A073000 -id:A073000 - cp's OEIS frontend

A195710 Decimal expansion of arccos(-sqrt(2/5)).

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

Offset: 1

Clark Kimberling, Sep 23 2011

			arccos(-sqrt(2/5)) = 2.25551552979717...

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

Cf. A195708, A195701.

[Arccos(-Sqrt(2/5))]; // G. C. Greubel, Nov 18 2017

r = Sqrt[1/5]; s = Sqrt[2/5];
N[ArcSin[r], 100]
RealDigits[%]  (* A073000 *)
N[ArcCos[r], 100]
RealDigits[%]  (* A105199 *)
N[ArcTan[r], 100]
RealDigits[%]  (* A188595 *)
N[ArcCos[-r], 100]
RealDigits[%]  (* A137218 *)
N[ArcSin[s], 100]
RealDigits[%]  (* A195701 *)
N[ArcCos[s], 100]
RealDigits[%]  (* A195708 *)
N[ArcTan[s], 100]
RealDigits[%]  (* A195709 *)
N[ArcCos[-s], 100]
RealDigits[%]  (* A195710 *)
RealDigits[ArcCos[-Sqrt[(2/5)]],10,120][[1]] (* Harvey P. Dale, Apr 06 2023 *)

acos(-sqrt(2/5)) \\ G. C. Greubel, Nov 18 2017

Equals Pi - arcsin(sqrt(3/5)) = Pi - arctan(sqrt(3/2)). - Amiram Eldar, Jul 08 2023

A254381 a(n) = 3^n(2n + 1)!/n!.

Original entry on oeis.org

1, 18, 540, 22680, 1224720, 80831520, 6304858560, 567437270400, 57878601580800, 6598160580211200, 831368233106611200, 114728816168712345600, 17209322425306851840000, 2787910232899709998080000, 485096380524549539665920000, 90227926777566214377861120000

Offset: 0

Peter Bala, Feb 04 2015

Cf. A002391, A034910, A073000, A093766, A254619, A254620.

Maple
```
seq(3^n*(2*n + 1)!/n!, n = 0..13);
```

Table[3^n(2n + 1)!/n!, {n, 0, 19}] (* Alonso del Arte, Feb 04 2015 *)

E.g.f.: 1/(1 - 12*x)^(3/2) = 1 + 18*x + 540*x^2/2! + 22680*x^3/3! + ....
Recurrence equation: a(n) = 6*(2*n + 1)*a(n-1) with a(0) = 1.
2nd order recurrence equation: a(n) = 8*(n + 1)*a(n-1) + 12*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 18.
Define a sequence b(n) := a(n)*sum {k = 0..n} (-1)^k/((2*k + 1)*3^k) beginning [1, 16, 492, 20544, 1111056, 73299456, 5718022848, ...]. It is not difficult to check that b(n) also satisfies the previous 2nd order recurrence equation (and so is an integer sequence). Using this observation we obtain the continued fraction expansion Pi/(2*sqrt(3)) = Sum {k >= 0} (-1)^k/( (2*k + 1)*3^k ) = 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). Cf. A254619 and A254620.

A086203 Decimal expansion of arctan(1/2)/Pi.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 12 2003

This constant is a transcendental number (Margolius, 2002; Smith, 2003). - Amiram Eldar, Aug 13 2020

			0.14758...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003. See pp. 430-433.

Jonathan M. Borwein and Roland Girgensohn, Addition Theorems and Binary Expansions, Canadian Journal of Mathematics, Vol. 47, No. 2 (1995), pp. 262-273.
Barbara H. Margolius, Plouffe's Constant is Transcendental, unpublished note, 2002.
Simon Plouffe, The computation of certain numbers using a ruler and compass, Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.3.
Warren D. Smith, Pythagorean triples, rational angles, and space-filling simplices, preprint, 2003.
Eric Weisstein's World of Mathematics, Plouffe's Constants
Index entries for transcendental numbers

Cf. A000796, A073000.

RealDigits[ArcTan[1/2]/Pi, 10, 100][[1]] (* Amiram Eldar, Aug 13 2020 *)

atan(1/2)/Pi \\ Michel Marcus, Jul 29 2015

A105533 Decimal expansion of arctan(1/7).

Original entry on oeis.org

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

Offset: 0

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 12 2005

			0.1418970546041639228128516171...

D. H. Lehmer, On Arccotangent Relations for π, The American Mathematical Monthly, Vol. 45, No. 10 (Dec., 1938), pp. 657-664.
Eric Weisstein's World of Mathematics, Machin-Like Formulas.
Index entries for transcendental numbers

Cf. A073000, A105531.

RealDigits[ArcTan[1/7],10,120][[1]] (* Harvey P. Dale, Oct 03 2012 *)

PARI
```
atan(1/7) \\ Michel Marcus, Mar 01 2020
```

2*A073000 - arctan(1/7) = 2*A105531 + arctan(1/7) = Pi/4.
5*arctan(1/7) + 2*arctan(3/79) = Pi/4. - Frank Ellermann, Mar 01 2020
Equals arcsin(1/(5*sqrt(2))) = arccos(7/(5*sqrt(2))). - Amiram Eldar, Jul 11 2023

A254620 a(n) = 9^n(2n + 1)!/n!.

Original entry on oeis.org

1, 54, 4860, 612360, 99202320, 19642059360, 4596241890240, 1240985310364800, 379741504971628800, 129871594700297049600, 49091462796712284748800, 20323865597838885886003200, 9145739519027498648701440000, 4444829406247364343268899840000

Offset: 0

Peter Bala, Feb 03 2015

Cf. A002162, A002391, A034910, A073000, A105531, A254381, A254619.

Maple
```
seq(9^n*(2*n + 1)!/n!, n = 0..14);
```

Table[9^n (2n+1)!/n!,{n,0,20}] (* Harvey P. Dale, Aug 13 2019 *)

E.g.f.: 1/(1 - 36*x)^(3/2) = 1 + 54*x + 4860*x^2/2! + 612360*x^3/3! + ....
Recurrence equation: a(n) = 18*(2*n + 1)*a(n-1) with a(0) = 1.
2nd order recurrence equation: a(n) = (40*n + 16)*a(n-1) - 36*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 54.
Define a sequence b(n) := a(n)*sum {k = 0..n} 1/((2*k + 1)*9^k) beginning [1, 56, 5052, 636672, 103142544, 20422253952, 4778808090048, ...]. It is not difficult to check that b(n) also satisfies the previous 2nd order recurrence equation (and so is an integer sequence). Using this observation we obtain the continued fraction expansion log(2) = 2/3*Sum {k >= 0} 1/((2*k + 1)*9^k) = 2/3*(1 + 2/(54 - 36*3^2/(96 - 36*5^2/(136 - ... - 36*(2*n - 1)^2/((40*n + 16) - ... ))))).
Alternative 2nd order recurrence equation: a(n) = (32*n + 20)*a(n-1) + 36*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 54.
Define now a sequence c(n) := a(n)*sum {k = 0..n} (-1)^k/((2*k + 1)*9^k) beginning [1, 52, 4692, 591072, 95755344, 18959527872, 4436530187328, ...], which, along with a(n), satisfies the alternative 2nd order recurrence equation. From this observation we find the continued fraction expansion arctan(1/3) = 1/3*Sum {k >= 0} (-1)^k/((2*k + 1)*9^k) = 1/3*(1 - 2/(54 + 36*3^2/(84 + 36*5^2/(116 + ... + 36*(2*n - 1)^2/((32*n + 20) + ... ))))). Cf. A254381 and A254619.

A344906 Decimal expansion of Sum_{k>=0} arctan(1/2^k).

Original entry on oeis.org

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

Offset: 1

Daniel Hoyt, Jun 01 2021

This number can be interpreted geometrically as the angle in radians of a fan made of stacked right triangles, with the length to height ratio doubling each successive triangle as seen in the illustration.

Since this angle exceeds Pi/2, the set of rotation angles used in the CORDIC algorithm covers an angle range sufficient to compute sine and cosine for any angle between 0 and Pi/2. This means the algorithm can converge to any angle in that range through appropriate combinations of these basic rotations. - Daniel Hoyt, Oct 25 2024

			1.743286620472340003...

Daniel Hoyt, Illustration of this angle's arctan relationship.
Wikipedia, CORDIC.

Cf. A003881, A065445, A073000, A195727, A195782.

Digits:= 140:
evalf(sum(arccot(2^k), k=0..infinity));  # Alois P. Heinz, Jun 02 2021

PARI
```
suminf(k=0, atan(1/2^k))
```

sumalt(k=1, ((-1)^(k+1))*2^(2*k-1)/((2^(2*k-1)-1)*(2*k-1)))

Equals Sum_{k>=1} (-1)^(k+1)*2^(2*k-1)/((2^(2*k-1)-1)*(2*k-1)).

A288443 a(n) = (2n + 1)2^(2n + 1); numbers k such that v(k)2^v(k) = k, where v(n) = A007814(n) is 2-adic valuation of n.

Original entry on oeis.org

2, 24, 160, 896, 4608, 22528, 106496, 491520, 2228224, 9961472, 44040192, 192937984, 838860800, 3623878656, 15569256448, 66571993088, 283467841536, 1202590842880, 5085241278464, 21440476741632, 90159953477632, 378231999954944, 1583296743997440, 6614661952700416, 27584547717644288, 114841790497947648

Offset: 0

Juri-Stepan Gerasimov, Jun 24 2017

Cf. A007814, A058962, A098713, A286683.
Odd bisection of A036289.
Cf. A073000, A156057.

Magma
```
[(2*n+1)*2^(2*n+1): n in [0..25]];
```

a(n) = (2*n+1)<<(2*n+1) \\ Charles R Greathouse IV, Jul 07 2017

a(n) = (2n + 1)*2^(2n + 1).
a(n) = A036289(2n + 1).
a(n) = A098713(n) + 1.
a(n) = 2*A058962(n). - Joerg Arndt, Jun 25 2017
From Amiram Eldar, Jul 03 2020: (Start)
Sum_{n>=0} 1/a(n) = arctanh(1/2) = log(3)/2 (A156057).
Sum_{n>=0} (-1)^n/a(n) = arctan(1/2) (A073000). (End)

A384806 Simple continued fraction expansion of arctan(1/2)/Pi.

Original entry on oeis.org

0, 6, 1, 3, 2, 5, 1, 6, 5, 3, 1, 1, 2, 1, 1, 2, 3, 1, 2, 3, 2, 2, 2, 2, 3, 2, 1, 1, 3, 1, 3, 2, 1, 1, 3, 1, 17, 1, 5, 1, 2, 1, 2, 1, 1, 1, 1, 4, 7, 11, 1, 2, 1, 583, 1, 2, 1, 1, 2, 22, 7, 3, 23, 2, 6, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 5, 1, 1, 2, 3, 17, 6, 2, 70, 1, 6

Offset: 0

Stefano Spezia, Jun 10 2025

			arctan(1/2)/Pi = 1/(6 + 1/(1 + 1/(3 + 1/(2 + 1/(5 + ... ))))).

Shigeki Akiyama, Emily R. Korfanty, and Yan-li Xu, Delone sets associated with badly approximable triangles, Journal of Number Theory, (2025).
Index entries for continued fractions for constants.

Cf. A000796, A073000, A086203 (decimal expansion).

Mathematica
```
ContinuedFraction[ArcTan[1/2]/Pi,90]
```

cp's OEIS Frontend

A195710 Decimal expansion of arccos(-sqrt(2/5)).

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A254381 a(n) = 3^n*(2*n + 1)!/n!.

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Maple

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Formula

A086203 Decimal expansion of arctan(1/2)/Pi.

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A105533 Decimal expansion of arctan(1/7).

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A254620 a(n) = 9^n*(2*n + 1)!/n!.

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Maple

Mathematica

Formula

A344906 Decimal expansion of Sum_{k>=0} arctan(1/2^k).

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Programs

Maple

PARI

PARI

Formula

A288443 a(n) = (2n + 1)*2^(2n + 1); numbers k such that v(k)*2^v(k) = k, where v(n) = A007814(n) is 2-adic valuation of n.

Views

Author

Keywords

Crossrefs

Programs

Magma

PARI

Formula

A384806 Simple continued fraction expansion of arctan(1/2)/Pi.

Views

Author

Keywords

A254381 a(n) = 3^n(2n + 1)!/n!.

A254620 a(n) = 9^n(2n + 1)!/n!.

A288443 a(n) = (2n + 1)2^(2n + 1); numbers k such that v(k)2^v(k) = k, where v(n) = A007814(n) is 2-adic valuation of n.