A117950 a(n) = n^2 + 3.

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

Offset: 0

Author

Eric W. Weisstein, Apr 04 2006

Keywords

• nonn
• easy

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 = (X-5)(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 n-node 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 (3n-1)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

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969. - _Emeric Deutsch_, Sep 24 2010
• Eric Weisstein's World of Mathematics, Near-Square Prime
• Eric Weisstein's World of Mathematics, Fan Graph. - _Emeric Deutsch_, Sep 24 2010
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A027688, A028560, A005563, A132111.

For primes in this sequence see A049422.

Programs

Mathematica

Table[n^2 + 3, {n, 0, 49}] (* Vladimir Joseph Stephan Orlovsky, Dec 15 2008 *)

PARI

a(n)=n^2+3 \\ Charles R Greathouse IV, Sep 24 2015

Sage

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

Formula

G.f.: (3 - 5*x + 4*x^2)/(1-x)^3. - R. J. Mathar, Nov 20 2007

a(n) = A000290(n) + 3. - Omar E. Pol, Dec 20 2008

a(n) = ((n-3)^2 + 3*(n+1)^2)/4. - Reinhard Zumkeller, Feb 13 2009

a(n) = A132111(n-1,2) for n>1. - Reinhard Zumkeller, Aug 10 2007

a(n) = ceiling((n+1/n)^2), n>0. - Vincenzo Librandi, Oct 19 2011

a(n) = 2*n + a(n-1) - 1 (with a(0)=3). - Vincenzo Librandi, Nov 13 2010

a(n)*a(n-1) - 3 = (a(n)-n)^2 = A027688(n-1)^2. - Bruno Berselli, Dec 08 2011

From Amiram Eldar, Jul 21 2020: (Start)

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)

From Amiram Eldar, Jan 29 2021: (Start)

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)

E.g.f.: exp(x)*(3 + x + x^2). - Stefano Spezia, Aug 26 2024

Extensions

Edited by N. J. A. Sloane Apr 15 2009 at the suggestion of Leroy Quet