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A158485
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a(n) = 14*n^2 - 1.
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5
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13, 55, 125, 223, 349, 503, 685, 895, 1133, 1399, 1693, 2015, 2365, 2743, 3149, 3583, 4045, 4535, 5053, 5599, 6173, 6775, 7405, 8063, 8749, 9463, 10205, 10975, 11773, 12599, 13453, 14335, 15245, 16183, 17149, 18143, 19165, 20215, 21293, 22399
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OFFSET
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1,1
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COMMENTS
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The identity (14*n^2-1)^2-(49*n^2-7)*(2*n)^2=1 can be written as a(n)^2-A158484(n)*A005843(n)^2=1.
Sequence found by reading the line from 13, in the direction 13, 55,..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 13 2011
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).
G.f: x*(-13-16*x+x^2)/(x-1)^3.
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(14))*cot(Pi/sqrt(14)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(14))*csc(Pi/sqrt(14)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(14))*csc(Pi/sqrt(14)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(14))*sin(Pi/sqrt(7))/sqrt(2). (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {13, 55, 125}, 50]
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PROG
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(Magma) I:=[13, 55, 125]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
(PARI) a(n) = 14*n^2-1
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CROSSREFS
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Cf. A005843, A158484.
Sequence in context: A027000 A198160 A029531 * A274973 A005902 A051798
Adjacent sequences: A158482 A158483 A158484 * A158486 A158487 A158488
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KEYWORD
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nonn,easy
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AUTHOR
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Vincenzo Librandi, Mar 20 2009
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STATUS
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approved
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