1,2

Also sum of the successive powers of all combinations of products of two different roots of the quintic pentanacci polynomial X^5-X^4-X^3-X^2-X-1; namely (X1 X2)^n + (X1 X3)^n + (X1 X4)^n + (X1 X5)^n + (X2 X3)^n+ (X2 X4)^n + (X2 X5)^n + (X3 X4)^n + (X3 X5)^n + (X4 X5)^n, where X1,X2,X3,X4,X5 are the roots. A074048 are the coefficients, with changed signs, of X^4 in the characteristic polynomials of the successive powers of the pentanacci matrix or (X1)^n+(X2)^n+(X3)^n+(X4)^n+(X5)^n.

a(5)=49 because characteristic polynomial of fifth power of pentanacci matrix M^5 is X^5-31X^4+49X^3-31X^2+9X-1 in which coefficent of X^3 is 49

with(linalg): M[1]:=matrix(5, 5, [1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0]): for n from 2 to 40 do M[n]:=multiply(M[n-1], M[1]) od: seq(coeff(charpoly(M[n], x), x, 3), n=1..40); (Deutsch)

f[n_] := CoefficientList[ CharacteristicPolynomial[ MatrixPower[{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, n], x], x][[4]]; Array[f, 36] (Robert G. Wilson v)

Cf. A074048.

Sequence in context: A055325 A134049 A123951 this_sequence A077451 A019829 A091528

Adjacent sequences: A123124 A123125 A123126 this_sequence A123128 A123129 A123130

sign

Artur Jasinski (grafix(AT)csl.pl), Sep 30 2006

Edited by njas, Oct 24 2006

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu) and Robert G. Wilson v, Oct 24 2006