A135503 a(n) = n*(n^2 - 1)/2.

0, 0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440

Offset: 0

Author

Cino Hilliard, Feb 09 2008

Keywords

• nonn
• easy

Comments

Previous name was: Integer values of sqrt(b) solving sqrt(d) + sqrt(b) = sqrt(c) with d^2 + b = c.

Squaring the first equation and setting the result equal to the second, we need d + b + 2*sqrt(d*b) = d^2+b -> d + 2*sqrt(d*b) = d^2 -> d^2 - d = 2*sqrt(d*b)

-> d^2*(d-1)^2 = 4*d*b -> b = d*(d-1)^2/4 -> sqrt(b) = (d-1)*sqrt(d)/2. Setting d = (n+1)^2 yields sqrt(b) = A027480(n).

This is the case k = 2 for FLTR, Fermat's Last Theorem with rational exponents 1/k: Consider x + y = x + y. Then (x^k)^(1/k) + (y^k)^(1/k) = ((x+y)^k)^(1/k).

For k > 2, there are infinitely many solutions to d^(1/k) + b^(1/k) = c^(1/k). E.g., 8^(1/3) + 27^(1/3) = 125^(1/3) at k = 3. However, in conjunction with d^2 + b = c, I could not find any nontrivial solutions.

A shifted version of A027480. - R. J. Mathar, Apr 07 2009

For n > 2, a(n) is the maximum value of the magic constant in a perimeter-magic n-gon of order n (see A342758). - Stefano Spezia, Mar 21 2021

a(n) is equal to the total number of P_3 edge-disjoint subgraphs of the complete graph on n vertices. - Samuel J. Bevins, May 09 2023

Examples

			For d = 9, b = 144, c = 225, 9^(1/2) + 144^(1/2) = 225^(1/2) and 9^2 + 144 = 225. So b^(1/2) = 12 is the 4th entry in the sequence.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• C. D. Bennet, A. M. W. Glass and G. J. Székely, Fermat's Last Theorem for Rational Exponents, Am. Math. Monthly 111 (2004), 322-329. - _R. J. Mathar_, Apr 21 2009
• Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20.

Crossrefs

Cf. A000217, A000578, A002411, A004188, A006003, A007531, A007588, A027480, A342758.

Programs

Mathematica

Array[# (#^2 - 1)/2 &, 42, 0] (* Michael De Vlieger, Feb 20 2018 *)

PARI

flt2(n,p) = { local(a,b); for(a=0,n, b = (a^3-a)/2; print1(b", ") ) }

Formula

a(n) = 3*A000292(n-1).

From R. J. Mathar Feb 20 2008: (Start)

O.g.f.: 3*x^2/(-1+x)^4.

a(n) = n*(n^2 - 1)/2 = A007531(n+1)/2. (End)

G.f.: 3*x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + (k+1)/(k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013

a(n) = A006003(n+1) - A000326(n+1). - J. M. Bergot, Dec 04 2014

E.g.f.: (1/2)* x^2 *(3 + x)*exp(x). - G. C. Greubel, Oct 15 2016

From Miquel Cerda, Dec 25 2016: (Start)

a(n) = A000578(n) - A006003(n).

a(n) = A004188(n) - A000578(n).

a(n) = A007588(n) - A004188(n). (End)

a(n) = A002411(n) - A000217(n). - Justin Gaetano, Feb 20 2018

From Amiram Eldar, Jan 09 2021: (Start)

Sum_{n>=2} 1/a(n) = 1/2.

Sum_{n>=2} (-1)^n/a(n) = 4*log(2) - 5/2. (End)

Extensions

Edited by R. J. Mathar, Apr 21 2009

New name using R. J. Mathar's formula, Joerg Arndt, Dec 05 2014