

A117950


a(n) = n^2 + 3.


32



3, 4, 7, 12, 19, 28, 39, 52, 67, 84, 103, 124, 147, 172, 199, 228, 259, 292, 327, 364, 403, 444, 487, 532, 579, 628, 679, 732, 787, 844, 903, 964, 1027, 1092, 1159, 1228, 1299, 1372, 1447, 1524, 1603, 1684, 1767, 1852, 1939, 2028, 2119, 2212, 2307, 2404, 2503
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

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
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
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
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


LINKS



FORMULA

G.f.: (3  5*x + 4*x^2)/(1x)^3.  R. J. Mathar, Nov 20 2007
Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/6.
Sum_{n>=0} (1)^n/a(n) = (1 + (sqrt(3)*Pi)*csch(sqrt(3)*Pi))/6. (End)
Product_{n>=0} (1 + 1/a(n)) = 2*csch(sqrt(3)*Pi)*sinh(2*Pi)/sqrt(3).
Product_{n>=0} (1  1/a(n)) = sqrt(2/3)*csch(sqrt(3)*Pi)*sinh(sqrt(2)*Pi). (End)


MATHEMATICA



PROG

(Sage) [lucas_number1(3, n, 3) for n in range(0, 51)] # Zerinvary Lajos, May 16 2009


CROSSREFS

For primes in this sequence see A049422.


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



