

A028872


a(n) = n^2  3.


27



1, 6, 13, 22, 33, 46, 61, 78, 97, 118, 141, 166, 193, 222, 253, 286, 321, 358, 397, 438, 481, 526, 573, 622, 673, 726, 781, 838, 897, 958, 1021, 1086, 1153, 1222, 1293, 1366, 1441, 1518, 1597, 1678, 1761, 1846, 1933, 2022, 2113, 2206, 2301
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OFFSET

2,2


COMMENTS

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
Number of vertices in the hexagonal triangle T(n2) (see the He et al. reference).  Emeric Deutsch, Nov 14 2014
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
Equals binomial transform of [1, 5, 2, 0, 0, 0, ...].  Gary W. Adamson, Apr 30 2008
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
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
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
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
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
Numbers represented as 141 in number base B: 141(5) = 46, 141(6) = 61 and, if 'digits' larger than (B1) are allowed, 141(2) = 13, 141(3) = 22, 141(4) = 33.  Ron Knott, Nov 14 2017


LINKS

G. C. Greubel, Table of n, a(n) for n = 2..5000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
Q. H. He, J. Z. Gu, S. J. Xu, and W. H. Chan, Hosoya polynomials of hexagonal triangles and trapeziums, MATCH, Commun. Math. Comput. Chem. 72, 2014, 835843.  Emeric Deutsch, Nov 14 2014
Ran Pan, Exercise V, Project P.
Eric Weisstein's World of Mathematics, NearSquare Prime.
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

From R. J. Mathar, Apr 28 2008: (Start)
O.g.f.: x^2*(1 + 3*x  2*x^2)/(1  x)^3.
a(n) = 3*a(n1)  3*a(n2) + a(n3). (End)
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
a(n) = a(n1) + 2*n1 (with a(2)=1).  Vincenzo Librandi, Nov 18 2010
a(n)*a(n1) + 3 = (a(n)  n)^2 = A014209(n2)^2.  Bruno Berselli, Dec 07 2011
a(n) = A000290(n)  3.  Michel Marcus, Nov 13 2013
Sum_{n>=2} 1/a(n) = 2/3  Pi*cot(sqrt(3)*Pi)/(2*sqrt(3)) = 1.476650189986093617... .  Vaclav Kotesovec, Apr 10 2016
E.g.f.: (x^2 + x  3)*exp(x) + 2*x + 3.  G. C. Greubel, Jul 19 2017
Sum_{n>=2} (1)^n/a(n) = (2 + sqrt(3)*Pi*cosec(sqrt(3)*Pi))/6 = 0.8826191087...  Amiram Eldar, Nov 04 2020
From Amiram Eldar, Jan 29 2021: (Start)
Product_{n>=2} (1 + 1/a(n)) = sqrt(6)*csc(sqrt(3)*Pi)*sin(sqrt(2)*Pi).
Product_{n>=3} (1  1/a(n)) = Pi*csc(sqrt(3)*Pi)/(4*sqrt(3)). (End)


MAPLE

A028872 := proc(n) n^23; end proc: # R. J. Mathar, Aug 23 2011


MATHEMATICA

Range[2, 60]^2  3 (* or *) LinearRecurrence[{3, 3, 1}, {1, 6, 13}, 60] (* Harvey P. Dale, May 09 2013 *)


PROG

(Sage) [lucas_number1(3, n, 3) for n in range(2, 50)] # Zerinvary Lajos, Jul 03 2008
(PARI) a(n)=n^23 \\ Charles R Greathouse IV, Aug 23 2011
(PARI) x='x+O('x^99); Vec(x^2*(13*x+2*x^2)/(1+x)^3) \\ Altug Alkan, Apr 10 2016


CROSSREFS

Essentially the same: A123968, A267874.
Cf. A117950, A132411, A132414, A002522.
Sequence in context: A101247 A243655 A072212 * A049718 A036707 A054311
Adjacent sequences: A028869 A028870 A028871 * A028873 A028874 A028875


KEYWORD

nonn,easy


AUTHOR

Patrick De Geest


STATUS

approved



