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%I #121 Mar 18 2022 20:49:02
%S 1,6,13,22,33,46,61,78,97,118,141,166,193,222,253,286,321,358,397,438,
%T 481,526,573,622,673,726,781,838,897,958,1021,1086,1153,1222,1293,
%U 1366,1441,1518,1597,1678,1761,1846,1933,2022,2113,2206,2301
%N a(n) = n^2 - 3.
%C Number of edges in the join of two star graphs, each of order n, S_n * S_n. - _Roberto E. Martinez II_, Jan 07 2002
%C Number of vertices in the hexagonal triangle T(n-2) (see the He et al. reference). - _Emeric Deutsch_, Nov 14 2014
%C Positive X values of solutions to the equation X^3 + (X - 3)^2 + X - 6 = Y^2. To prove that X = n^2 + 4n + 1: 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 = 1 or (X + 3) must be a perfect square, so X = k^2 - 3 with k >= 2. we can put: k = n + 2, which gives: X = n^2 + 4n + 1 and Y = (n + 2)(n^2 + 4n). - _Mohamed Bouhamida_, Nov 29 2007
%C Equals binomial transform of [1, 5, 2, 0, 0, 0, ...]. - _Gary W. Adamson_, Apr 30 2008
%C Let C = 2 + sqrt(3) = 3.732...; and 1/C = 0.267...; then a(n) = (n - 2 + C) * (n - 2 + 1/C). Example: a(5) = 46 = (5 + 3.732...)*(5 + 0.267...). - _Gary W. Adamson_, Jul 29 2009
%C a(n), n >= 0, with a(0) = -3 and a(1) = -2, gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 12 for b = 2*n. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - _Wolfdieter Lang_, Aug 15 2013
%C If A(n) is a 3 X 3 Khovanski matrix having 1 below the main diagonal, n on the main diagonal, and n^3 above the main diagonal, then (Det(A(n)) - 2*n^3) / n^4 = a(n). - _Gary Detlefs_, Nov 12 2013
%C Imagine a large square containing four smaller square "holes" of equal size: Let x = large square side and y = smaller square side; considering instances where the area of this shape [x^2 - 4*y^2] equals the length of its perimeter, [4*(x + 4*y)]. When y is an integer n, the above equation is satisfied by x = 2 + 2*sqrt(a(n)). - _Peter M. Chema_, Apr 10 2016
%C a(n+1) is the number of distinct linear partitions of 2 X n grid points. A linear partition is a way to partition given points by a line into two nonempty subsets. Details can be found in Pan's link. - _Ran Pan_, Jun 06 2016
%C Numbers represented as 141 in number base B: 141(5) = 46, 141(6) = 61 and, if 'digits' larger than (B-1) are allowed, 141(2) = 13, 141(3) = 22, 141(4) = 33. - _Ron Knott_, Nov 14 2017
%H G. C. Greubel, <a href="/A028872/b028872.txt">Table of n, a(n) for n = 2..5000</a>
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/consemor.htm">Palindromic Quasipronics of the form n(n+x)</a>.
%H Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match72/n3/match72n3_835-843.pdf">Hosoya polynomials of hexagonal triangles and trapeziums</a>, MATCH, Commun. Math. Comput. Chem. 72, 2014, 835-843. - _Emeric Deutsch_, Nov 14 2014
%H Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/warmups/eV.html">Exercise V</a>, Project P.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F From _R. J. Mathar_, Apr 28 2008: (Start)
%F O.g.f.: x^2*(1 + 3*x - 2*x^2)/(1 - x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F a(n+1) = floor((n^4 + 2*n^3)/(n^2 + 1)). - _Gary Detlefs_, Feb 20 2010, corrected by _Charles R Greathouse IV_, Mar 18 2022
%F a(n) = a(n-1) + 2*n-1 (with a(2)=1). - _Vincenzo Librandi_, Nov 18 2010
%F a(n)*a(n-1) + 3 = (a(n) - n)^2 = A014209(n-2)^2. - _Bruno Berselli_, Dec 07 2011
%F a(n) = A000290(n) - 3. - _Michel Marcus_, Nov 13 2013
%F Sum_{n>=2} 1/a(n) = 2/3 - Pi*cot(sqrt(3)*Pi)/(2*sqrt(3)) = 1.476650189986093617... . - _Vaclav Kotesovec_, Apr 10 2016
%F E.g.f.: (x^2 + x - 3)*exp(x) + 2*x + 3. - _G. C. Greubel_, Jul 19 2017
%F Sum_{n>=2} (-1)^n/a(n) = -(2 + sqrt(3)*Pi*cosec(sqrt(3)*Pi))/6 = 0.8826191087... - _Amiram Eldar_, Nov 04 2020
%F From _Amiram Eldar_, Jan 29 2021: (Start)
%F Product_{n>=2} (1 + 1/a(n)) = sqrt(6)*csc(sqrt(3)*Pi)*sin(sqrt(2)*Pi).
%F Product_{n>=3} (1 - 1/a(n)) = -Pi*csc(sqrt(3)*Pi)/(4*sqrt(3)). (End)
%p A028872 := proc(n) n^2-3; end proc: # _R. J. Mathar_, Aug 23 2011
%t Range[2, 60]^2 - 3 (* or *) LinearRecurrence[{3, -3, 1}, {1, 6, 13}, 60] (* _Harvey P. Dale_, May 09 2013 *)
%o (Sage) [lucas_number1(3,n,3) for n in range(2,50)] # _Zerinvary Lajos_, Jul 03 2008
%o (PARI) a(n)=n^2-3 \\ _Charles R Greathouse IV_, Aug 23 2011
%o (PARI) x='x+O('x^99); Vec(x^2*(-1-3*x+2*x^2)/(-1+x)^3) \\ _Altug Alkan_, Apr 10 2016
%Y Essentially the same: A123968, A267874.
%Y Cf. A117950, A132411, A132414, A002522.
%K nonn,easy
%O 2,2
%A _Patrick De Geest_
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