OFFSET
0,2
COMMENTS
Arises from middle column of following triangle: 4^2, 12^2, 24^2,...:
....................... 3^2 + 4^2 = 5^2
............... 10^2 + 11^2 + 12^2 = 13^2 + 14^2
........ 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2
. 36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2, etc.
REFERENCES
C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, pp. 90-92.
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).
FORMULA
G.f.: 16*x*(1+4*x+x^2)/(1-x)^5. [Colin Barker, Apr 22 2012]
a(n) = 16 * A000537(n) = 16 * (n*(n+1)/2)^2 = 16 * A000217(n)^2 = A046092(n)^2. - Bruce J. Nicholson, Jun 05 2017
a(n) = Integral_{x=1..2*n+1} (x^3-x) dx. - César Aguilera, Jun 27 2020
MATHEMATICA
CoefficientList[Series[16 x (1 + 4 x + x^2) / (1 - x)^5, {x, 0, 33}], x] (* Vincenzo Librandi, Nov 18 2016 *)
Table[(2n(n+1))^2, {n, 0, 30}] (* Harvey P. Dale, Jan 19 2019 *)
PROG
(PARI) { for (n=0, 1000, write("b060300.txt", n, " ", (2*n*(n + 1))^2); ) } \\ Harry J. Smith, Jul 03 2009
(Magma) [(2*n*(n+1))^2: n in [0..30]]; // Vincenzo Librandi, Nov 18 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jason Earls, Mar 25 2001
EXTENSIONS
Corrected the definition from 2n(n+1)^2 to (2n(n+1))^2. - Harry J. Smith, Jul 03 2009
STATUS
approved