A016687 -id:A016687 - cp's OEIS frontend

A074378 Even triangular numbers halved.

0, 3, 5, 14, 18, 33, 39, 60, 68, 95, 105, 138, 150, 189, 203, 248, 264, 315, 333, 390, 410, 473, 495, 564, 588, 663, 689, 770, 798, 885, 915, 1008, 1040, 1139, 1173, 1278, 1314, 1425, 1463, 1580, 1620, 1743, 1785, 1914, 1958, 2093, 2139, 2280, 2328, 2475

Offset: 0

W. Neville Holmes, Sep 04 2002

Set of integers k such that k + (1 + 2 + 3 + 4 + ... + x) = 3*k, where x is sufficiently large. For example, 203 is a term because 203 + (1 + 2 + 3 + 4 + ... +28) = 609 and 609 = 3*203. - Gil Broussard, Sep 01 2008
Set of all m such that 16*m+1 is a perfect square. - Gary Detlefs, Feb 21 2010
Integers of the form Sum_{k=0..n} k/2. - Arkadiusz Wesolowski, Feb 07 2012
Numbers of the form h*(4*h + 1) for h = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018
Numbers whose distance to nearest square equals their distance to nearest oblong; that is, numbers k such that A053188(k) = A053615(k). - Lamine Ngom, Oct 27 2020
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-3))*(1 + q^(8*n-5)) = 1 + q^3 + q^5 + q^14 + q^18 + .... - Peter Bala, Dec 30 2024

David A. Corneth, Table of n, a(n) for n = 0..9999
Neville Holmes, More Geometric Integer Sequences
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A011848, A014493, A014494, A074377, A035294, A274757.
Cf. A010709, A047522. [Vincenzo Librandi, Feb 14 2009]
Cf. A266883 (numbers n such that 16*n-15 is a square).
Cf. A016687, A153799.
Cf. A053615, A053188.

f:=func; [0] cat [f(n*m): m in [-1,1], n in [1..25]]; // Bruno Berselli, Nov 13 2012

Maple

a:=n->(2*n+1)*floor((n+1)/2): seq(a(n),n=0..50); # Muniru A Asiru, Feb 01 2019

Mathematica

1/2 * Select[PolygonalNumber@ Range[0, 100], EvenQ] (* Michael De Vlieger, Jun 01 2017, Version 10.4 *) Select[Accumulate[Range[0,100]],EvenQ]/2 (* Harvey P. Dale, Feb 15 2025 *)

PARI

a(n)=(2*n+1)*(n-n\2)

Formula

Sum_{n>=0} q^a(n) = (Prod_{n>0} (1-q^n))*(Sum_{n>=0} A035294(n)*q^n).

a(n) = n*(n + 1)/4 where n*(n + 1)/2 is even.

G.f.: x*(3 + 2*x + 3*x^2)/((1 - x)*(1 - x^2)^2).

From Benoit Jubin, Feb 05 2009: (Start)

a(n) = (2*n + 1)*floor((n + 1)/2).

a(2*k) = k*(4*k+1); a(2*k+1) = (k+1)*(4*k+3). (End)

a(2*n) = A007742(n), a(2*n-1) = A033991(n). - Arkadiusz Wesolowski, Jul 20 2012

a(n) = (4*n + 1 - (-1)^n)*(4*n + 3 - (-1)^n)/4^2. - Peter Bala, Jan 21 2019

a(n) = (2*n+1)*(n+1)*(1+(-1)^(n+1))/4 + (2*n+1)*(n)*(1+(-1)^n)/4. - Eric Simon Jacob, Jan 16 2020

From Amiram Eldar, Jul 03 2020: (Start)

Sum_{n>=1} 1/a(n) = 4 - Pi (A153799).

Sum_{n>=1} (-1)^(n+1)/a(n) = 6*log(2) - 4 (See A016687). (End)

a(n) = A014494(n)/2 = A274757(n)/3 = A266883(n) - 1. - Hugo Pfoertner, Dec 31 2024

A016492 Continued fraction for log(64).

Original entry on oeis.org

4, 6, 3, 2, 2, 18, 1, 1, 1, 64, 1, 3, 1, 10, 1, 2, 1, 7, 2, 5, 3, 1, 2, 2, 3, 2, 1, 2, 1, 4, 1, 3, 3, 2, 1, 1, 1, 4, 1, 1, 7, 2, 1, 27, 2, 1, 1, 4, 4, 1, 1, 1, 4, 1, 1, 6, 20, 1, 8, 2, 2, 1, 5, 1, 11, 1, 10, 1, 5, 1, 1, 2, 5, 9, 1, 5, 1, 2, 2

Offset: 1

N. J. A. Sloane

			4.158883083359671856503392728... = 4 + 1/(6 + 1/(3 + 1/(2 + 1/(2 + ...)))). - _Harry J. Smith_, May 22 2009

Harry J. Smith, Table of n, a(n) for n = 1..20000

Cf. A016687 (decimal expansion).

ContinuedFraction[Log[64], 100] (* Paolo Xausa, Mar 27 2024 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(64)); for (n=1, 20000, write("b016492.txt", n, " ", x[n])); } \\ Harry J. Smith, May 22 2009

A383824 Decimal expansion of 12log(2)/(6log(2) - 3).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, May 11 2025

Upper bound for the irrationality measure of 2-adic analog of zeta(3) (see Lai et al., 2025 at p. 3).

			7.177398899124179661610768863884179976261011824...

Li Lai, Johannes Sprang, and Wadim Zudilin, A note on the irrationality of zeta_2(5), arXiv:2505.05005 [math.NT], 2025.
Eric Weisstein's World of Mathematics, Irrationality Measure.
Wikipedia, Irrationality measure.

Cf. A002162, A016687, A382497, A382778, A383822.

RealDigits[12Log[2]/(6Log[2]-3),10,100][[1]]

A091477 Decimal expansion of (3CatalanPi)/4 - Pi^3/64 + (Pi^2log(64))/64 - (105zeta(3))/64.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 13 2004

			0.342947449816831475186873451421175415636919316094040413221830284...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Definite Integral

Cf. A000796, A002117, A006752, A016687.

SetDefaultRealField(RealField(100)); R:=RealField(); L:=RiemannZeta();  (48*Catalan(R)*Pi(R) - Pi(R)^3 + Pi(R)^2*Log(64) - 105*Evaluate(L,3))/64; // G. C. Greubel, Aug 25 2018

RealDigits[(48*Catalan*Pi - Pi^3 + Pi^2*Log[64] - 105*Zeta[3])/64, 10, 100][[1]] (* G. C. Greubel, Aug 25 2018 *)

default(realprecision, 100); (48*Catalan*Pi - Pi^3 + Pi^2*log(64) - 105*zeta(3))/64 \\ G. C. Greubel, Aug 25 2018

Equals Integral_{t=0..Pi/4} t^3/sin(t)^2 dt.

A365523 Decimal expansion of 6*log(2) - 4.

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Sep 08 2023

This sequence is also the decimal expansion of Sum_{k>=1} (-1)^(k+1)*f(k), where f(k) = (4*k^2 - 2*k)/(k^2 + k) is the ratio between the k-th hexagonal and triangular numbers.

			0.15888308335967185650339272874905940845300080616153...

Michael Ian Shamos, A catalog of the real numbers (2011), p. 219.
Wikipedia, Polygonal number.
Index entries for transcendental numbers

Cf. A002162. Essentially the same as A016687.

Cf. A000217, A000384.

RealDigits[6*Log[2] - 4, 10 , 100][[1]] (* Amiram Eldar, Sep 08 2023 *)

PARI
```
6*log(2)-4
```

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

Equals Sum_{k>=1} (-1)^(k+1)*(4*k^2 - 2*k)/(k^2 + k).

cp's OEIS Frontend

A074378 Even triangular numbers halved.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A016492 Continued fraction for log(64).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A383824 Decimal expansion of 12*log(2)/(6*log(2) - 3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A091477 Decimal expansion of (3*Catalan*Pi)/4 - Pi^3/64 + (Pi^2*log(64))/64 - (105*zeta(3))/64.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A365523 Decimal expansion of 6*log(2) - 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383824 Decimal expansion of 12log(2)/(6log(2) - 3).

A091477 Decimal expansion of (3CatalanPi)/4 - Pi^3/64 + (Pi^2log(64))/64 - (105zeta(3))/64.