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A067900
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a(n) = 14*a(n-1) - a(n-2); a(0) = 0, a(1) = 8.
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5
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0, 8, 112, 1560, 21728, 302632, 4215120, 58709048, 817711552, 11389252680, 158631825968, 2209456310872, 30773756526240, 428623135056488, 5969950134264592, 83150678744647800, 1158139552290804608
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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, verify the triangle relation (a^2 + b^2 + c^2 + d^2)^2 = 3*(a^4 + b^4 + c^4 + d^4). Thus d corresonds 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.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
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FORMULA
| a(n)=-(1/3)*sqrt(3)*[7-4*sqrt(3)]^n+(1/3)*sqrt(3)*[7+4*sqrt(3)]^n, with n>=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Oct 02 2008]
G.f.: 8x/(1-14*x+x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2008]
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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:
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CROSSREFS
| Cf. A067902.
First differences of A045899.
Equals 8 * A007655(n+1).
Sequence in context: A075851 A053536 A139741 * A067414 A034689 A010041
Adjacent sequences: A067897 A067898 A067899 * A067901 A067902 A067903
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KEYWORD
| nonn
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AUTHOR
| Lekraj Beedassy (blekraj(AT)yahoo.com), May 13 2003
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EXTENSIONS
| More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net) and James A. Sellers (sellersj(AT)math.psu.edu), May 19, 2003
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