A060300 a(n) = (2*n*(n+1))^2.

0, 16, 144, 576, 1600, 3600, 7056, 12544, 20736, 32400, 48400, 69696, 97344, 132496, 176400, 230400, 295936, 374544, 467856, 577600, 705600, 853776, 1024144, 1218816, 1440000, 1690000, 1971216, 2286144, 2637376, 3027600, 3459600, 3936256, 4460544, 5035536, 5664400

Offset

0,2

Comments

Arises from middle column 4^2, 12^2, 24^2, ... of following triangle: :

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

...

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) = 4*A035287(n+1) = 4*A002378(n)^2. - Michel Marcus, May 24 2016

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

From Elmo R. Oliveira, Aug 14 2025: (Start)

E.g.f.: 4*x*(2 + x)*(2 + 6*x + x^2)*exp(x).

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = 2*A254371(n) = 8*A163102(n). (End)

From Amiram Eldar, Feb 03 2026: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/12 - 3/4.

Sum_{n>=1} (-1)^(n+1)/a(n) = 3/4 - log(2). (End)

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) a(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

Cf. A000217, A000537, A002378, A035287, A046092, A163102, A254371.

Sequence in context: A017114 A331741 A092820 * A128985 A341369 A004409

Adjacent sequences: A060297 A060298 A060299 * A060301 A060302 A060303

Keyword

nonn,easy

Author

Jason Earls, Mar 25 2001

Extensions

Name corrected by Harry J. Smith, Jul 03 2009

Status

approved