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 A135503 a(n) = n*(n^2 - 1)/2. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 is an infinite number of 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 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 C. D. Bennet, A. M. W. Glass and G. J. SzĂ©kely, Fermat's Last Theorem for Rational Exponents, Am. Math. Monthly 111 (2004), 322-329. - R. J. Mathar, Apr 21 2009 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 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 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

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Last modified February 15 22:28 EST 2019. Contains 320138 sequences. (Running on oeis4.)