A330298 -id:A330298 - cp's OEIS frontend

A006002 a(n) = n*(n+1)^2/2.

0, 2, 9, 24, 50, 90, 147, 224, 324, 450, 605, 792, 1014, 1274, 1575, 1920, 2312, 2754, 3249, 3800, 4410, 5082, 5819, 6624, 7500, 8450, 9477, 10584, 11774, 13050, 14415, 15872, 17424, 19074, 20825, 22680, 24642, 26714, 28899, 31200, 33620, 36162, 38829, 41624

Offset: 0

N. J. A. Sloane

a(n) is the largest number that is not the sum of distinct numbers of form kn+1, k >= 0. - David W. Wilson, Dec 11 1999
Sum of the nontriangular numbers between successive triangular numbers. 1, (2), 3, (4, 5), 6, (7, 8, 9), 10, (11, 12, 13, 14), 15, ... Sum of the terms in brackets. Or sum of n consecutive integers beginning with T(n) + 1, where T(n) = n(n+1)/2. - Amarnath Murthy, Aug 27 2005
Apparently this is also the splittance (as defined by Hammer & Simeone, 1977) of the Kneser graphs of the form K(n+3,2). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 13 2009
Row sums of triangle A159797. - Omar E. Pol, Jul 24 2009
The same results occur when one plots the points (1,3), (3,6), (6,10), (10,15), and so on, for all the triangular numbers and finds the area beneath. Take three consecutive triangular numbers and label them a, b, c; the area created is simply (b-a)*(b+c)/2. Thus for 6,10,15 the area beneath the line defined by the points (6,10) and (10,15) is (10-6)*(10+15)/2 = 50. - J. M. Bergot, Jun 28 2011
Let P = ab where a and b are nonequal prime numbers > 1. Let Q be the product of all divisors of P^n. Q can be expressed as P^k, where k = n*(n+1)^2/2. This follows from the fact that all divisors are of the form a^i*b^j, for i,j from 0 to n. An example is given below. In the more general case, where P is the product of m nonequal prime numbers, k = n*(n+1)^m/2. When m=3, the sequence is the same as A092364. - James A. Raymond & Douglas Raymond, Dec 04 2011
For n > 0: sum of n-th row in A014132, seen as a triangle read by rows. - Reinhard Zumkeller, Dec 12 2012
Partial sums of A005449. - Omar E. Pol, Jan 12 2013
a(n) is the sum of x (or y) coordinates for an n X n square lattice in the upper right quadrant of Z^2 whose corner points are (0, 0), (0, n), (n, 0), and (n, n). - Joseph Wheat, Feb 03 2018
a(n) is the number of permutations of [n+2] that contain exactly 2 elements which are not left-to-right minimal. E.g., the 9 permutations comprising a(2) are 2134, 2143, 3124, 3142, 4123, 4132, 2314, 2413, 3412. - Andy Niedermaier, May 07 2022

			Let P^n=6^2. The product of the divisors of 36 = 10077796 = 6^9, i.e., for n=2, k=9. - _James A. Raymond_ & Douglas Raymond, Dec 04 2011

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Quang T. Bach, Roshil Paudyal and Jeffrey B. Remmel, A Fibonacci analogue of Stirling numbers, arXiv preprint arXiv:1510.04310 [math.CO], 2015.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Dexter Jane L. Indong and Gilbert R. Peralta, Inversions of permutations in Symmetric, Alternating, and Dihedral Gropus, JIS, Vol. 11 (2008), Article 08.4.3.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Index entries for two-way infinite sequences.

Cf. A002411: -a(-1-n).
Cf. A000217, A000292, A000330, A014132, A035006, A045991, A092364, A159797, A163274.
Cf. A000914 (partial sums), A005449 (first differences).
Cf. similar sequences of the type n*(n+1)*(n+k)/2 listed in A267370.
Cf. A002378, A027480.
A bisection of A330298.

List([0..10^3],n->n*(n+1)^2/2); # Muniru A Asiru, Feb 04 2018

a006002 n = n * (n + 1) ^ 2 `div` 2  -- Reinhard Zumkeller, Dec 12 2012

[n*(n+1)^2/2 : n in [0..50]]; // Wesley Ivan Hurt, Nov 17 2014

seq(binomial(n+1,2)*(n+1), n=0..36); # Zerinvary Lajos, Apr 25 2007

Table[(n^3-n^2)/2, {n, 41}] (* Zerinvary Lajos, Mar 21 2007 *)
LinearRecurrence[{4,-6,4,-1}, {0,2,9,24}, 40] (* Harvey P. Dale, Aug 14 2012 *)
Accumulate @ # (# + 1) & [Range[0,50]] (* Waldemar Puszkarz, Jan 24 2015 *)

PARI
```
a(n)=n*(n+1)^2/2
```

G.f.: x*(x + 2)/(1 - x)^4. - Michael Somos, Jan 30 2004
a(n) = (n + 1) * binomial(n+1, 2). - Zerinvary Lajos, Jan 10 2006
a(n) = A035006(n+1)/4. - Johannes W. Meijer, Feb 04 2010
a(n) = 2*binomial(n+1, 2) + 3*binomial(n+1, 3). - Gary Detlefs, Jun 06 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Aug 14 2012
a(n) = A000292(n) + A000330(n). - Omar E. Pol, Jan 11 2013
a(n) = A045991(n+1)/2. - J. M. Bergot, Aug 10 2013
a(n) = Sum_{j=1..n} Sum_{i=1..j} (2*j - i + 1). - Wesley Ivan Hurt, Nov 17 2014
a(n) = Sum_{i=0..n} n*(n - i) + i. - Bruno Berselli, Jan 13 2016
a(n) = t(n, A000217(n)), where t(h,k) = A000217(h) + h*k. - Bruno Berselli, Feb 28 2017
Sum_{n>0} 1/a(n) = 4 - Pi^2/3. - Jaume Oliver Lafont, Jul 11 2017 [corrected by Amiram Eldar, Jan 28 2022]
E.g.f.: exp(x)*x*(4 + 5*x + x^2)/2. - Stefano Spezia, May 21 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/6 + 4*log(2) - 4. - Amiram Eldar, Jan 28 2022
From J.S. Seneschal, Jun 27 2024: (Start)
a(n) = (A002378(n)^2/2)/n = (n+1)/2 * A002378(n).
a(n) = A027480(n) - A000217(n). (End)

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

Original entry on oeis.org

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

A330299 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 3 even numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 4, 16, 20, 50, 60, 120, 140, 245, 280, 448, 504, 756, 840, 1200, 1320, 1815, 1980, 2640, 2860, 3718, 4004, 5096, 5460, 6825, 7280, 8960, 9520, 11560, 12240, 14688, 15504, 18411, 19380, 22800, 23940, 27930, 29260, 33880, 35420, 40733, 42504, 48576, 50600, 57500

Offset: 0

Enrique Navarrete, Feb 29 2020

The general formula for the number of subsets of {1..n} that contain exactly k odd and j even numbers is binomial(ceiling(n/2), k) * binomial(floor(n/2), j).

			a(7)=4 and the 4 subsets are {1,2,4,6}, {2,3,4,6}, {2,4,5,6}, {2,4,6,7}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A330298, A330300.

a[n_] := Ceiling[n/2] * Binomial[Floor[n/2], 3]; Array[a, 51, 0] (* Amiram Eldar, Mar 01 2020 *)

a(n) = ceil(n/2) * binomial(floor(n/2), 3) \\ Andrew Howroyd, Mar 01 2020

concat([0,0,0,0,0,0], Vec(x^6*(3 + x) / ((1 - x)^5*(1 + x)^4) + O(x^40))) \\ Colin Barker, Mar 02 2020

a(n) = ceiling(n/2) * binomial(floor(n/2), 3).
From Colin Barker, Mar 01 2020: (Start)
G.f.: x^6*(3 + x) / ((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n>8.
(End)
E.g.f.: (x*(15 - 3*x - 2*x^2 + x^3)*cosh(x) + (-15 + 3*x - 3*x^2 + x^4)*sinh(x))/96. - Stefano Spezia, Mar 02 2020

A330300 a(n) is the number of subsets of {1..n} that contain exactly 2 odd and 3 even numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 6, 24, 40, 100, 150, 300, 420, 735, 980, 1568, 2016, 3024, 3780, 5400, 6600, 9075, 10890, 14520, 17160, 22308, 26026, 33124, 38220, 47775, 54600, 67200, 76160, 92480, 104040, 124848, 139536, 165699, 184110, 216600, 239400, 279300, 307230, 355740, 389620, 448063

Offset: 0

Enrique Navarrete, Feb 29 2020

The general formula for the number of subsets of {1..n} that contain exactly k odd and j even numbers is binomial(ceiling(n/2), k) * binomial(floor(n/2), j).

			a(7) = 6 and the 6 subsets are {1,2,3,4,6}, {1,2,4,5,6}, {1,2,4,6,7}, {2,3,4,5,6}, {2,3,4,6,7}, {2,4,5,6,7}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Cf. A330298, A330299.

a[n_] := Binomial[Ceiling[n/2], 2] * Binomial[Floor[n/2], 3]; Array[a, 47, 0] (* Amiram Eldar, Mar 01 2020 *)

a(n) = binomial(ceil(n/2), 2) * binomial(floor(n/2), 3) \\ Andrew Howroyd, Mar 01 2020

concat([0,0,0,0,0,0], Vec(x^6*(3 + 3*x + 3*x^2 + x^3) / ((1 - x)^6*(1 + x)^5) + O(x^40))) \\ Colin Barker, Mar 02 2020

a(n) = binomial(ceiling(n/2), 2) * binomial(floor(n/2), 3).
From Colin Barker, Mar 01 2020: (Start)
G.f.: x^6*(3 + 3*x + 3*x^2 + x^3) / ((1 - x)^6*(1 + x)^5).
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) - 10*a(n-4) + 10*a(n-5) + 10*a(n-6) - 10*a(n-7) - 5*a(n-8) + 5*a(n-9) + a(n-10) - a(n-11) for n>10.
(End)
E.g.f.: (x*(-15 + 3*x - 7*x^2 + 2*x^3 + x^4)*cosh(x) + (15 - 3*x + 12*x^2 - 3*x^3 + x^4 + x^5)*sinh(x))/384. - Stefano Spezia, Mar 02 2020

A333321 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 4 even numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 4, 5, 25, 30, 90, 105, 245, 280, 560, 630, 1134, 1260, 2100, 2310, 3630, 3960, 5940, 6435, 9295, 10010, 14014, 15015, 20475, 21840, 29120, 30940, 40460, 42840, 55080, 58140, 73644, 77520, 96900, 101745, 125685, 131670, 160930, 168245, 203665, 212520

Offset: 0

Enrique Navarrete, Mar 14 2020

The general formula for the number of subsets of {1..n} that contain exactly k odd and j even numbers is binomial(ceiling(n/2), k) * binomial(floor(n/2), j).

			a(9)=5 and the 5 subsets are {1,2,4,6,8}, {2,3,4,6,8}, {2,4,5,6,8}, {2,4,6,7,8}, {2,4,6,8,9}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-10,10,10,-10,-5,5,1,-1).

Cf. A330298, A330299, A333320.

Array[Binomial[Ceiling[#], 1] Binomial[Floor[#], 4] &[#/2] &, 48, 0] (* Michael De Vlieger, Mar 14 2020 *)

concat([0,0,0,0,0,0,0,0], Vec(x^8*(4 + x) / ((1 - x)^6*(1 + x)^5) + O(x^45))) \\ Colin Barker, Mar 17 2020

a(n) = ceiling(n/2) * binomial(floor(n/2), 4).
From Colin Barker, Mar 17 2020: (Start)
G.f.: x^8*(4 + x) / ((1 - x)^6*(1 + x)^5).
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3) - 10*a(n-4) + 10*a(n-5) + 10*a(n-6) - 10*a(n-7) - 5*a(n-8) + 5*a(n-9) + a(n-10) - a(n-11) for n>10.
(End)

cp's OEIS Frontend

A006002 a(n) = n*(n+1)^2/2.

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

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A330299 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 3 even numbers.

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A330300 a(n) is the number of subsets of {1..n} that contain exactly 2 odd and 3 even numbers.

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Mathematica

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PARI

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A333321 a(n) is the number of subsets of {1..n} that contain exactly 1 odd and 4 even numbers.

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Comments

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Programs

Mathematica

PARI

Formula

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