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%I #94 Sep 08 2022 08:45:10
%S 0,6,36,120,300,630,1176,2016,3240,4950,7260,10296,14196,19110,25200,
%T 32640,41616,52326,64980,79800,97020,116886,139656,165600,195000,
%U 228150,265356,306936,353220,404550,461280,523776,592416,667590,749700,839160,936396
%N a(n) = n^2 * (n^2 - 1)/2.
%C Triangular numbers t_n as n runs through the squares.
%C 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
%C Second leg of Pythagorean triangles with smallest side a cube: A000578(n)^2 + a(n)^2 = A037270(n)^2. - _Martin Renner_, Nov 12 2011
%C a(n) is the number of segments on an n X n grid or geoboard. - _Martin Renner_, Apr 17 2014
%C 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
%D Albert H. Beiler, Recreations in the theory of numbers, New York: Dover, (2nd ed.) 1966, p. 106, table 55.
%H Vincenzo Librandi, <a href="/A083374/b083374.txt">Table of n, a(n) for n = 1..10000</a>
%H Somaya Barati, Beáta Bényi, Abbas Jafarzadeh and Daniel Yaqubi, <a href="https://arxiv.org/abs/1812.02955">Mixed restricted Stirling numbers</a>, arXiv:1812.02955 [math.CO], 2018.
%H Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = (n + 1) * A006002(n).
%F a(n) = A047928(n)/2 = A002415(n+1)*6 = A006011(n+1)*2 = A008911(n+1)*3. - _Zerinvary Lajos_, May 09 2007
%F a(n) = binomial(n^2,2), n>=1. - _Zerinvary Lajos_, Jan 07 2008
%F 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
%F G.f.: -6*x^2*(1+x)/(x-1)^5. - _R. J. Mathar_, Apr 10 2009
%F Sum_{n>1} 1/a(n) = (21 - 2*Pi^2)/6. - _Enrique Pérez Herrero_, Apr 01 2013
%F a(n) = Sum_{k=0..n-1} k*A000217(2*k+1). - _Bruno Berselli_, Sep 04 2013
%F a(n) = 2*A000217(n-1)*A000217(n). - _Gionata Neri_, May 04 2015
%F a(n) = Sum_{i=1..n^2-1} i. - _Wesley Ivan Hurt_, Nov 24 2015
%F E.g.f.: exp(x)*x^2*(6 + 6*x + x^2)/2. - _Stefano Spezia_, Jun 06 2021
%F Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 - 3/2. - _Amiram Eldar_, Nov 02 2021
%p A083374 := proc(n) n^2*(n^2-1)/2 ; end proc: # _R. J. Mathar_, Aug 23 2011
%t Table[n^2*(n^2-1)/2, {n,40}] (* _T. D. Noe_, Oct 25 2006 *)
%o (PARI) a(n)=binomial(n^2,2) \\ _Charles R Greathouse IV_, Aug 23 2011
%o (Magma) [n^2*(n^2-1)/2: n in [1..40]]; // _Vincenzo Librandi_, Sep 14 2011
%o (Magma) A000217:=func<i | i*(i+1)/2>; [&+[k*A000217(2*k+1): k in [0..n-1]]: n in [1..40]]; // _Bruno Berselli_, Sep 04 2013
%Y Cf. A000217, A000578, A002415, A006002, A006011, A008911, A037270, A047928, A055112.
%K easy,nonn
%O 1,2
%A Alan Sutcliffe (alansut(AT)ntlworld.com), Jun 05 2003
%E Corrected and extended by _T. D. Noe_, Oct 25 2006
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