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A011945 Area of triangles with integral side lengths m-1, m, m+1 and integral area. 12
0, 6, 84, 1170, 16296, 226974, 3161340, 44031786, 613283664, 8541939510, 118973869476, 1657092233154, 23080317394680, 321467351292366, 4477462600698444, 62363009058485850, 868604664218103456, 12098102289994962534, 168504827395711372020, 2346969481249964245746 (list; graph; refs; listen; history; text; internal format)
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.

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

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. A003500, A102341, A103974, A103975, A016064.

Sequence in context: A248338 A144514 A244284 * A113888 A163947 A277304

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

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Last modified November 19 11:04 EST 2017. Contains 294936 sequences.