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A074193
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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)).
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2
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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
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OFFSET
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0,1
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COMMENTS
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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)
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LINKS
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FORMULA
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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).
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MATHEMATICA
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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]
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PROG
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(PARI) polsym(x^6 + x^5 + 2*x^4 + 2*x^3 - 2*x^2 + x - 1, 44) \\ Joerg Arndt, Oct 22 2020
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Aug 20 2002
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STATUS
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approved
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