A016639 -id:A016639 - cp's OEIS frontend

A027480 a(n) = n(n+1)(n+2)/2.

0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440

Offset: 0

Olivier Gérard and Ken Knowlton (kcknowlton(AT)aol.com)

Write the integers in groups: 0; 1,2; 3,4,5; 6,7,8,9; ... and add the groups: a(n) = Sum_{j=0..n} (A000217(n)+j), row sums of the triangular view of A001477. - Asher Auel, Jan 06 2000
With offset = 2, a(n) is the number of edges of the line graph of the complete graph of order n, L(K_n). - Roberto E. Martinez II, Jan 07 2002
Also the total number of pips on a set of dominoes of type n. (A "3" domino set would have 0-0, 0-1, 0-2, 0-3, 1-1, 1-2, 1-3, 2-2, 2-3, 3-3.) - Gerard Schildberger, Jun 26 2003. See A129533 for generalization to n-armed "dominoes". - N. J. A. Sloane, Jan 06 2016
Common sum in an (n+1) X (n+1) magic square with entries (0..n^2-1).
Alternate terms of A057587. - Jeremy Gardiner, Apr 10 2005
If Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of 4-subsets of X which have exactly one element in common with Y. Also, if Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-5)-subsets of X which have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
These numbers, starting with 3, are the denominators of the power series f(x) = (1-x)^2 * log(1/(1-x)), if the numerators are kept at 1. This sequence of denominators starts at the term x^3/3. - Miklos Bona, Feb 18 2009
a(n) is the number of triples (w,x,y) having all terms in {0..n} and satisfying at least one of the inequalities x+y < w, y+w < x, w+x < y. - Clark Kimberling, Jun 14 2012
From Martin Licht, Dec 04 2016: (Start)
Let b(n) = (n+1)(n+2)(n+3)/2 (the same sequence, but with a different offset). Then (see Arnold et al., 2006):
b(n) is the dimension of the Nédélec space of the second kind of polynomials of order n over a tetrahedron.
b(n-1) is the dimension of the curl-conforming Nédélec space of the first kind of polynomials of order n with tangential boundary conditions over a tetrahedron.
b(n) is the dimension of the divergence-conforming Nédélec space of the first kind of polynomials of order n with normal boundary conditions over a tetrahedron. (End)
After a(0), the digital root has period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9]. - Peter M. Chema, Jan 19 2017

			Row sums of n consecutive integers, starting at 0, seen as a triangle:
.
    0 |  0
    3 |  1  2
   12 |  3  4  5
   30 |  6  7  8  9
   60 | 10 11 12 13 14
  105 | 15 16 17 18 19 20

T. D. Noe, Table of n, a(n) for n = 0..1000
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica 15 (2006), 1-155.
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Solomon Gartenhaus, Odd Order Pandiagonal Latin and Magic Cubes in Three and Four Dimensions, arXiv:math/0210275 [math.CO], 2002.
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/beta(n, 3) in A061928.
Cf. A057587, A006003, A254407.
A row of array in A129533.
Cf. similar sequences of the type n*(n+1)*(n+k)/2 listed in A267370.
Similar sequences are listed in A316224.
Cf. A056923, A281258.
Third column of A003506.
Cf. A000217, A002378, A006002.
A bisection of A330298.

[n*(n+1)*(n+2)/2: n in [0..40]]; // Vincenzo Librandi, Nov 14 2014

[seq(3*binomial(n+2,3),n=0..37)]; # Zerinvary Lajos, Nov 24 2006
a := n -> add((j+n)*(n+2)/3,j=0..n): seq(a(n),n=0..35); # Zerinvary Lajos, Dec 17 2006

Table[(m^3 - m)/2, {m, 36}] (* Zerinvary Lajos, Mar 21 2007 *)
LinearRecurrence[{4,-6,4,-1},{0,3,12,30},40] (* Harvey P. Dale, Oct 10 2012 *)
CoefficientList[Series[3 x / (x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Nov 14 2014 *)
With[{nn=50},Total/@TakeList[Range[0,(nn(nn+1))/2-1],Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jun 02 2019 *)

a(n)=3*binomial(n+2,3) \\ Charles R Greathouse IV, May 23 2011

def a(n): return (n**3+3*n**2+2*n)//2 # _Torlach Rush, Jun 16 2024

a(n) = a(n-1) + A050534(n) = 3*A000292(n-1) = A050534(n) - A050534(n-1).
a(n) = n*binomial(2+n, 2). - Zerinvary Lajos, Jan 10 2006
a(n) = A007531(n+2)/2. - Zerinvary Lajos, Jul 17 2006
Starting with offset 1 = binomial transform of [3, 9, 9, 3, 0, 0, 0]. - Gary W. Adamson, Oct 25 2007
From R. J. Mathar, Apr 07 2009: (Start)
G.f.: 3*x/(x-1)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
a(n) = Sum_{i=0..n} n*(n - i) + 2*i. - Bruno Berselli, Jan 13 2016
From Ilya Gutkovskiy, Aug 07 2016: (Start)
E.g.f.: x*(6 + 6*x + x^2)*exp(x)/2.
a(n) = Sum_{k=0..n} A045943(k).
Sum_{n>=1} 1/a(n) = 1/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (8*log(2) - 5)/2 = 0.2725887222397812... = A016639/10. (End)
a(n-1) = binomial(n^2,2)/n for n > 0. - Jonathan Sondow, Jan 07 2018
For k > 1, Sum_{i=0..n^2-1} (k+i)^2 = (k*n + a(k-1))^2 + A126275(k). - Charlie Marion, Apr 23 2021

A188859 Decimal expansion of 2 - log(4).

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Apr 12 2011

Limit as n increases without bound of the probability that n mod m is less than m/2, with m chosen uniformly at random from 1..n. (As usual, 0 <= n mod m < m.)

			0.61370563888010938116553575708364686384899973127949...

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.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.236.5).
Index entries for transcendental numbers

Cf. A016639, A023416, A244009.

Mathematica
```
RealDigits[2 - Log[4], 10, 120][[1]]
```

vecextract(eval(Vec(Str(2-log(4)))),"3..")

From Amiram Eldar, Aug 15 2020: (Start)
Equals Sum_{k>=1} 1/(2*k^2 + k).
Equals -Integral_{x=0..1} log(1-x^2) dx. (End)
Equals Sum_{k>=1} A023416(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021
Equals 1/(1 + 2/(3 + 1^2/(4 + 3^2/(5 + 2^2/(6 + 4^2/(7 + 3^2/(8 + 5^2/(9 + 4^2/(10 + 6^2/(11 + ... + (n-1)^2/((2*n) + (n+1)^2/((2*n+1) + ... )))))))))))). Cf. A016639. - Peter Bala, Mar 04 2024
Equals 1/2 + Sum_{k>=1} 1/(k*(4*k^2-1)^2). - Sean A. Irvine, Apr 06 2025

A358517 Decimal expansion of 4*log(2) - 11/4.

Original entry on oeis.org

0, 2, 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

Offset: 0

Claude H. R. Dequatre, Nov 20 2022

			0.0225887222397812376689284858327062723020005374...

Michael Ian Shamos, A catalog of the real numbers (2011), p. 63.
Index entries for transcendental numbers

Cf. A016639, A035287, A242023, A242024, A358516, A358519.

Cf. A002162.

Join[{0}, RealDigits[4*Log[2] - 11/4, 10, 120][[1]]] (* Amiram Eldar, Nov 21 2022 *)

PARI
```
4*log(2) - 11/4
```

Equals Sum_{k>=1} (-1)^(k+1) * k/((k + 1)*(k + 2)*(k + 3)) [Shamos].
Equals Sum_{k>=1} (-1)^(k+1)/((k+1)^2*(k+2)^2).
Equals Sum_{k>=3} (-1)^(k+1)/A035287(k). - Amiram Eldar, Nov 21 2022

A016444 Continued fraction for log(16).

Original entry on oeis.org

2, 1, 3, 2, 1, 1, 14, 3, 9, 43, 1, 1, 7, 1, 1, 3, 1, 3, 2, 1, 2, 1, 3, 1, 1, 1, 4, 6, 1, 1, 6, 3, 1, 1, 21, 3, 1, 2, 1, 2, 4, 1, 8, 4, 1, 1, 3, 3, 28, 1, 6, 1, 4, 5, 3, 3, 5, 1, 19, 7, 1, 1, 70, 11, 8, 6, 1, 1, 3, 4, 1, 2, 7, 1, 5, 3, 2, 34

Offset: 1

N. J. A. Sloane

			2.772588722239781237668928485... = 2 + 1/(1 + 1/(3 + 1/(2 + 1/(1 + ...)))). - _Harry J. Smith_, May 17 2009

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

Cf. A016639 Decimal expansion. - Harry J. Smith, May 17 2009

ContinuedFraction[Log[16],80] (* Harvey P. Dale, May 13 2012 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(16)); for (n=1, 20000, write("b016444.txt", n, " ", x[n])); } \\ Harry J. Smith, May 17 2009

A185280 Decimal expansion of a constant appearing in the solution of Polya's 2D drunkard problem.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 23 2013

			0.882542400610606373585825728471990763930758994918621881957052934828487068186...

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; p. 425-426.

Cf. A016639, A163973.

1+(4*Log[2]-Pi)/Pi // N[#, 100]& // RealDigits // First

4*log(2)/Pi \\ Michel Marcus, Jul 28 2016

1 + Sum_{n>=1} binomial(2*n, n)^2/16^n - 1/(Pi*n).
Equals 1 + (4*log(2) - Pi)/Pi.
Equals 4*log(2)/Pi. - Michel Marcus, Jul 28 2016

a(99) corrected by Georg Fischer, Jul 12 2021

A303658 Decimal expansion of the alternating sum of the reciprocals of the triangular numbers.

Original entry on oeis.org

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

Offset: 0

Jon E. Schoenfield, Apr 28 2018

			1/1 - 1/3 + 1/6 - 1/10 + 1/15 - 1/21 + ... = 0.77258872223978123766892848583270627230200053744102...

Cf. A000217 (triangular numbers), A057711.

Apart from leading digit the same as A016639 (log(16)).

RealDigits[4*Log[2] - 2, 10, 100][[1]] (* Amiram Eldar, Aug 19 2020 *)
RealDigits[Log[16]-2,10,120][[1]] (* Harvey P. Dale, Apr 30 2022 *)

sumalt(n=1, (-1)^(n+1)*2/(n*(n+1))) \\ Michel Marcus, Apr 28 2018

PARI
```
log(16)-2 \\ Altug Alkan, May 07 2018
```

Equals log(16/e^2) = log(16) - 2.
Equals Sum_{k>=0} 1/((k+2)*2^k) = Sum_{k>=2} 1/A057711(k). - Amiram Eldar, Aug 19 2020
Equals 1 - Sum_{k>=1} 1/(k*(k+1)*(2*k+1)). - Davide Rotondo, May 24 2025

A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Jan 04 2023

			0.276427204245986573092639825616889946783740795...

John M. Campbell, Applications of a class of transformations of complex sequences, arXiv:2212.13305 [math.NT], 2022. See p. 6.

Cf. A000796, A001008, A002805, A002897, A016639, A016813, A063448, A068465, A068466, A091925, A203142, A203143.

Cf. A359532.

First[RealDigits[N[(Gamma[1/8]Gamma[3/8]/(Gamma[1/4]Gamma[3/4]))^2/(6Sqrt[2]Pi)-4Log[2]/Pi,100]]]

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8)/(Gamma(1/4)*Gamma(3/4)))^2/(6*sqrt(2)*Pi).

Equals 4*log(2)/Pi - (Gamma(1/8)*Gamma(3/8))^2/(12*sqrt(2)*Pi^3).

A365524 Decimal expansion of 4*log(2) - 5/2.

Original entry on oeis.org

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

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) = (3*k^2 - k)/(k^2 + k) is the ratio between the k-th pentagonal and triangular numbers.

			0.27258872223978123766892848583270627230200053744102...

Michael Ian Shamos, A catalog of the real numbers (2011), p. 330.
Wikipedia, Polygonal number.
Index entries for transcendental numbers

Cf. A002162.
Cf. A000217, A000326, A358517.
Essentially the same as A016639 and A358517.

RealDigits[4*Log[2] - 5/2, 10 , 100][[1]] (* Amiram Eldar, Sep 08 2023 *)

PARI
```
4*log(2)-5/2
```

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

Equals Sum_{k>=1} (-1)^(k+1)*(3*k^2 - k)/(k^2 + k).

cp's OEIS Frontend

A027480 a(n) = n*(n+1)*(n+2)/2.

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A188859 Decimal expansion of 2 - log(4).

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A358517 Decimal expansion of 4*log(2) - 11/4.

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A016444 Continued fraction for log(16).

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A185280 Decimal expansion of a constant appearing in the solution of Polya's 2D drunkard problem.

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A303658 Decimal expansion of the alternating sum of the reciprocals of the triangular numbers.

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A359533 Decimal expansion of Sum_{k>=0} (-1/64)^k*binomial(2*k, k)^3*(4*k + 1)*H_k, where H_k is the k-th harmonic number (negated).

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A027480 a(n) = n(n+1)(n+2)/2.

A359533 Decimal expansion of Sum_{k>=0} (-1/64)^kbinomial(2k, k)^3(4k + 1)*H_k, where H_k is the k-th harmonic number (negated).