A074193 Sum of determinants of 2nd order principal minors of powers of the matrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).

6, -1, -3, -1, 17, -16, -15, 13, 81, -127, -58, 175, 329, -885, -31, 1424, 833, -5543, 2181, 9233, -2298, -31025, 27893, 49495, -54879, -150416, 245697, 204965, -526887, -570895, 1801670, 407711, -3882303, -946397, 11542929, -3442672, -24121039, 10317745, 64959629, -56727711, -127083514

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 20 2002

Keywords

• easy
• sign

Comments

From Kai Wang, Oct 21 2020: (Start)

Let f(x) = x^4 - x^3 - x^2 - x - 1 be the characteristic polynomial for Tetranacci numbers (A000078). Let {x1,x2,x3,x4} be the roots of f(x). Then a(n) = (x1*x2)^n + (x1*x3)^n + (x1*x4)^n + (x2*x3)^n + (x2*x4)^n + (x3*x4)^n.

Let g(y) = y^6 + y^5 + 2*y^4 + 2*y^3 - 2*y^2 + y - 1 be the characteristic polynomial for a(n). Let {y1,y2,y3,y4,y5,y6} be the roots of g(y). Then a(n) = y1^n + y2^n + y3^n + y4^n + y5^n + y6^n. (End)

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..5052
• Kai Wang, Identities, generating functions and Binet formula for generalized k-nacci sequences, 2020.
• Index entries for linear recurrences with constant coefficients, signature (-1,-2,-2,2,-1,1).

Crossrefs

Cf. A073817, A073937, A000078, A074081.

Programs

Mathematica

CoefficientList[Series[(6+5*x+8*x^2+6*x^3-4*x^4+x^5)/(1+x+2*x^2+2*x^3-2*x^4+x^5-x^6), {x, 0, 50}], x]

PARI

polsym(x^6 + x^5 + 2*x^4 + 2*x^3 - 2*x^2 + x - 1,44) \\ Joerg Arndt, Oct 22 2020

Formula

a(n) = -a(n-1)-2a(n-2)-2a(n-3)+2a(n-4)-a(n-5)+a(n-6).

G.f.: (6+5x+8x^2+6x^3-4x^4+x^5)/(1+x+2x^2+2x^3-2x^4+x^5-x^6).

abs(a(n)) = abs(A074453(n)). - Joerg Arndt, Oct 22 2020