A083374 a(n) = n^2 * (n^2 - 1)/2.

0, 6, 36, 120, 300, 630, 1176, 2016, 3240, 4950, 7260, 10296, 14196, 19110, 25200, 32640, 41616, 52326, 64980, 79800, 97020, 116886, 139656, 165600, 195000, 228150, 265356, 306936, 353220, 404550, 461280, 523776, 592416, 667590, 749700, 839160, 936396

Offset: 1

Author

Alan Sutcliffe (alansut(AT)ntlworld.com), Jun 05 2003

Keywords

• easy
• nonn

Comments

Triangular numbers t_n as n runs through the squares.

Partial sums of A055112: If one generated Pythagorean primitive triangles from n, n+1, then the collective areas of n of them would be equal to the numbers in this sequence. The sum of the first three triangles is 6+30+84 = 120 which is the third nonzero term of the sequence. - J. M. Bergot, Jul 14 2011

Second leg of Pythagorean triangles with smallest side a cube: A000578(n)^2 + a(n)^2 = A037270(n)^2. - Martin Renner, Nov 12 2011

a(n) is the number of segments on an n X n grid or geoboard. - Martin Renner, Apr 17 2014

Consider the partitions of 2n into two parts (p,q). Then a(n) is the total volume of the family of rectangular prisms with dimensions p, q and |q-p|. - Wesley Ivan Hurt, Apr 15 2018

References

• Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 55.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Somaya Barati, Beáta Bényi, Abbas Jafarzadeh and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
• Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A000578, A002415, A006002, A006011, A008911, A037270, A047928, A055112.

Programs

Magma

[n^2*(n^2-1)/2: n in [1..40]]; // Vincenzo Librandi, Sep 14 2011

Magma

A000217:=func; [&+[k*A000217(2*k+1): k in [0..n-1]]: n in [1..40]]; // Bruno Berselli, Sep 04 2013

Maple

A083374 := proc(n) n^2*(n^2-1)/2 ; end proc: # R. J. Mathar, Aug 23 2011

Mathematica

Table[n^2*(n^2-1)/2, {n,40}] (* T. D. Noe, Oct 25 2006 *)

PARI

a(n)=binomial(n^2,2) \\ Charles R Greathouse IV, Aug 23 2011

Formula

a(n) = (n + 1) * A006002(n).

a(n) = A047928(n)/2 = A002415(n+1)*6 = A006011(n+1)*2 = A008911(n+1)*3. - Zerinvary Lajos, May 09 2007

a(n) = binomial(n^2,2), n>=1. - Zerinvary Lajos, Jan 07 2008

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5. - R. J. Mathar, Apr 10 2009

G.f.: -6*x^2*(1+x)/(x-1)^5. - R. J. Mathar, Apr 10 2009

Sum_{n>1} 1/a(n) = (21 - 2*Pi^2)/6. - Enrique Pérez Herrero, Apr 01 2013

a(n) = Sum_{k=0..n-1} k*A000217(2*k+1). - Bruno Berselli, Sep 04 2013

a(n) = 2*A000217(n-1)*A000217(n). - Gionata Neri, May 04 2015

a(n) = Sum_{i=1..n^2-1} i. - Wesley Ivan Hurt, Nov 24 2015

E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/2. - Stefano Spezia, Jun 06 2021

Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 - 3/2. - Amiram Eldar, Nov 02 2021

Extensions

Corrected and extended by T. D. Noe, Oct 25 2006