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

Author

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

Keywords

• easy
• sign

Comments

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)).

References

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

Links

• 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).

Crossrefs

Cf. A073817, A073937.

Programs

Mathematica

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

PARI

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

Formula

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)