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A154357
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25n^2 - 14n + 2.
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7
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2, 13, 74, 185, 346, 557, 818, 1129, 1490, 1901, 2362, 2873, 3434, 4045, 4706, 5417, 6178, 6989, 7850, 8761, 9722, 10733, 11794, 12905, 14066, 15277, 16538, 17849, 19210, 20621, 22082, 23593, 25154, 26765, 28426, 30137, 31898, 33709, 35570, 37481, 39442, 41453, 43514
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| The identity (1250n^2-700n+99)^2-(25n^2-14n+2)*(250n-70)^2=1 can be written as A154359(n)^2-a(n)*A154361(n)^2=1.
Numbers of the form (4n-1)^2+(3n-1)^2. - Bruno Berselli, Dec 11 2011
Contribution from Bruno Berselli, Dec 13 2011: (Start)
More generally, considering together this sequence and A154355, A154358-A154361, for
r = (1/4)*(1250*(n-1)*(n-2)+75*(2*n-3)(-1)^n+321) with n>=0, i.e. the interleaving of A154358 and A154359 (649, 99, 99, 649, 2049, 3699,...)
s = (5/2)*(50*n+3*(-1)^n-75), the interleaving of A154360 and A154361 (-180, -70, 70, 180, 320, 430,...)
t = (1/8)*(50*(n-1)*(n-2)+3*(2*n-3)*(-1)^n+13), the interleaving of A154355 and A154357 (13, 2, 2, 13, 41, 74,...)
we verify that r^2-t*s^2=1.
For n even we obtain (1250n^2-1800n+649)^2-(25n^2-36n+13)*(250n-180)^2=1; for n odd we have the identity shown in the first comment. (End)
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000 Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| G.f.: -(2+7*x+41*x^2)/(x-1)^3. - R. J. Mathar, Jan 05 2011
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Feb 08 2012
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MATHEMATICA
| LinearRecurrence[{3, -3, 1}, {2, 13, 74}, 40] (* Vincenzo Librandi, Feb 08 2012 *)
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PROG
| (PARI) a(n)=25*n^2-14*n+2 \\ Charles R Greathouse IV, Dec 23 2011
(MAGMA) I:=[2, 13, 74]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 08 2012
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CROSSREFS
| Cf. A154355, A154358-A154361.
Sequence in context: A109112 A163190 A004027 * A161130 A192700 A007509
Adjacent sequences: A154354 A154355 A154356 * A154358 A154359 A154360
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 08 2009
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EXTENSIONS
| One entry and offset corrected from R. J. Mathar, Jan 05 2011
First comment rewritten from Bruno Berselli, Dec 11 2011
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