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

1,2

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).

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

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 *)

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.

Sequence in context: A248338 A144514 A244284 * A113888 A163947 A306244

Adjacent sequences: A011942 A011943 A011944 * A011946 A011947 A011948

Keyword

nonn,easy

Author

E. K. Lloyd

Extensions

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

Status

approved