A316224 -id:A316224 - 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

A055112 a(n) = n(n+1)(2*n+1).

Original entry on oeis.org

0, 6, 30, 84, 180, 330, 546, 840, 1224, 1710, 2310, 3036, 3900, 4914, 6090, 7440, 8976, 10710, 12654, 14820, 17220, 19866, 22770, 25944, 29400, 33150, 37206, 41580, 46284, 51330, 56730, 62496, 68640, 75174, 82110, 89460, 97236, 105450

Offset: 0

Henry Bottomley, Jun 15 2000

Original name: Areas of Pythagorean triangles (X, Y, Z = Y + 1) with X^2 + Y^2 = Z^2.
a(n) is the set of possible y values for 4*x^3 + x^2 = y^2 with the x values being A002378(n). - Gary Detlefs, Feb 22 2010
This sequence is related to A028896 by a(n) = n*A028896(n) - Sum_{i = 0..n-1} A028896(i) and this is the case d = 3 in the identity n*(d*(d+1)*n*(n+1)/4) - Sum_{i = 0..n-1} d*(d+1)*i*(i+1)/4 = d*(d+1)*n*(n+1)*(2*n+1)/12. - Bruno Berselli, Mar 31 2012
Also sums of rows of natural numbers (cf. A001477) seen as triangle with an odd numbers of terms per row, see example. - Reinhard Zumkeller, Jan 24 2013
Without mentioning the connection to Pythagorean triangles, Bolker (1967) gives it as an exercise to prove that these numbers are always divisible by 6. This is easy to prove from the formula that he gives, n(n - 1)(2n - 1): obviously either n or (n - 1) must be even; then, if n is congruent to 2 mod 3 it means that (2n - 1) is a multiple of 3, otherwise either n or (n - 1) is a multiple of 3; thus both prime divisors of 6 are accounted for in a(n). - Alonso del Arte, Oct 13 2013
a(n) = n*(n+1)*(n+(n+1)) is the product of two consecutive integers multiplied by the sum of those two consecutive integers. - Charles Kusniec, Sep 04 2022

			.  n   A001477(n) as triangle with row lengths = 2*n+1   Row sums = a(n)
.  0                         0                                  0
.  1                      1  2  3                               6
.  2                   4  5  6  7  8                           30
.  3                9 10 11 12 13 14 15                        84
.  4            16 17 18 19 20 21 22 23 24                    180
.  5         25 26 27 28 29 30 31 32 33 34 35                 330
.  6      36 37 38 39 40 41 42 43 44 45 46 47 48              546
.  7   49 50 51 52 53 54 55 56 57 58 59 60 61 62 63           840 .
- _Reinhard Zumkeller_, Jan 24 2013

Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach. Mineola, New York: Dover Publications (1969, reprinted 2007): p. 7, Problem 6.5.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. Roy, The discovery of the series formula for π by Leibniz, Gregory and Nilakantha, Mathematics Magazine, 63 (5) 1990, 291-306.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005408 (X values), A046092 (Y values), A001844 (Z values), A002939 (perimeter), A033581.
Similar sequences are listed in A316224.
Cf. A000217, A000290, A005898, A083374.

Table[n(n + 1)(2n + 1), {n, 0, 39}] (* Vladimir Joseph Stephan Orlovsky, Nov 21 2010 *)
LinearRecurrence[{4,-6,4,-1},{0,6,30,84},50] (* Harvey P. Dale, Oct 02 2024 *)

PARI
```
a(n)=n*(n+1)*(2*n+1);
```

def A055112(n): return n*(n*((n<<1) + 3) + 1) # Chai Wah Wu, Nov 14 2022

a(n) = n*(n+1)*(2*n+1).
G.f.: 6*x*(1+x)/(1-x)^4. - Bruno Berselli, Mar 31 2012
From Benoit Cloitre, Apr 30 2002: (Start)
a(n) = 6*A000330(n) = A007531(2*n)/4 = 3*A000292(2*n-1)/2 = A005408(n)*A046092(n)/2 = A005408(n)*(A001844(n)-1)/2.
Sum_{n > 0} 1/a(n) = 3 - 4*log(2). (End)
a(n) = Sum_{i = 1..n} A033581(i). - Jonathan Vos Post, Mar 15 2006
a(n) = A000217(2*n)*A000217(2*n+1)/(2*n+1). - Charlie Marion, Feb 17 2012
a(n) = Sum_{i = 1..2*n + 1} (n^2 + (i-1)). - Charlie Marion, Sep 14 2012
Sum_{n >= 1} (-1)^(n+1)/a(n) = Pi - 3, due to Nilakantha, circa 1500. See Roy p. 304. - Peter Bala, Feb 19 2015
a(n) = A002378(n) * (2n+1). - Bruce J. Nicholson, Aug 31 2017
a(n) = Sum_{k=n^2..(n+1)^2-1} k. - Darío Clavijo, Jan 31 2025
E.g.f.: exp(x)*x*(6 + 9*x + 2*x^2). - Stefano Spezia, Feb 02 2025
a(n) = A005898(n) - A005408(n) = A083374(n+1) - A083374(n). - J.S. Seneschal, Jul 08 2025

A228888 a(n) = binomial(3*n + 2, 3).

Original entry on oeis.org

10, 56, 165, 364, 680, 1140, 1771, 2600, 3654, 4960, 6545, 8436, 10660, 13244, 16215, 19600, 23426, 27720, 32509, 37820, 43680, 50116, 57155, 64824, 73150, 82160, 91881, 102340, 113564, 125580, 138415, 152096, 166650, 182104, 198485, 215820, 234136, 253460

Offset: 1

Peter Bala, Sep 09 2013

			From _Bruno Berselli_, Jun 26 2018: (Start)
Including 0, row sums of the triangle:
| 0|   .................................................................. 0
| 1|   2   3   4   ..................................................... 10
| 5|   6   7   8   9  10  11   ......................................... 56
|12|  13  14  15  16  17  18  19  20  21   ............................ 165
|22|  23  24  25  26  27  28  29  30  31  32  33  34   ................ 364
|35|  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50   .... 680
...
in the first column of which we have the pentagonal numbers (A000326).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A006566 (binomial(3*n,3)) and A228887 (binomial(3*n + 1,3)).
Cf. A228889.
Similar sequences are listed in A316224.

[Binomial(3*n + 2, 3): n in [1..40]]; // Vincenzo Librandi, Sep 09 2013

Maple
```
seq(binomial(3*n+2,3), n = 1..38);
```

Table[(Binomial[3 n + 2, 3]), {n, 1, 40}] (* Vincenzo Librandi, Sep 09 2013 *)

a(n) = binomial(3*n + 2, 3) = 1/6*(3*n)*(3*n + 1)*(3*n + 2).
a(-n) = - A006566(n).
a(n) = 1/6*A228889(n).
G.f.: (10*x + 16*x^2 + x^3)/(1 - x)^4 = 10*x + 56*x^2 + 165*x^3 + ....
Sum {n >= 1} 1/a(n) = 9/2 - 3/2*log(3) - 1/2*sqrt(3)*Pi.
Sum {n >= 1} (-1)^n/a(n) = 9/2 - 4*log(2) - 1/3*sqrt(3)*Pi.

cp's OEIS Frontend

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

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A055112 a(n) = n*(n+1)*(2*n+1).

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A228888 a(n) = binomial(3*n + 2, 3).

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Examples

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Programs

Magma

Maple

Mathematica

Formula

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

A055112 a(n) = n(n+1)(2*n+1).