A006002 - OEIS

%I M1920 #225 Jul 07 2024 21:24:54

%S 0,2,9,24,50,90,147,224,324,450,605,792,1014,1274,1575,1920,2312,2754,

%T 3249,3800,4410,5082,5819,6624,7500,8450,9477,10584,11774,13050,14415,

%U 15872,17424,19074,20825,22680,24642,26714,28899,31200,33620,36162,38829,41624

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

%C 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

%C 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

%C 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

%C Row sums of triangle A159797. - _Omar E. Pol_, Jul 24 2009

%C 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

%C 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

%C For n > 0: sum of n-th row in A014132, seen as a triangle read by rows. - _Reinhard Zumkeller_, Dec 12 2012

%C Partial sums of A005449. - _Omar E. Pol_, Jan 12 2013

%C 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

%C 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

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

%H T. D. Noe, <a href="/A006002/b006002.txt">Table of n, a(n) for n = 0..1000</a>

%H Quang T. Bach, Roshil Paudyal and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1510.04310">A Fibonacci analogue of Stirling numbers</a>, arXiv preprint arXiv:1510.04310 [math.CO], 2015.

%H Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.

%H Dexter Jane L. Indong and Gilbert R. Peralta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Peralta/peralta6.html">Inversions of permutations in Symmetric, Alternating, and Dihedral Gropus</a>, JIS, Vol. 11 (2008), Article 08.4.3.

%H S. M. Losanitsch, <a href="http://dx.doi.org/10.1002/cber.189703002144">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber., Vol. 30 (1897), pp. 1917-1926.

%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.

%F G.f.: x*(x + 2)/(1 - x)^4. - _Michael Somos_, Jan 30 2004

%F a(n) = (n + 1) * binomial(n+1, 2). - _Zerinvary Lajos_, Jan 10 2006

%F a(n) = A035006(n+1)/4. - _Johannes W. Meijer_, Feb 04 2010

%F a(n) = 2*binomial(n+1, 2) + 3*binomial(n+1, 3). - _Gary Detlefs_, Jun 06 2010

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Harvey P. Dale_, Aug 14 2012

%F a(n) = A000292(n) + A000330(n). - _Omar E. Pol_, Jan 11 2013

%F a(n) = A045991(n+1)/2. - _J. M. Bergot_, Aug 10 2013

%F a(n) = Sum_{j=1..n} Sum_{i=1..j} (2*j - i + 1). - _Wesley Ivan Hurt_, Nov 17 2014

%F a(n) = Sum_{i=0..n} n*(n - i) + i. - _Bruno Berselli_, Jan 13 2016

%F a(n) = t(n, A000217(n)), where t(h,k) = A000217(h) + h*k. - _Bruno Berselli_, Feb 28 2017

%F Sum_{n>0} 1/a(n) = 4 - Pi^2/3. - _Jaume Oliver Lafont_, Jul 11 2017 [corrected by _Amiram Eldar_, Jan 28 2022]

%F E.g.f.: exp(x)*x*(4 + 5*x + x^2)/2. - _Stefano Spezia_, May 21 2021

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/6 + 4*log(2) - 4. - _Amiram Eldar_, Jan 28 2022

%F From _J.S. Seneschal_, Jun 27 2024: (Start)

%F a(n) = (A002378(n)^2/2)/n = (n+1)/2 * A002378(n).

%F a(n) = A027480(n) - A000217(n). (End)

%e 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

%p seq(binomial(n+1,2)*(n+1), n=0..36); # _Zerinvary Lajos_, Apr 25 2007

%t Table[(n^3-n^2)/2, {n, 41}] (* _Zerinvary Lajos_, Mar 21 2007 *)

%t LinearRecurrence[{4,-6,4,-1}, {0,2,9,24}, 40] (* _Harvey P. Dale_, Aug 14 2012 *)

%t Accumulate @ # (# + 1) & [Range[0,50]] (* _Waldemar Puszkarz_, Jan 24 2015 *)

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

%o (Haskell) a006002 n = n * (n + 1) ^ 2 `div` 2  -- _Reinhard Zumkeller_, Dec 12 2012

%o (Magma) [n*(n+1)^2/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Nov 17 2014

%o (GAP) List([0..10^3],n->n*(n+1)^2/2); # _Muniru A Asiru_, Feb 04 2018

%Y Cf. A002411: -a(-1-n).

%Y Cf. A000217, A000292, A000330, A014132, A035006, A045991, A092364, A159797, A163274.

%Y Cf. A000914 (partial sums), A005449 (first differences).

%Y Cf. similar sequences of the type n*(n+1)*(n+k)/2 listed in A267370.

%Y Cf. A002378, A027480

%K nonn,nice,easy

%O 0,2

%A _N. J. A. Sloane_