%I
%S 3,4,7,12,19,28,39,52,67,84,103,124,147,172,199,228,259,292,327,364,
%T 403,444,487,532,579,628,679,732,787,844,903,964,1027,1092,1159,1228,
%U 1299,1372,1447,1524,1603,1684,1767,1852,1939,2028,2119,2212,2307,2404,2503
%N a(n) = n^2 + 3.
%C Nonnegative X values of solutions to the equation X^3  (X + 3)^2 + X + 6 = Y^2. To prove that X = n^2 + 3: Y^2 = X^3  (X + 3)^2 + X + 6 = X^3  X^2  5X  3 = (X  3)(X^2 + 2X + 1) = (X  3)*(X + 1)^2 it means: (X  3) must be a perfect square, so X = n^2 + 3 and Y = n(n^2 + 4).  _Mohamed Bouhamida_, Nov 12 2007
%C An equivalent technique of integer factorization would work, for example, for the equation X^3  3*X^2  9*X  5 = (X5)(X+1)^2 = Y^2, looking for perfect squares of the form X  5 = n^2.  _R. J. Mathar_, Nov 20 2007
%C Take a square array of (n+1) X (n+1) dots (which correspond to the vertices of a grid of n X n squares). Connect the dots with vertical and horizontal line segments of any length so that each dot is connected to each of its orthogonal neighbors, and so that no line segment crosses any previously drawn line segment. Then the minimum number of line segments is a(n), for n >= 1.  _Leroy Quet_, Apr 12 2009
%C a(n) is also the Wiener index of the double fan graph F(n). The double fan graph F(n) is defined as the graph obtained by joining each node of an nnode path graph with two additional nodes. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph. The Wiener polynomial of the graph F(n) is (3n1)t + (1/2)(n^2  3n + 4)t^2. Example: a(3)=12 because the corresponding double fan graph is the wheel graph on 5 nodes OABCD, O being the center of the wheel. Its Wiener index = number of edges + AC +BD = 8 + 2 + 2 = 12.  _Emeric Deutsch_, Sep 24 2010
%H Ivan Panchenko, <a href="/A117950/b117950.txt">Table of n, a(n) for n = 0..1000</a>
%H B. E. Sagan, YN. Yeh and P. Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097461X(1996)60:5<959::AIDQUA2>3.0.CO;2W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959969.  _Emeric Deutsch_, Sep 24 2010
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NearSquarePrime.html">NearSquare Prime</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FanGraph.html">Fan Graph</a>.  _Emeric Deutsch_, Sep 24 2010
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,1).
%F G.f.: (3  5*x + 4*x^2)/(1x)^3.  _R. J. Mathar_, Nov 20 2007
%F a(n) = A000290(n) + 3.  _Omar E. Pol_, Dec 20 2008
%F a(n) = ((n3)^2 + 3*(n+1)^2)/4.  _Reinhard Zumkeller_, Feb 13 2009
%F a(n) = A132111(n1,2) for n>1.  _Reinhard Zumkeller_, Aug 10 2007
%F a(n) = ceiling((n+1/n)^2), n>0.  _Vincenzo Librandi_, Oct 19 2011
%F a(n) = 2*n + a(n1)  1 (with a(0)=3).  _Vincenzo Librandi_, Nov 13 2010
%F a(n)*a(n1)  3 = (a(n)n)^2 = A027688(n1)^2.  _Bruno Berselli_, Dec 08 2011
%F From _Amiram Eldar_, Jul 21 2020: (Start)
%F Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/6.
%F Sum_{n>=0} (1)^n/a(n) = (1 + (sqrt(3)*Pi/2)*coth(sqrt(3)*Pi/2))/6. (End)
%F From _Amiram Eldar_, Jan 29 2021: (Start)
%F Product_{n>=0} (1 + 1/a(n)) = 2*csch(sqrt(3)*Pi)*sinh(2*Pi)/sqrt(3).
%F Product_{n>=0} (1  1/a(n)) = sqrt(2/3)*csch(sqrt(3)*Pi)*sinh(sqrt(2)*Pi). (End)
%t Table[n^2 + 3, {n, 0, 49}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 15 2008 *)
%o (Sage) [lucas_number1(3,n,3) for n in range(0, 51)] # _Zerinvary Lajos_, May 16 2009
%o (PARI) a(n)=n^2+3 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000290, A027688, A028560, A005563, A132111.
%Y For primes in this sequence see A049422.
%K nonn,easy
%O 0,1
%A _Eric W. Weisstein_, Apr 04 2006
%E Edited by _N. J. A. Sloane_ Apr 15 2009 at the suggestion of _Leroy Quet_
