A028872 a(n) = n^2 - 3.

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

Offset: 2

Author

Patrick De Geest

Keywords

• nonn
• easy

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(n-2) (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 (B-1) 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, 835-843. - _Emeric Deutsch_, Nov 14 2014
• Ran Pan, Exercise V, Project P.
• Eric Weisstein's World of Mathematics, Near-Square Prime.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Essentially the same: A123968, A267874.

Cf. A117950, A132411, A132414, A002522.

Programs

Maple

A028872 := proc(n) n^2-3; 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 *)

PARI

a(n)=n^2-3 \\ Charles R Greathouse IV, Aug 23 2011

PARI

x='x+O('x^99); Vec(x^2*(-1-3*x+2*x^2)/(-1+x)^3) \\ Altug Alkan, Apr 10 2016

Sage

[lucas_number1(3,n,3) for n in range(2,50)] # Zerinvary Lajos, Jul 03 2008

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(n-1) - 3*a(n-2) + a(n-3). (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(n-1) + 2*n-1 (with a(2)=1). - Vincenzo Librandi, Nov 18 2010

a(n)*a(n-1) + 3 = (a(n) - n)^2 = A014209(n-2)^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)