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

-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

1,1

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).

Links

Table of n, a(n) for n=1..50.

Eric W. Weisstein, Chromatic Polynomial.

Wikipedia, Chromatic polynomial.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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)

Example

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.

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}]

Prog

(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
(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)

Crossrefs

Essentially the same: A028872, A267874.

Sequence in context: A002562 A218492 A136456 * A282329 A343806 A372254

Adjacent sequences: A123965 A123966 A123967 * A123969 A123970 A123971

Keyword

sign,easy,changed

Author

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

Extensions

Edited and extended by Klaus Brockhaus, Nov 13 2010

Definition simplified by M. F. Hasler, Nov 12 2010

Status

approved