] (* Harvey P. Dale, Mar 02 2012 *)

[lucas_number1(n,12,29) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 12*a(n-1) - 29*a(n-2) for n > 1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 12*x + 29*x^2). (End)
E.g.f.: sinh(sqrt(7)*x)*exp(6*x)/sqrt(7). - Ilya Gutkovskiy, Sep 08 2016

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

A154249 a(n) = ( (8 + sqrt(7))^n - (8 - sqrt(7))^n )/(2*sqrt(7)).

Original entry on oeis.org

1, 16, 199, 2272, 25009, 270640, 2904727, 31049152, 331216993, 3529670224, 37595354983, 400334476960, 4262416397329, 45379597170544, 483115820080951, 5143216082574208, 54753855576573121, 582898372518440080

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = 8 + sqrt(7) = 10.6457513110....

G. C. Greubel, Table of n, a(n) for n = 1..750
Index entries for linear recurrences with constant coefficients, signature (16, -57).

Cf. A010465 (decimal expansion of square root of 7).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((8+r)^n-(8-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

seq(expand((8+sqrt(7))^n-(8-sqrt(7))^n)/sqrt(28), n = 1 .. 20); # Emeric Deutsch, Jan 08 2009

Join[{a=1,b=16},Table[c=16*b-57*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
LinearRecurrence[{16,-57},{1,16},25] (* or *) Table[( (8 + sqrt(7))^n - (8 - sqrt(7))^n )/(2*sqrt(7)), {n,1,25}] (* G. C. Greubel, Sep 08 2016 *)

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 16*a(n-1)-57*a(n-2) for n>1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 16*x + 57*x^2). (End)
E.g.f.: (1/sqrt(7))*exp(8*x)*sinh(sqrt(7)*x). - G. C. Greubel, Sep 08 2016

Extended by Emeric Deutsch and Klaus Brockhaus, Jan 08 2009

Edited by Klaus Brockhaus, Oct 06 2009

A154250 a(n) = ( (9 + sqrt(7))^n - (9 - sqrt(7))^n )/(2*sqrt(7)).

Original entry on oeis.org

1, 18, 250, 3168, 38524, 459000, 5411224, 63436032, 741418000, 8651257632, 100857705376, 1175245632000, 13690951178176, 159468944439168, 1857310612720000, 21630889140461568, 251915019187028224

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

nonn

lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(7) = 11.6457513110....

Harvey P. Dale, Table of n, a(n) for n = 1..938
Index entries for linear recurrences with constant coefficients, signature (18,-74).

Cf. A010465 (decimal expansion of square root of 7).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

Join[{a=1,b=18},Table[c=18*b-74*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011*)
LinearRecurrence[{18,-74},{1,18},20] (* Harvey P. Dale, Feb 16 2014 *)

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 18*a(n-1) - 74*a(n-2) for n>1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 18*x + 74*x^2). (End)

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 06 2009

A171545 Decimal expansion of sqrt(2/7).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 3/2 ; 0 -3/2 | 7/2 -3/2>.

			sqrt(2/7) = sqrt(14)/7 = 0.53452248382484876936910696175950...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Table of Clebsch-Gordan coefficients

SetDefaultRealField(RealField(100)); Sqrt(2/7); // G. C. Greubel, Oct 02 2018

RealDigits[N[Sqrt[2/7],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

default(realprecision, 100); sqrt(2/7) \\ G. C. Greubel, Oct 02 2018

Equals A002193/A010465 = 2/A010471 = A010467/A010490.

A171547 Decimal expansion of sqrt(3/14).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 2 ; -2 0 | 4 -2>.

			sqrt(3/14) = sqrt(42)/14 = 0.462910049886275730783283388291999761264...

Daniel Starodubtsev, Table of n, a(n) for n = 0..9998
Wikipedia, Table of Clebsch-Gordan coefficients

Cf. A002194, A010471, A115754, A010465, A171537, A002193.

RealDigits[Sqrt[3/14],10,120][[1]] (* Harvey P. Dale, Sep 23 2011 *)

Equals A002194/A010471 = A115754/A010465 = A171537/A002193.

A171548 Decimal expansion of 2*sqrt(2/35).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient < j1 j2; m1 m2 | J M > = < 2 2; -1 1 | 4 0 >.

			Sqrt(8/35) = 2*sqrt(70)/35 = 0.478091443733757455987526871877249993..

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Table of Clebsch-Gordan coefficients.

Cf. A010465, A010466, A010490, A171543.

SetDefaultRealField(RealField(100)); 2*Sqrt(2/35); // G. C. Greubel, Oct 02 2018

RealDigits[2*Sqrt[2/35], 10, 100][[1]] (* G. C. Greubel, Oct 02 2018 *)

2*sqrt(2/35) \\ Michel Marcus, Dec 23 2015

Equals A010466/A010490 = A171543*A010465.

A176054 Decimal expansion of (7+3*sqrt(7))/7.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (7+3*sqrt(7))/7 is A010697.

			(7+3*sqrt(7))/7 = 2.13389341902768168164...

Cf. A010465 (decimal expansion of sqrt(7)), A010697 (repeat 2, 7).

A176434 Decimal expansion of (7+3*sqrt(7))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (7+3*sqrt(7))/2 is A010697 preceded by 7.

			(7+3*sqrt(7))/2 = 7.46862696659688588575...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010465 (decimal expansion of sqrt(7)), A010697 (repeat 2, 7).

A177016 Decimal expansion of sqrt(16926).

Original entry on oeis.org

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

Offset: 3

Klaus Brockhaus, May 01 2010

Continued fraction expansion of sqrt(16926) is 130 followed by (repeat 10, 260).

sqrt(16926) = sqrt(2)*sqrt(3)*sqrt(7)*sqrt(13)*sqrt(31).

			sqrt(16926) = 130.09996156801891999550...

Cf. A002193 (decimal expansion of sqrt(2)), A002194 (decimal expansion of sqrt(3)), A010465 (decimal expansion of sqrt(7)), A010470 (decimal expansion of sqrt(13)), A010486 (decimal expansion of sqrt(31)), A177015 (decimal expansion of (124+sqrt(16926))/25).

cp's OEIS Frontend

A154246 a(n) = ( (5 + sqrt(7))^n - (5 - sqrt(7))^n )/(2*sqrt(7)).

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A154247 a(n) = ( (6 + sqrt(7))^n - (6 - sqrt(7))^n )/(2*sqrt(7)).

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A154249 a(n) = ( (8 + sqrt(7))^n - (8 - sqrt(7))^n )/(2*sqrt(7)).

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A154250 a(n) = ( (9 + sqrt(7))^n - (9 - sqrt(7))^n )/(2*sqrt(7)).

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A171545 Decimal expansion of sqrt(2/7).

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A171547 Decimal expansion of sqrt(3/14).

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A171548 Decimal expansion of 2*sqrt(2/35).

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