cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Patrick De Geest

Keywords

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

Crossrefs

Essentially the same: A123968, A267874.
Cf. A117950, A132411, A132414, A002522.

Programs

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)

A123968 a(n) = n^2 - 3, starting at n=1.

Original entry on oeis.org

-2, 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, 2398, 2497
Offset: 1

Views

Author

Gary W. Adamson and Roger L. Bagula, Oct 29 2006

Keywords

Comments

Essentially the same as A028872 (n^2-3 with offset 2).
a(n) is the constant term of the quadratic factor of the characteristic polynomial of the 5 X 5 tridiagonal matrix M_n with M_n(i,j) = n for i = j, M_n(i,j) = -1 for i = j+1 and i = j-1, M_n(i,j) = 0 otherwise.
The characteristic polynomial of M_n is (x-(n-1))*(x-n)*(x-(n+1))*(x^2-2*n*x+c) with c = n^2-3.
The characteristic polynomials are related to chromatic polynomials, cf. links. They have roots n+sqrt(3).

Examples

			The quadratic factors of the characteristic polynomials of M_n for n = 1..6 are
  x^2 -  2*x -  2,
  x^2 -  4*x +  1,
  x^2 -  6*x +  6,
  x^2 -  8*x + 13,
  x^2 - 10*x + 22,
  x^2 - 12*x + 33.

Links

Crossrefs

Essentially the same: A028872, A267874.

Programs

  • Magma
    mat:=func< n | Matrix(IntegerRing(), 5, 5, [< i, j, i eq j select n else (i eq j+1 or i eq j-1) select -1 else 0 > : i, j in [1..5] ]) >; [ Coefficients(Factorization(CharacteristicPolynomial(mat(n)))[4][1])[1]:n in [1..50] ]; // Klaus Brockhaus, Nov 13 2010
  • Maple
    with(combinat):seq(fibonacci(3, i)-4,i=1..55); # Zerinvary Lajos, Mar 20 2008
  • Mathematica
    M[n_] := {{n, -1, 0, 0, 0}, {-1, n, -1, 0, 0}, {0, -1, n, -1, 0}, {0, 0, -1, n, -1}, {0, 0, 0, -1, n}}; p[n_, x_] = Factor[CharacteristicPolynomial[M[n], x]] Table[ -3 + n^2, {n, 1, 25}]
  • PARI
    A123968(n) = n^2-3   /* or: */
  • PARI
    a(n)=polcoeff(factor(charpoly(matrix(5,5,i,j,if(abs(i-j)>1,0,if(i==j,n,-1)))))[4,1], 0)

Formula

a(n) = 2*n + a(n-1) - 1. - Vincenzo Librandi, Nov 12 2010
G.f.: x*(-2+x)*(1-3*x)/(1-x)^3. - Colin Barker, Jan 29 2012
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(x^2 + x - 3) + 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Extensions

Edited and extended by Klaus Brockhaus, Nov 13 2010
Definition simplified by M. F. Hasler, Nov 12 2010
Showing 1-2 of 2 results.

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