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-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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| 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).
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LINKS
| Eric W. Weisstein, Chromatic Polynomial
Wikipedia, Chromatic polynomial
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| a(n) = 2*n+a(n-1)-1 (with a(1)=-2). [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it)]
G.f.: x*(-2+x)*(1-3*x)/(1-x)^3. [Colin Barker, Jan 29 2012]
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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.
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MAPLE
| with(combinat):seq(fibonacci(3, i)-4, i=1..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008
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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}]
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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] ]; [From 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)
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CROSSREFS
| Essentially the same as A028872 (n^2-3 with offset 2).
Sequence in context: A128534 A002562 A136456 * A068797 A049951 A025263
Adjacent sequences: A123965 A123966 A123967 * A123969 A123970 A123971
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KEYWORD
| sign,easy
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AUTHOR
| Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 29 2006
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EXTENSIONS
| Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 13 2010
Definition simplified by M. F. Hasler (univ-ag.fr/~mhasler), Nov 12 2010
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