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A152811
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a(n) = 2*(n^2 + 2*n - 2).
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9
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2, 12, 26, 44, 66, 92, 122, 156, 194, 236, 282, 332, 386, 444, 506, 572, 642, 716, 794, 876, 962, 1052, 1146, 1244, 1346, 1452, 1562, 1676, 1794, 1916, 2042, 2172, 2306, 2444, 2586, 2732, 2882, 3036, 3194, 3356, 3522, 3692, 3866, 4044, 4226, 4412, 4602
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OFFSET
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1,1
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COMMENTS
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Positive numbers k such that 2*k+12 is a square. [Comment simplified by Zak Seidov, Jan 14 2009]
Sequence gives positive x values of solutions (x, y) to the Diophantine equation 2*x^3+12*x^2 = y^2. Corresponding y values are 8*A154560. There are three other solutions: (0, 0), (-4, 8) and (-6, 0).
From a(2) onwards, third subdiagonal of triangle defined in A144562.
Also, nonnegative numbers of the form (m+sqrt(-3))^2 + (m-sqrt(-3))^2. [Bruno Berselli, Mar 13 2015]
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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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G.f.: 2*(1 + x^3 - 2*x^2)/(1-x)^3.
a(n) = 2*A028872(n+1).
a(n) = a(n-1) + 4*n + 2 for n>1, a(1)=2.
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EXAMPLE
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a(4) = 2*(4^2 + 2*4 - 2) = 44 = 2*22 = 2*A028872(5); 2*44^3 + 12*44^2 = 193600 = 440^2 is a square.
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MATHEMATICA
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a[n_]:=Sqrt[2*n^3+12*n^2]; lst={}; Do[If[Floor[a[n]]==a[n], AppendTo[lst, n]], {n, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 13 2009 *)
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PROG
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(MAGMA) [ 2*(n^2+2*n-2) : n in [1..47] ];
(PARI) {m=4700; for(n=1, m, if(issquare(2*n^3+12*n^2), print1(n, ", ")))}
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CROSSREFS
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Cf. A028872 (n^2-3), A154560 ((n+3)^2*n/2+1), A144562 (triangle T(m,n) = 2m*n+m+n-1).
Sequence in context: A031048 A294554 A098707 * A294552 A294170 A102960
Adjacent sequences: A152808 A152809 A152810 * A152812 A152813 A152814
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KEYWORD
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nonn,easy
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AUTHOR
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Vincenzo Librandi, Dec 17 2008
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EXTENSIONS
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Edited and extended by Klaus Brockhaus, Jan 12 2009
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STATUS
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approved
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