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A152811
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a(n) = 2*(n^2 + 2*n - 2).
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14
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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
a(n-1) is the maximum Zagreb index over all maximal 2-degenerate graphs with n vertices. The extremal graphs are 2-stars, so the bound also applies to 2-trees. (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024
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LINKS
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FORMULA
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G.f.: 2*(1 + x^3 - 2*x^2)/(1-x)^3.
a(n) = a(n-1) + 4*n + 2 for n>1, a(1)=2.
Sum_{n>=1} 1/a(n) = 1/3 - cot(sqrt(3)*Pi)*Pi/(4*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = -(2 + sqrt(3)*Pi*cosec(sqrt(3)*Pi))/12. (End)
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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.
The graph K_3 has 3 degree 2 vertices, so a(3-1) = 3*4 = 12.
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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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KEYWORD
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nonn,easy,less,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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