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Search: id:A117950
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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; listen)
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OFFSET
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0,1
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COMMENT
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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 (q1qq2qqq3qqqq(AT)yahoo.com), Apr 12 2009]
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LINKS
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Eric Weisstein's World of Mathematics, Near-Square Prime
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FORMULA
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G.f.: -(3-5*x+4*x^2)/(-1+x)^3 = -2/(-1+x)^3-3/(-1+x)^2-4/(-1+x) . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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)=2*n+a(n-1)-3 (with a(1)=3) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]
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EXAMPLE
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For n=2, a(2)=2*2+3-3=4; n=3, a(3)=2*3+4-3=7; n=4, a(4)=2*4+7-3=12 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]
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MATHEMATICA
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a[n_]:=n^2+3; [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 2008]
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PROGRAM
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(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
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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 this_sequence A025047 A050342 A108700
Adjacent sequences: A117947 A117948 A117949 this_sequence A117951 A117952 A117953
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KEYWORD
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nonn,easy
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AUTHOR
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Eric Weisstein (eric(AT)weisstein.com), Apr 04, 2006
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EXTENSIONS
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Edited by N. J. A. Sloane Apr 15 2009 at the suggestion of Leroy Quet
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