A011945 Areas of almost-equilateral Heronian triangles (integral side lengths m-1, m, m+1 and integral area).

0, 6, 84, 1170, 16296, 226974, 3161340, 44031786, 613283664, 8541939510, 118973869476, 1657092233154, 23080317394680, 321467351292366, 4477462600698444, 62363009058485850, 868604664218103456, 12098102289994962534, 168504827395711372020, 2346969481249964245746

Offset: 1

Author

E. K. Lloyd

Keywords

• nonn
• easy

Comments

Corresponding m's are in A016064. Corresponding values of lesser side give A016064.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..890
• Tanya Khovanova, Recursive Sequences
• E. Keith Lloyd, The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles, Math. Gaz. vol 81 (1997), 231-243.
• Eric Weisstein's World of Mathematics, Heronian Triangle
• Wikipedia, Heronian triangle
• P. Yiu, Heron triangles with consecutive sides, Recreational Mathematics, Chap. 9.3, pp. 80/360. (This is a download of 360 pages.)
• Index entries for linear recurrences with constant coefficients, signature (14,-1).

Crossrefs

Equals 6 * A007655(n+1).

Cf. this sequence (areas), A334277 (perimeters).

Cf. A003500 (middle side lengths), A016064 (smallest side lengths), A335025 (largest side lengths).

Cf. A102341, A103974, A103975.

Programs

Mathematica

CoefficientList[Series[6 x/(1 - 14 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 15 2013 *)
LinearRecurrence[{14,-1},{0,6},20] (* Harvey P. Dale, Jan 24 2015 *)

Formula

s(n) = floor((a+1)/4)*sqrt(3*(a+3)*(a-1)), where a = A016064(n). - Zak Seidov, Feb 23 2005

a(n) = 14*a(n-1) - a(n-2); a(1) = 0, a(2) = 6.

G.f.: 6*x^2/(1 - 14*x + x^2). - Philippe Deléham, Nov 17 2008

a(n) = (s/4)*((7 + 4*s)^n - (7 - 4*s)^n), where s = sqrt(3). - Zak Seidov, Apr 02 2014

E.g.f.: 6 - exp(7*x)*(12*cosh(4*sqrt(3)*x) - 7*sqrt(3)*sinh(4*sqrt(3)*x))/2. - Stefano Spezia, Dec 12 2022

Extensions

Entry revised by N. J. A. Sloane, Feb 03 2007