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A067900 a(n) = 14*a(n-1) - a(n-2); a(0) = 0, a(1) = 8. 6
0, 8, 112, 1560, 21728, 302632, 4215120, 58709048, 817711552, 11389252680, 158631825968, 2209456310872, 30773756526240, 428623135056488, 5969950134264592, 83150678744647800, 1158139552290804608 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Solves for y in x^2 - 3*y^2 = 4. Quadruples (a=b-y, b, c=b+y, d), with b=y^2 + 1 and d=x*y, where (x, y) solves x^2 - 3*y^2 = 4, satisfy the triangle relation (a^2 + b^2 + c^2 + d^2)^2 = 3*(a^4 + b^4 + c^4 + d^4). Thus d corresponds to the distance sum of the Fermat (or Torriccelli) point from its vertices in a triangle whose sides are in A.P. with middle side b and common difference y.

LINKS

Table of n, a(n) for n=0..16.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (14,-1).

FORMULA

a(n) = -(1/3)*sqrt(3)*[7-4*sqrt(3)]^n+(1/3)*sqrt(3)*[7+4*sqrt(3)]^n, with n>=0. [Paolo P. Lava, Oct 02 2008]

G.f.: 8x/(1-14*x+x^2). [Philippe Deléham, Nov 17 2008]

MAPLE

a := proc(n) option remember: if n=0 then RETURN(0) fi: if n=1 then RETURN(8) fi: 14*a(n-1)-a(n-2): end: for n from 0 to 30 do printf(`%d, `, a(n)) od:

MATHEMATICA

LinearRecurrence[{14, -1}, {0, 8}, 17] (* Jean-François Alcover, Sep 19 2017 *)

CROSSREFS

Cf. A067902.

First differences of A045899.

Equals 8 * A007655(n+1).

Sequence in context: A270111 A053536 A139741 * A067414 A265665 A219184

Adjacent sequences:  A067897 A067898 A067899 * A067901 A067902 A067903

KEYWORD

nonn,easy

AUTHOR

Lekraj Beedassy, May 13 2003

STATUS

approved

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Last modified November 21 14:44 EST 2017. Contains 295002 sequences.