A016655 -id:A016655 - cp's OEIS frontend

A005843 The nonnegative even numbers: a(n) = 2n.

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 0

N. J. A. Sloane

-2, -4, -6, -8, -10, -12, -14, ... are the trivial zeros of the Riemann zeta function. - Vivek Suri (vsuri(AT)jhu.edu), Jan 24 2008
If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-2) is the number of 2-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
A134452(a(n)) = 0; A134451(a(n)) = 2 for n > 0. - Reinhard Zumkeller, Oct 27 2007
Omitting the initial zero gives the number of prime divisors with multiplicity of product of terms of n-th row of A077553. - Ray Chandler, Aug 21 2003
A059841(a(n))=1, A000035(a(n))=0. - Reinhard Zumkeller, Sep 29 2008
(APSO) Alternating partial sums of (a-b+c-d+e-f+g...) = (a+b+c+d+e+f+g...) - 2*(b+d+f...), it appears that APSO(A005843) = A052928 = A002378 - 2*(A116471), with A116471=2*A008794. - Eric Desbiaux, Oct 28 2008
A056753(a(n)) = 1. - Reinhard Zumkeller, Aug 23 2009
Twice the nonnegative numbers. - Juri-Stepan Gerasimov, Dec 12 2009
The number of hydrogen atoms in straight-chain (C(n)H(2n+2)), branched (C(n)H(2n+2), n > 3), and cyclic, n-carbon alkanes (C(n)H(2n), n > 2). - Paul Muljadi, Feb 18 2010
For n >= 1; a(n) = the smallest numbers m with the number of steps n of iterations of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = m. See A175126 and A175127. A175126(a(n)) = A175126(A175127(n)) = n. Example (a(4)=8): 8-2=6, 6-2=4, 4-2=2, 2-2=0; iterations has 4 steps and number 8 is the smallest number with such result. - Jaroslav Krizek, Feb 15 2010
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is not integer. A040001(a(n)) > 1. See A145051 and A040001. - Jaroslav Krizek, May 28 2010
Union of A179082 and A179083. - Reinhard Zumkeller, Jun 28 2010
a(k) is the (Moore lower bound on and the) order of the (k,4)-cage: the smallest k-regular graph having girth four: the complete bipartite graph with k vertices in each part. - Jason Kimberley, Oct 30 2011
For n > 0: A048272(a(n)) <= 0. - Reinhard Zumkeller, Jan 21 2012
Let n be the number of pancakes that have to be divided equally between n+1 children. a(n) is the minimal number of radial cuts needed to accomplish the task. - Ivan N. Ianakiev, Sep 18 2013
For n > 0, a(n) is the largest number k such that (k!-n)/(k-n) is an integer. - Derek Orr, Jul 02 2014
a(n) when n > 2 is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
It appears that for n > 2, a(n) = A020482(n) + A002373(n), where all sequences are infinite. This is consistent with Goldbach's conjecture, which states that every even number > 2 can be expressed as the sum of two prime numbers. - Bob Selcoe, Mar 08 2015
Number of partitions of 4n into exactly 2 parts. - Colin Barker, Mar 23 2015
Number of neighbors in von Neumann neighborhood. - Dmitry Zaitsev, Nov 30 2015
Unique solution b( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - Clark Kimberling, Nov 21 2017
Also the maximum number of non-attacking bishops on an (n+1) X (n+1) board (n>0). (Cf. A000027 for rooks and queens (n>3), A008794 for kings or A030978 for knights.) - Martin Renner, Jan 26 2020
Integer k is even positive iff phi(2k) > phi(k), where phi is Euler's totient (A000010) [see reference De Koninck & Mercier]. - Bernard Schott, Dec 10 2020
Number of 3-permutations of n elements avoiding the patterns 132, 213, 312 and also number of 3-permutations avoiding the patterns 213, 231, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
a(n) gives the y-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The x-value is given by 2*n^2 + 1 (see Tattersall). - Stefano Spezia, Jul 24 2025

			G.f. = 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 28.
J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 529a pp. 71 and 257, Ellipses, 2004, Paris.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
Kevin Fagan, Drabble cartoon, Jun 15 1987: Intelligence Test
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012, J. Int. Seq. 15 (2012) # 12.6.2
Milan Janjic, Two Enumerative Functions
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Even Number
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros
Wikipedia, Alkane
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

a(n)=2*A001477(n). - Juri-Stepan Gerasimov, Dec 12 2009
Cf. A000027, A002061, A005408, A001358, A077553, A077554, A077555, A002024, A087112, A157888, A157889, A140811, A157872, A157909, A157910, A165900.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), this sequence (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A231200 (boustrophedon transform).

a005843 = (* 2)
a005843_list = [0, 2 ..]  -- Reinhard Zumkeller, Feb 11 2012

Magma
```
[ 2*n : n in [0..100]];
```

A005843 := n->2*n;
A005843:=2/(z-1)**2; # Simon Plouffe in his 1992 dissertation

Range[0,120,2] (* Harvey P. Dale, Aug 16 2011 *)

PARI
```
A005843(n) = 2*n
```

def a(n): return 2*n # Martin Gergov, Oct 20 2022

R
```
seq(0,200,2)
```

G.f.: 2*x/(1-x)^2.
E.g.f.: 2*x*exp(x). - Geoffrey Critzer, Aug 25 2012
G.f. with interpolated zeros: 2x^2/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*sinh(x). - Geoffrey Critzer, Aug 25 2012
Inverse binomial transform of A036289, n*2^n. - Joshua Zucker, Jan 13 2006
a(0) = 0, a(1) = 2, a(n) = 2a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
a(n) = Sum_{k=1..n} floor(6n/4^k + 1/2). - Vladimir Shevelev, Jun 04 2009
a(n) = A034856(n+1) - A000124(n) = A000217(n) + A005408(n) - A000124(n) = A005408(n) - 1. - Jaroslav Krizek, Sep 05 2009
a(n) = Sum_{k>=0} A030308(n,k)*A000079(k+1). - Philippe Deléham, Oct 17 2011
Digit sequence 22 read in base n-1. - Jason Kimberley, Oct 30 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Dec 23 2011
a(n) = 2*n = Product_{k=1..2*n-1} 2*sin(Pi*k/(2*n)), n >= 0 (undefined product := 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013
From Ilya Gutkovskiy, Aug 19 2016: (Start)
Convolution of A007395 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 = (1/2)*A002162 = (1/10)*A016655. (End)
From Bernard Schott, Dec 10 2020: (Start)
Sum_{n>=1} 1/a(n)^2 = Pi^2/24 = A222171.
Sum_{n>=1} (-1)^(n+1)/a(n)^2 = Pi^2/48 = A245058. (End)

A003056 n appears n+1 times. Also the array A(n,k) = n+k (n >= 0, k >= 0) read by antidiagonals. Also inverse of triangular numbers.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 0

N. J. A. Sloane

Also triangle read by rows: T(n,k), n>=0, k>=0, in which n appears n+1 times in row n. - Omar E. Pol, Jul 15 2012
The PARI functions t1, t2 can be used to read a triangular array T(n,k) (n >= 0, 0 <= k <= n-1) by rows from left to right: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002
Number of terms in partition of n with greatest number of distinct terms. - Amarnath Murthy, May 20 2001
Summation table for (x+y) = (0+0),(0+1),(1+0),(0+2),(1+1),(2+0), ...
Also the number of triangular numbers less than or equal to n, not counting 0 as triangular. - Robert G. Wilson v, Oct 21 2005
Permutation of A116939: a(n) = A116939(A116941(n)), a(A116942(n)) = A116939(n). - Reinhard Zumkeller, Feb 27 2006
Maximal size of partitions of n into distinct parts, see A000009. - Reinhard Zumkeller, Jun 13 2009
Also number of digits of A000462(n). - Reinhard Zumkeller, Mar 27 2011
Also the maximum number of 1's contained in the list of hook-lengths of a partition of n. E.g., a(4)=2 because hooks of partitions of n=4 comprise {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} where the number of 1's in each is 1,2,1,2,1. Hence the maximum is 2. - T. Amdeberhan, Jun 03 2012
Fan, Yang, and Yu (2012) prove a conjecture of Amdeberhan on the generating function of a(n). - Jonathan Sondow, Dec 17 2012
Also the number of partitions of n into distinct parts p such that max(p) - min(p) <= length(p). - Clark Kimberling, Apr 18 2014
Also the maximum number of occurrences of any single value among the previous terms. - Ivan Neretin, Sep 20 2015
Where records occur gives A000217. - Omar E. Pol, Nov 05 2015
Also number of peaks in the largest Dyck path of the symmetric representation of sigma(n), n >= 1. Cf. A237593. - Omar E. Pol, Dec 19 2016

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...
As triangle, the sequence starts
  0;
  1, 1;
  2, 2, 2;
  3, 3, 3, 3;
  4, 4, 4, 4, 4;
  5, 5, 5, 5, 5, 5;
  6, 6, 6, 6, 6, 6, 6;
  7, 7, 7, 7, 7, 7, 7, 7;
  8, 8, 8, 8, 8, 8, 8, 8, 8;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Anna R. B. Fan, Harold R. L. Yang, and Rebecca T. Yu, On the Maximum Number of k-Hooks of Partitions of n, arXiv:1212.3505 [math.CO], 2012.
Michael Somos, Sequences used for indexing triangular or square arrays.

a(n) = A002024(n+1)-1.
Cf. A000196, A000217, A000462, A001227, A001462, A001614, A004247 (multiplication table), A006463 (partial sums), A016655, A050600, A050602, A048645, A122797, A131507, A238005.
Partial sums of A073424.
Cf. A116939, A116941, A116942,

a003056 = floor . (/ 2) . (subtract 1) .
                  sqrt . (+ 1) . (* 8) . fromIntegral
a003056_row n = replicate (n + 1) n
a003056_tabl = map a003056_row [0..]
a003056_list = concat $ a003056_tabl
-- Reinhard Zumkeller, Aug 02 2014, Oct 17 2010

[Floor((Sqrt(1+8*n)-1)/2): n in [0..80]]; // Vincenzo Librandi, Oct 23 2011

A003056 := (n,k) -> n: # Peter Luschny, Oct 29 2011
a := [ 0 ]: for i from 1 to 15 do for j from 1 to i+1 do a := [ op(a),i ]; od: od: a;
A003056 := proc(n)
    floor((sqrt(1+8*n)-1)/2) ;
end proc: # R. J. Mathar, Jul 10 2015

f[n_] := Floor[(Sqrt[1 + 8n] - 1)/2]; Table[ f[n], {n, 0, 87}] (* Robert G. Wilson v, Oct 21 2005 *)
Table[x, {x, 0, 13}, {y, 0, x}] // Flatten
T[ n_, k_] := If[ n >= k >= 0, n, 0]; (* Michael Somos, Dec 22 2016 *)
Flatten[Table[PadRight[{},n+1,n],{n,0,12}]] (* Harvey P. Dale, Jul 03 2021 *)

A003056(n)=(sqrtint(8*n+1)-1)\2  \\ M. F. Hasler, Oct 08 2011

t1(n)=floor(-1/2+sqrt(2+2*n)) /* A003056 */

t2(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2) /* A002262 */

from math import isqrt
def A003056(n): return (k:=isqrt(m:=n+1<<1))+int((m<<2)>(k<<2)*(k+1)+1)-1 # Chai Wah Wu, Jul 26 2022

a(n) = floor((sqrt(1+8*n)-1)/2). - Antti Karttunen
a(n) = floor(-1/2 + sqrt(2*n+b)) with 1/4 <= b < 9/4 or a(n) = floor((sqrt(8*n+b)-1)/2) with 1 <= b < 9. - Michael A. Childers (childers_moof(AT)yahoo.com), Nov 11 2001
a(n) = f(n,0) with f(n,k) = k if n <= k, otherwise f(n-k-1, k+1). - Reinhard Zumkeller, May 23 2009
a(n) = 2*n + 1 - A001614(n+1) = n + 1 - A122797(n+1). - Reinhard Zumkeller, Feb 12 2012
a(n) = k if k*(k+1)/2 <= n < (k+1)*(k+2)/2. - Jonathan Sondow, Dec 17 2012
G.f.: (1-x)^(-1)*Sum_{n>=1} x^(n*(n+1)/2) = (Theta_2(0,x^(1/2)) - 2*x^(1/8))/(2*x^(1/8)*(1-x)) where Theta_2 is a Jacobi Theta function. - Robert Israel, May 21 2015
a(n) = floor((A000196(1+8*n)-1)/2). - Pontus von Brömssen, Dec 10 2018
a(n+1) = a(n-a(n)) + 1, a(0) = 0. - Rok Cestnik, Dec 29 2020
a(n) = A001227(n) + A238005(n), n >= 1. - Omar E. Pol, Sep 30 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 (cf. A016655). - Amiram Eldar, Sep 24 2023
G.f. as array: (x + y - 2*x*y)/((1 - x)^2*(1 - y)^2). - Stefano Spezia, Dec 20 2023 [corrected by Stefano Spezia, Apr 22 2024]

Definition clarified by N. J. A. Sloane, Dec 08 2020

A016639 Decimal expansion of log(16) = 4*log(2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.77258872223978123766892848583270627230200053744102101648272... - _Harry J. Smith_, May 17 2009

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter Bala, A continued fraction expansion for the constant log(16)
Eric Weisstein's World of Mathematics, Madelung Constants.
Index entries for transcendental numbers

Equals 4*A002162.
Equals (4/5)*A016655.
Equals A303658 + 2.
Cf. A016444 (continued fraction).
Cf. A001008, A002805, A188859.

Log(16); // Vincenzo Librandi, Feb 20 2015

RealDigits[Log[16], 10, 120][[1]] (* Harvey P. Dale, Jun 12 2012 *)

default(realprecision, 20080); x=log(16); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016639.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009, corrected May 19 2009

Equals 4*A002162.
Equals Sum_{k=1..4} (-1)^(k+1) gamma(0, k/4) where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018
Equals -2 + Sum_{k>=1} H(k)*(k+1)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, May 28 2021
Equals 1 + Limit_{n -> infinity} (1/n)*Sum_{k = 1..n} (2*n + k)/(2*n - k) = 2*( 1 + Limit_{n -> infinity} (1/n)*Sum_{k = 1..n} (n - k)/(n + k) ). - Peter Bala, Oct 10 2021
Equals 2 + 1/(1 + 1/(3 + 2/(4 + 6/(5 + 6/(6 + 12/(7 + 12/(8 + ... + n*(n-1)/(2*n-1 + n*(n-1)/(2*n + ...))))))))). Cf. A188859. - Peter Bala, Mar 04 2024

A180447 n appears 3n+1 times.

Original entry on oeis.org

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

Offset: 0

William A. Tedeschi, Sep 07 2010

			a(5) = floor((sqrt(24*5+1)+1)/6) = 2.

Kevin Ryde, Table of n, a(n) for n = 0..10000

Cf. A000326 (indices of run starts), A016655, A180446.

f[n_] := Floor[(Sqrt[24 n + 1] + 1)/6]; Array[f, 105, 0] (* Robert G. Wilson v, Sep 10 2010 *)

a(n) = (sqrtint(24*n+1)+1)\6; \\ Kevin Ryde, Apr 21 2021

l = [floor((sqrt(24*n+1)+1)/6) for n in range(0,101)]

from math import isqrt
def A180447(n): return (m:=isqrt((k:=n<<1)//3))+(k>m*(3*m+5)) # Chai Wah Wu, Nov 04 2024

a(n) = floor((sqrt(24n+1)+1)/6).
a(n) = m+1 if 2n>m*(3m+5) and a(n) = m otherwise where m = floor(sqrt(2n/3)). For n>0, a(n) = k+1 if 2n>=(k+1)(3k+2) and a(n) = k otherwise where k = floor(sqrt(2(n-1)/3)). - Chai Wah Wu, Nov 04 2024
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 (= A016655 / 10). - Amiram Eldar, Jun 30 2025

More terms from Robert G. Wilson v, Sep 10 2010

A196521 Decimal expansion of Pi/4-log(2)/2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

			0.438824573117475654907044785090787437011542282663648828183396143330257...

L. B. W. Jolley, Summation of series, Dover Publications Inc., New York, 1961, p. 14 (eq. 72).

Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.

Cf. A003881, A016655 (10*log(2)/2), A033264.

Cf. A231902 (Pi/4+log(2)/2), A342316.

Digits:=100; evalf(Pi/4-log(2)/2); # Wesley Ivan Hurt, Dec 06 2013

RealDigits[Pi/4 - Log[2]/2, 10, 100] (* Wesley Ivan Hurt, Dec 06 2013 *)

Pi/4-log(2)/2 \\ Altug Alkan, Dec 14 2015

Equals 1 - 1/2 - 1/3 + 1/4 + 1/5 - ....
Equals Sum_{n>=0} 2/((4*n+2)*(4*n+3)). - Peter Luschny, Dec 06 2013
Equals Sum_{n>=1} (-1)^(n+1)/((2*n-1)*(2*n)). - Robert FERREOL, Dec 14 2015
Equals Integral_{x=0..1} (arctan(x)) dx = Integral_{x=0..Pi/4} (x / cos(x)^2) dx = Integral_{x=0..1/sqrt(2)} (arcsin(x)/(1-x^2)^(3/2)) dx. - Robert FERREOL, Dec 14 2015
Equals Integral_{x>=0} (exp(x) - 1)/(exp(2*x) + 1) dx. - Peter Bala, Nov 01 2019
From Bernard Schott, Sep 07 2020: (Start)
Equals Sum_{n>=1} (-1)^(n*(n-1)/2) / n [compare with A231902 formula].
Equals Sum_{n>=0} (8*n+5) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)
Equals Sum_{k>=1} A033264(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021
From Peter Bala, Mar 04 2025: (Start)
Equals (1/2) * A342316.
Equals Integral_{x = 0..1} x/(x^2 - 2*x + 2) = Integral_{x = 0..1} x*(1 + x)/(2 - x^2*(1 - x)) dx.
Equals (5/2)*Sum_{n >= 1} 1/(n*binomial(3*n, n)*2^n). The first 10 terms of the series gives the approximate value 0.43882457311(68...), correct to 11 decimal places. (End)

A195697 First denominator and then numerator in a fraction expansion of log(2) - Pi/8.

Original entry on oeis.org

2, 1, 3, -1, 12, 1, 30, 1, 35, -1, 56, 1, 90, 1, 99, -1, 132, 1, 182, 1, 195, -1, 240, 1, 306, 1, 323, -1, 380, 1, 462, 1, 483, -1, 552, 1, 650, 1, 675, -1, 756, 1, 870, 1, 899, -1, 992, 1, 1122, 1, 1155, -1, 1260

Offset: 1

Mohammad K. Azarian, Sep 25 2011

The minus sign in front of a fraction is considered the sign of the numerator.

			1/2 - 1/3 + 1/12 + 1/30 - 1/35 + 1/56 + 1/90 - 1/99 + 1/132 + 1/182 - 1/195 + 1/240 + ... = [(1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + (1/9 - 1/10) + (1/11 - 1/12) + ... ] - (1/2)*[(1 - 1/3) + (1/5 - 1/7) + (1/9 - 1/11) + (1/13 - 1/15) + ... ] = log(2) - Pi/8.

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Cf. A164833, A195908, A195909, A195913, A016655, A019675, A075549, A098289, A118324, A144981, A161685, A168056, A004772.

log(2) - Pi/8 = Sum_{n>=1} (-1)^(n+1)*(1/n) + (-1/2)*Sum_{n>=0} (-1)^n*(1/(2*n+1)).

Empirical g.f.: x*(2+x+x^2-2*x^3+9*x^4+2*x^5+14*x^6-2*x^7+3*x^8+2*x^9+3*x^10-2*x^11+x^13) / ((1-x)^3*(1+x)^3*(1-x+x^2)^2*(1+x+x^2)^2). - Colin Barker, Dec 17 2015

A195909 First numerator and then denominator in a fraction expansion of log(2) - Pi/8.

Original entry on oeis.org

1, 2, -1, 3, 1, 12, 1, 30, -1, 35, 1, 56, 1, 90, -1, 99, 1, 132, 1, 182, -1, 195, 1, 240, 1, 306, -1, 323, 1, 380, 1, 462, -1, 483, 1, 552, 1, 650, -1, 675, 1, 756, 1, 870, -1, 899, 1, 992, 1, 1122, -1, 1155, 1

Offset: 1

Mohammad K. Azarian, Sep 26 2011

			1/2 - 1/3 + 1/12 + 1/30 - 1/35 + 1/56 + 1/90 - 1/99 + 1/132 + 1/182 - 1/195 + 1/240 + ... = [(1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + (1/9 - 1/10) + (1/11 - 1/12) + ... ] - (1/2)*[(1 - 1/3) + (1/5 - 1/7) + (1/9 - 1/11) + (1/13 - 1/15) + ... ] = log(2) - Pi/8.

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Cf. A195913, A195697, A195947, A164833, A118324, A098289, A075549, A016655, A019675, A161685, A144981, A168056, A004772.

log(2) - Pi/8 = Sum_{n>=1} (-1)^(n+1)*(1/n) + (-1/2)*Sum_{n>=0} (-1)^n*(1/(2*n+1)).

Empirical g.f.: x*(1+2*x-2*x^2+x^3+2*x^4+9*x^5-2*x^6+14*x^7+2*x^8+3*x^9-2*x^10+3*x^11+x^12) / ((1-x)^3*(1+x)^3*(1-x+x^2)^2*(1+x+x^2)^2). - Colin Barker, Dec 17 2015

A195913 The denominator in a fraction expansion of log(2)-Pi/8.

Original entry on oeis.org

2, 3, 12, 30, 35, 56, 90, 99, 132, 182, 195, 240, 306, 323, 380, 462, 483, 552, 650, 675, 756, 870, 899, 992, 1122, 1155, 1260, 1406, 1443, 1560, 1722, 1763, 1892, 2070, 2115, 2256, 2450, 2499, 2652, 2862, 2915

Offset: 1

Mohammad K. Azarian, Sep 25 2011

The minus sign in front of a fraction is considered the sign of the numerator and hence the sign of the fraction does not appear in this sequence.

			1/2 - 1/3 + 1/12 + 1/30 - 1/35 + 1/56 + 1/90 - 1/99 + 1/132 + 1/182 - 1/195 + 1/240 + ... = [(1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + (1/9 - 1/10) + (1/11 - 1/12) + ...] - (1/2)*[(1 - 1/3) + (1/5 - 1/7) + (1/9 - 1/11) + (1/13 - 1/15) + ... ] = log(2) - Pi/8.

Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

Cf. A195909, A195697, A195947, A164833, A118324, A098289, A075549, A016655, A019675, A161685, A144981, A168056, A004772.

log(2) - Pi/8 = Sum_{n>=1} (-1)^(n+1)*(1/n) + (-1/2)*Sum_{n>=0} (-1)^n*(1/(2*n+1)).
Empirical g.f.: x*(2+x+9*x^2+14*x^3+3*x^4+3*x^5) / ((1-x)^3*(1+x+x^2)^2). - Colin Barker, Dec 17 2015
From Bernard Schott, Aug 11 2019: (Start)
k >= 1, a(3*k) = (4*k-1) * 4*k,
k >= 0, a(3*k+1) = (4*k+1) * (4*k+2),
k >= 0, a(3*k+2) = (4*k+1) * (4*k+3).
The even terms a(3*k) and a(3*k+1) come from log(2) and the odd terms a(3*k+2) come from - Pi/8. (End)

A164833 Decimal expansion of Pi/8 - log(2)/2.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Aug 27 2009

Digits and formula given at Waldschmidt, p. 4.

			0.0461254914187515000992143621808495764868961077417606...
1/(2*3*4) + 1/(6*7*8) + 1/(10*11*12) + 1/(14*15*16) + ... [_Bruno Berselli_, Mar 17 2014]

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 46 (series n. 251).
A. J. Van Der Poorten, Effectively computable bounds for the solutions of certain Diophantine equations, Acta Arith., 33 (1977), pp. 195-207.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michel Waldschmidt, Perfect Powers: Pillai's works and their developments, Aug 27, 2009.

Cf. A001597, A019675, A016655.
Cf. A195909, A195913, A195697. - Mohammad K. Azarian, Oct 11 2011
Cf. A239362: Sum_{k>=1} 1/((3k-2)*(3k-1)*(3k)).

SetDefaultRealField(RealField(130)); R:= RealField(); (Pi(R)-4*Log(2))/8; // G. C. Greubel, Aug 11 2019

evalf[130]((Pi - 4*log(2))/8 ); # G. C. Greubel, Aug 11 2019

Join[{0},RealDigits[Pi/8-Log[2]/2,10,120][[1]]] (* Harvey P. Dale, Nov 13 2012 *)

default(realprecision, 130); (Pi - 4*log(2))/8 \\ G. C. Greubel, Aug 11 2019

numerical_approx((pi-4*log(2))/8, digits=130) # G. C. Greubel, Aug 11 2019

Equals Sum_{n>=0} Sum_{m>=0} 1/((4*n+3)^(2*m+1)).

Equals Sum_{k>=1} 1/( (4*k-2)*(4*k-1)*(4*k) ). - Bruno Berselli, Mar 17 2014

Normalized offset and leading zeros - R. J. Mathar, Sep 27 2009

A231902 Decimal expansion of Pi/4 + log(2)/2.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Nov 15 2013

			1.131971753677420964324276906548964005087042417023904082304076152823650...

L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 28 (formula 154).
Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.15, p. 269.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Michael Penn, A nice integral, YouTube video, 2022.

Cf. A003881 (Pi/4), A016655 (10*(log(2)/2)), A072691 (Pi^2/12).
Cf. A006752 (Catalan's constant)
Cf. A196521 (Pi/4-log(2)/2).
Cf. A037800.

SetDefaultRealField(RealField(100)); R:=RealField(); (Pi(R) + 2*Log(2))/4; // G. C. Greubel, Aug 24 2018

RealDigits[Pi/4 + Log[2]/2, 10, 90][[1]]

default(realprecision, 100); (Pi + 2*log(2))/4 \\ G. C. Greubel, Aug 24 2018

Equals 1 + Sum_{m>=1} -(-1)^m/(2*m*(2*m+1)) = 1 + 1/(2*3) - 1/(4*5) + 1/(6*7) - 1/(8*9) + ... .
From Amiram Eldar, Jul 16 2020: (Start)
Equals Integral_{x=1..oo} arctan(x)/x^2 dx.
Equals 1 + Integral_{x=0..1/2} log(4*x^2 + 1) dx. (End)
From Bernard Schott, Sep 07 2020: (Start)
Equals -Sum_{n>=1} (-1)^(n*(n+1)/2) / n [compare with A196521 formula].
Equals Sum_{n>=0} (32*n^2+40*n+11) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)
Equals 1 + Sum_{k>=1} A037800(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

cp's OEIS Frontend

A005843 The nonnegative even numbers: a(n) = 2n.

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A003056 n appears n+1 times. Also the array A(n,k) = n+k (n >= 0, k >= 0) read by antidiagonals. Also inverse of triangular numbers.

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A016639 Decimal expansion of log(16) = 4*log(2).

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A180447 n appears 3n+1 times.

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A196521 Decimal expansion of Pi/4-log(2)/2.

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A195697 First denominator and then numerator in a fraction expansion of log(2) - Pi/8.

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