A073937 -id:A073937 - cp's OEIS frontend

A074058 Reflected tetranacci numbers A073817.

4, -1, -1, -1, 7, -6, -1, -1, 15, -19, 4, -1, 31, -53, 27, -6, 63, -137, 107, -39, 132, -337, 351, -185, 303, -806, 1039, -721, 791, -1915, 2884, -2481, 2303, -4621, 7683, -7846, 7087, -11545, 19987, -23375, 22020, -30177, 51519, -66737, 67415, -82374, 133215, -184993, 201567, -232163, 348804

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 16 2002

Also a(n) is the trace of A^(-n), where A is the 4 X 4 matrix ((1,1,0,0), (1,0,1,0), (1,0,0,1), (1,0,0,0)).

R. L. Graham, D. E. Knuth and O. Patashnik, "Concrete Mathematics", Addison-Wesley, Reading, MA, 1998.

Indranil Ghosh, Table of n, a(n) for n = 0..8998
A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,1).

Cf. A073817, A073937.

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

polsym(polrecip(1+x+x^2+x^3-x^4), 55) \\ Joerg Arndt, Jan 21 2023

a(n) = -a(n-1)-a(n-2)-a(n-3)+a(n-4), a(0)=4, a(1)=-1, a(2)=-1, a(3)=-1.
G.f.: (4+3x+2x^2+x^3)/(1+x+x^2+x^3-x^4).
From Peter Bala, Jan 19 2023: (Start)
a(n) = (-1)^n*A073937(n).
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. (End)

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)).

Original entry on oeis.org

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

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

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)

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).

Cf. A073817, A073937, A000078, A074081.

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]

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

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

A074453 Sum of determinants of 2nd order principal minors of powers of inverse of the matrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).

Original entry on oeis.org

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

Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2002

a(n) is the reflected (A074058) sequence of sequence A074193.

Index entries for linear recurrences with constant coefficients, signature (1, -2, 2, 2, 1, 1).

Cf. A073817, A073937, A074058, A074081, A074193.

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, 40}], x]
LinearRecurrence[{1,-2,2,2,1,1},{6,1,-3,1,17,16},50] (* Harvey P. Dale, Mar 16 2012 *)

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(A074193(n)). - Joerg Arndt, Oct 22 2020

cp's OEIS Frontend

A074058 Reflected tetranacci numbers A073817.

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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)).

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A074453 Sum of determinants of 2nd order principal minors of powers of inverse of the matrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).

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