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A117950
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a(n) = n^2 + 3.
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14
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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; internal format)
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OFFSET
| 0,1
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COMMENTS
| Sequence allows us to find the solutions of 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 (bhmd95(AT)yahoo.fr), 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 (mathar(AT)strw.leidenuniv.nl), Nov 20 2007
Take a square array of (n+1)-by-(n+1) dots (which correspond to the vertices of a grid of n-by-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. [From Leroy Quet, Apr 12 2009]
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 24 2010: (Start)
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.
(End)
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REFERENCES
| B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 24 2010]
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LINKS
| Eric Weisstein's World of Mathematics, Near-Square Prime
Eric Weisstein's World of Mathematics, Fan Graph. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 24 2010]
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| G.f.: -(3-5*x+4*x^2)/(-1+x)^3. - R. J. Mathar, Nov 20 2007
a(n) = ((n-3)^2 + 3*(n+1)^2)/4. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 13 2009]
a(n) = A132111(n-1,2) for n>1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 13 2010]
a(n)*a(n-1)-3 = (a(n)-n)^2 = A027688(n-1)^2. - Bruno Berselli, Dec 08 2011
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MATHEMATICA
| Table[n^2 + 3, {n, 0, 49}] (* From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 2008 *)
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PROG
| (Other) sage: [lucas_number1(3, n, -3) for n in xrange(0, 51)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]
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CROSSREFS
| a(n) = A000290(n)+3. [From Omar E. Pol (info(AT)polprimos.com), Dec 20 2008]
Cf. A028560, A005563.
For primes in this sequence see A049422.
Sequence in context: A130324 A020677 A158237 * A025047 A050342 A108700
Adjacent sequences: A117947 A117948 A117949 * A117951 A117952 A117953
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KEYWORD
| nonn,easy
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Apr 04, 2006
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EXTENSIONS
| Edited by N. J. A. Sloane Apr 15 2009 at the suggestion of Leroy Quet
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