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A083374 a(n) = n^2 * (n^2 - 1)/2. 38
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
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.
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
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 *)
PROG
(PARI) a(n)=binomial(n^2, 2) \\ Charles R Greathouse IV, Aug 23 2011
(Magma) [n^2*(n^2-1)/2: n in [1..40]]; // Vincenzo Librandi, Sep 14 2011
(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
CROSSREFS
Sequence in context: A262467 A225262 A253652 * A264366 A061707 A253945
KEYWORD
easy,nonn
AUTHOR
Alan Sutcliffe (alansut(AT)ntlworld.com), Jun 05 2003
EXTENSIONS
Corrected and extended by T. D. Noe, Oct 25 2006
STATUS
approved

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Last modified February 28 03:01 EST 2024. Contains 370379 sequences. (Running on oeis4.)