Seiichi Kirikami - cp's OEIS frontend

A209084 a(n) = 2a(n-1) + 4a(n-2) with n>1, a(0)=0, a(1)=4.

0, 4, 8, 32, 96, 320, 1024, 3328, 10752, 34816, 112640, 364544, 1179648, 3817472, 12353536, 39976960, 129368064, 418643968, 1354760192, 4384096256, 14187233280, 45910851584, 148570636288, 480784678912, 1555851902976, 5034842521600, 16293092655104

Offset: 0

Seiichi Kirikami, Mar 06 2012

a(n)/A063727(n) are convergents for A134972.

Abs(Sum_{i=0..n} C(n,n-i)*a(i)-(sqrt(5)-1)* A033887(n))->0. - Seiichi Kirikami, Jan 20 2016

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

Bruno Berselli, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (2,4).

Cf. A063727, A033887, A085449, A103435, A134972, A269991.

Cf. A086344 (this sequence with signs).

I:=[0,4]; [n le 2 select I[n] else 2*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 16 2016

RecurrenceTable[{a[n]==2*a[n-1]+4*a[n-2], a[0]==0, a[1]==4}, a, {n, 30}]
LinearRecurrence[{2, 4}, {0, 4}, 40] (* Vincenzo Librandi, Jan 16 2016 *)

concat(0, Vec(4*x/(1-2*x-4*x^2) + O(x^40))) \\ Michel Marcus, Jan 16 2016

a(n) = (2/sqrt(5))*((1+sqrt(5))^n-(1-sqrt(5))^n).
G.f.: 4*x/(1-2*x-4*x^2). - Bruno Berselli, Mar 08 2012
a(n) = 4*A085449(n) = 2*A103435(n). - Bruno Berselli, Mar 08 2012
Sum_{n>=1} 1/a(n) = (1/4) * A269991. - Amiram Eldar, Feb 01 2021

A206532 a(n) = (2(n+1)(2n+1)-1) * a(n-1) + 2n(2n-1) * a(n-2), a(0) = 1, a(1) = 11.

Original entry on oeis.org

1, 11, 331, 18535, 1668151, 220195931, 40075659443, 9618158266319, 2943156429493615, 1118399443207573699, 516700542761899048939, 285218699604568275014327, 185392154742969378759312551, 140156468985684850342040288555

Offset: 0

Seiichi Kirikami, Feb 11 2012

nonn

The denominators of the fractions limiting to the value of A206533.

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc.,1966.

Robert Israel, Table of n, a(n) for n = 0..223

Cf. A002939, A082108, A206531, A206533.

f:= gfun:-rectoproc({a(n) = (2*(n+1)*(2*n+1)-1) * a(n-1) + 2*n*(2*n-1) * a(n-2), a(0) = 1, a(1) = 11},a(n),remember):
map(f, [$0..40]); # Robert Israel, Sep 16 2018

RecurrenceTable[{a[n]==(2(n+1)(2n+1)-1)a[n-1]+2n(2n-1)a[n-2],a[0]==1,a[1]==11},a,{n,15}]

a(n) = A082108*a(n-1) + A002939*a(n-2), a(0) = 1, a(1) = 11.

a(n) = -4*n*(-1)^n*(n+1)*LommelS1(2*n+1/2, 3/2, 1)-2*(-1)^n*(n+1)*LommelS1(2*n+3/2, 1/2, 1)+(1-cos(1))*(2*n+2)!+(-1)^n. - Robert Israel, Sep 16 2018

A206531 a(n) = (2(n+1)(2n+1)-1)a(n-1) + 2n(2n-1)a(n-2), a(0)=0, a(1)=2.

Original entry on oeis.org

0, 2, 58, 3250, 292498, 38609738, 7026972314, 1686473355362, 516060846740770, 196103121761492602, 90599642253809582122, 50011002524102889331346, 32507151640666878065374898, 24575406640344159817423422890

Offset: 0

Seiichi Kirikami, Feb 11 2012

nonn

The numerators of the fractions limiting to the value of A206533.

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

G. C. Greubel, Table of n, a(n) for n = 0..220

Cf. A002939, A082108, A206532, A206533.

[n le 2 select 2*(n-1) else (2*n*(2*n-1)-1)*Self(n-1) + 2*(n-1)*(2*n-3)*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 21 2022

RecurrenceTable[{a[n]==(2(n+1)(2n+1)-1)a[n-1]+2n(2n-1)a[n-2],a[0]==0,a[1]==2},a,{n,15}]

@CachedFunction
def a(n): # a = A206531
    if (n<2): return 2*n
    else: return (2*(n+1)*(2*n+1)-1)*a(n-1) + 2*n*(2*n-1)*a(n-2)
[a(n) for n in range(31)] # G. C. Greubel, Dec 21 2022

a(n) = A082108(n)*a(n-1) + A002939(n)*a(n-2), a(0) = 0, a(1) = 2.

A206308 a(n) = ((2n+2)(2n+3) - 1)a(n-1) + 2n(2n+1)a(n-2), a(0)=1, a(1)=19.

Original entry on oeis.org

1, 19, 799, 57527, 6327971, 987163475, 207304329751, 56386777692271, 19284277970756683, 8099396747717806859, 4098294754345210270655, 2458976852607126162392999, 1726201750530202565999885299, 1401675821430524483591906862787

Offset: 0

Seiichi Kirikami, Feb 11 2012

The denominators of the fractions limiting to the value of A206530.

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

G. C. Greubel, Table of n, a(n) for n = 0..220

Cf. A002943, A125202, A206307, A206530.

[n le 2 select 19^(n-1) else (4*n^2+2*n-1)*Self(n-1) + 2*(n-1)*(2*n-1)*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 21 2022

RecurrenceTable[{a[n]==((2n+3)(2n+2)-1)a[n-1]+2n(2n+1)a[n-2], a[0]==1, a[1]==19}, a, {n,15}]

@CachedFunction # a = A206308
def a(n): return 19^n if (n<2) else (4*n^2+10*n+5)*a(n-1) + 2*n*(2*n+1)*a(n-2)
[a(n) for n in range(31)] # G. C. Greubel, Dec 21 2022

a(n) = A125202(n+2)*a(n-1) + A002943(n)*a(n-2), with a(0) = 1, a(1) = 19.

A206307 a(n) = ((2n+2)(2n+3) - 1)a(n-1) + 2n(2n+1)a(n-2), a(0)=0, a(1)=6.

Original entry on oeis.org

0, 6, 246, 17718, 1948974, 304039950, 63848389494, 17366761942374, 5939432584291902, 2494561685402598846, 1262248212813715016070, 757348927688229009642006, 531658947237136764768688206, 431707065156555052992174823278

Offset: 0

Seiichi Kirikami, Feb 11 2012

nonn

The numerators of the fractions limiting to the value of A206530.

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

G. C. Greubel, Table of n, a(n) for n = 0..220

Cf. A002943, A125202, A206308, A206530.

[n le 2 select 6*(n-1) else (4*n^2+2*n-1)*Self(n-1) + 2*(n-1)*(2*n-1)*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 20 2022

RecurrenceTable[{a[n]==((2n+3)(2n+2)-1)a[n-1]+2n(2n+1)a[n-2],a[0]==0,a[1]==6},a,{n,15}]

@CachedFunction
def a(n): return 6*n if (n<2) else (4*n^2+10*n+5)*a(n-1) + 2*n*(2*n+1)*a(n-2)
[a(n) for n in range(31)] # G. C. Greubel, Dec 20 2022

a(n) = A125202(n+2)*a(n-1) + A002943(n)*a(n-2), a(0) = 0, a(1) = 6.

A206530 Decimal expansion of 1/(1-sin(1)).

Original entry on oeis.org

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

Offset: 1

Seiichi Kirikami, Feb 11 2012

The value of the limit of (A206307+6*A206308) / (A206308).

			6.3079935164437400275135217398...

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002943, A049469, A125202, A206307, A206308.

SetDefaultRealField(RealField(150)); 1/(1-Sin(1)); // G. C. Greubel, Dec 20 2022

Mathematica
```
RealDigits[N[1/(1-Sin[1]), 150]][[1]]
```

numerical_approx(1/(1-sin(1)), digits=150) # G. C. Greubel, Dec 20 2022

Equals 1/(1-A049469).
A206307/A206308 + 6 -> 1/(1-A049469).
Abs(A206308/(1-sin(1)) - (A206307 + 6*A206308)) -> 0.

A206533 Decimal expansion of 1/(1-cos(1)).

Original entry on oeis.org

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

Offset: 1

Seiichi Kirikami, Feb 11 2012

The value of the fractional limit of the numerators(A206531+2*A206532) and the denominators(A206532).

Abs(A206532/(1-cos(1)) - (A206531+2*A206532)) -> 0.

			2.17534264967002141077678675965606997584844...

E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

Cf. A002939, A082108, A206531, A206532, A049470, A371936.

Mathematica
```
RealDigits[N[1/(1-Cos[1]), 150]][[1]]
```

1/(1 - cos(1)) \\ Stefano Spezia, Apr 21 2025

Equals 1/(1-A049470).
A206531/A206532+2 -> 1/(1-A049470).
Equals 1/A371936. - Hugo Pfoertner, Apr 21 2025

Incorrect a(86)=9 removed by Georg Fischer, Apr 04 2020

A196878 Decimal expansion of (Pi/8)(6zeta(3)+Pi^2log(2)+4log(2)^3).

Original entry on oeis.org

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

Offset: 1

Seiichi Kirikami, Oct 07 2011

The absolute value of the integral {x=0..Pi/2} log(sin(x))^3 dx. The absolute value of m=3 of sqrt(Pi)/2*(d^m/da^m(gamma((a+1)/2)/gamma(a/2+1))) at a=0. - Seiichi Kirikami and Peter J. C. Moses, Oct 07 2011

			6.041882909775093522150424130675995...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.621.1

G. C. Greubel, Table of n, a(n) for n = 1..5000
K. S. Kolbig, On the integral int_0^Pi/2 log^n cos x log^p sin x dx, Math. Comp. 40 (162) (1983) 565-570, r_{3,0}

Cf. A173623, A196877.

Pi/8*(6*Zeta(3)+Pi^2*log(2)+4*log(2)^3) ; evalf(%) ; # R. J. Mathar, Oct 08 2011

RealDigits[N[Pi/8 (6 Zeta[3] + Pi^2 Log[2] + 4 Log[2]^3), 150]][[1]]
Sqrt[Pi]/2*Derivative[3][Gamma[(#+1)/2]/Gamma[#/2+1]&][0] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 25 2013 *)

Pi/8*(6*zeta(3)+Pi^2*log(2)+4*log(2)^3) \\ G. C. Greubel, Feb 12 2017

Equals A019675*(6*A002117 + A002388*A002162 + 4*A002162^3).

A196877 Decimal expansion of Pi/2*(Pi^2/12 + (log(2))^2).

Original entry on oeis.org

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

Offset: 1

Seiichi Kirikami, Oct 07 2011

The value of the integral_{x=0..Pi/2} log(sin(x))^2 dx. The value of sqrt(Pi)/2*(d^2/da^2(gamma((a+1)/2)/gamma(a/2+1))) at a=0. - Seiichi Kirikami and Peter J. C. Moses, Oct 07 2011

			2.04662202447274064616964100817...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.621.1

G. C. Greubel, Table of n, a(n) for n = 1..10000
K. S. Kolbig, On the integral int_0^Pi/2 log^n cos x log^p sin x dx, Math. Comp. 40 (162) (1983) 565-570, r_{2,0}

Cf. A002162, A019669, A072691, A173623, A196878.

RealDigits[N[Pi/2 (Pi^2/12 + Log[2]^2),150]][[1]]

Pi/2*(Pi^2/12+(log(2))^2) \\ Michel Marcus, Jan 13 2015

Equals A019669*(A072691 + A002162^2).

Equals Integral_{x=0..1} log(x)^2/sqrt(1-x^2) dx. - Amiram Eldar, May 27 2023

A193712 Decimal expansion of Pi*zeta(3)/4.

Original entry on oeis.org

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

Offset: 0

Seiichi Kirikami, Aug 31 2011

The absolute value of Integral_{x=0..Pi/2} x^2*log(2*cos(x)) dx.
The absolute value of (d/db(d^2/da^2(Integral_{x=0..Pi/2} cos(ax)*(2*cos(x))^b dx))).
The absolute value of (Pi/2)*(d/db(d^2/da^2(gamma(b+1)/gamma((b+a)/2+1)/gamma((b-a)/2+1))) at a=0 and b=0. - Seiichi Kirikami and Peter J. C. Moses

			0.94409328404076973180...

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 3.631.9

Masato Kobayashi, Integral representations for zeta(3) with the inverse sine function, arXiv:2108.01247 [math.NT], 2021.

Cf. A000796, A002117, A152584, A193713, A194655.

Mathematica
```
RealDigits[ N[Pi Zeta[3]/4, 150]][[1]]
```

Equals A000796*A002117/4.

Equals 2 * Integral_{x=0..1} arcsin(x)^2*arccos(x)/x dx (Kobayashi, 2021). - Amiram Eldar, Jun 23 2023

cp's OEIS Frontend

User: Seiichi Kirikami

A209084 a(n) = 2*a(n-1) + 4*a(n-2) with n>1, a(0)=0, a(1)=4.

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A206532 a(n) = (2(n+1)(2n+1)-1) * a(n-1) + 2n(2n-1) * a(n-2), a(0) = 1, a(1) = 11.

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A206531 a(n) = (2*(n+1)*(2*n+1)-1)*a(n-1) + 2*n*(2*n-1)*a(n-2), a(0)=0, a(1)=2.

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A206308 a(n) = ((2*n+2)*(2*n+3) - 1)*a(n-1) + 2*n*(2*n+1)*a(n-2), a(0)=1, a(1)=19.

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A206307 a(n) = ((2*n+2)*(2*n+3) - 1)*a(n-1) + 2*n*(2*n+1)*a(n-2), a(0)=0, a(1)=6.

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A206530 Decimal expansion of 1/(1-sin(1)).

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A209084 a(n) = 2a(n-1) + 4a(n-2) with n>1, a(0)=0, a(1)=4.

A206531 a(n) = (2(n+1)(2n+1)-1)a(n-1) + 2n(2n-1)a(n-2), a(0)=0, a(1)=2.

A206308 a(n) = ((2n+2)(2n+3) - 1)a(n-1) + 2n(2n+1)a(n-2), a(0)=1, a(1)=19.

A206307 a(n) = ((2n+2)(2n+3) - 1)a(n-1) + 2n(2n+1)a(n-2), a(0)=0, a(1)=6.

A196878 Decimal expansion of (Pi/8)(6zeta(3)+Pi^2log(2)+4log(2)^3).