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

0,3

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

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.

Formula

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)

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A164013 A057671 A027480 * A048088 A064181 A281434

Adjacent sequences: A135500 A135501 A135502 * A135504 A135505 A135506

Keyword

nonn,easy

Author

Cino Hilliard, Feb 09 2008

Extensions

Edited by R. J. Mathar, Apr 21 2009

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

Status

approved