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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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

Examples

			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
		

References

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

Links

• 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.

Crossrefs

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.

Programs

GAP

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

Haskell

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

Magma

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

Maple

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

Mathematica

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

Formula

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)