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A078371
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(2*n+5)*(2*n+1).
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14
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5, 21, 45, 77, 117, 165, 221, 285, 357, 437, 525, 621, 725, 837, 957, 1085, 1221, 1365, 1517, 1677, 1845, 2021, 2205, 2397, 2597, 2805, 3021, 3245, 3477, 3717, 3965, 4221, 4485, 4757, 5037, 5325, 5621, 5925, 6237, 6557, 6885, 7221, 7565, 7917, 8277, 8645
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OFFSET
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0,1
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COMMENTS
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This is the generic form of D in the (nontrivially) solvable Pell equation x^2 - D*y^2 = +4. See A077428 and A078355.
Consider all primitive Pythagorean triples (a,b,c) with c-a=8, sequence gives values of a. (Corresponding values for b are A017113(n), while c follows A078370(n).) - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 19 2004
a(n)=A061037(2n+1)=(2n+3)^2-4. For A061037: a(2n+1)=(2n+1)*(2n+5)=(2n+3)^2-4. From Balmer spectrum of hydrogen. [From Paul Curtz, Sep 24 2008]
Contribution from Vincenzo Librandi, Aug 08 2010: (Start)
The identity (4n^3+18n^2+24n+9)^2-(4n^2+12n+5)*(2n^2+6n+4)^2=1 (see Ramasamy's paper in link) can be written as A141530(n+2)^2-a(n)*A046092(n+1)^2 = 1.
a(n)^3+6*a(n)^2+9*a(n)+4 is a square: in fact a(n)^3+6*a(n)^2+9*a(n)+4 = (a(n)+1)^2*(a(n)+4) with a(n)=(2n+3)^2-4 (see Paul Curtz above). (End)
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LINKS
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Bruno Berselli, Table of n, a(n) for n = 0..1000
A. M. S. Ramasamy, Polynomial solutions for the Pell's equation, Indian Journal of Pure and Applied Mathematics 25 (1994), p. 578.
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = (2*n+5)*(2*n+1) = 8*(binomial(n+2, 2)-1)+5, hence subsequence of A004770 (5 (mod 8) numbers).
G.f.: (5+6*x-3*x^2)/(1-x)^3.
a(n) = 8*(n+1)+a(n-1), (with a(0)=5). - Vincenzo Librandi, Aug 08 2010
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MAPLE
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seq((2*n+5)*(2*n+1), n=0..48); (Deutsch)
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CROSSREFS
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Subsequence of A077425 (D values (not a square) for which Pell x^2 - D*y^2 = +4 is solvable in positive integers).
Cf. A017113, A078370.
Sequence in context: A054286 A031292 A147331 * A049741 A166010 A146846
Adjacent sequences: A078368 A078369 A078370 * A078372 A078373 A078374
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Nov 29 2002
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EXTENSIONS
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More terms from Emeric Deutsch, Feb 24 2005
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STATUS
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approved
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