A190965 a(n) = 4*a(n-1) - 6*a(n-2), with a(0)=0, a(1)=1.

0, 1, 4, 10, 16, 4, -80, -344, -896, -1520, -704, 6304, 29440, 79936, 143104, 92800, -487424, -2506496, -7101440, -13366784, -10858496, 36766720, 212217856, 628271104, 1239777280, 1189482496, -2680733696, -17859829760, -55354916864, -114260688896

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

• sign
• easy

Comments

For the quaternion Q = 2+j+k, Q^n = r(n) + a(n)*(j+k). The sequence of real-parts r(n) is A266046. - Stanislav Sykora, Dec 20 2015

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..1000
• Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
• Index entries for linear recurrences with constant coefficients, signature (4,-6).

Crossrefs

Cf. A190958 (index to generalized Fibonacci sequences).

Cf. A088137 (Inv. Bin. Trans.), A168175, A213421, A266046.

Programs

Magma

[n le 2 select n-1 else 4*Self(n-1) -6*Self(n-2): n in [1..41]]; // G. C. Greubel, Jan 10 2024

Maple

w := I*sqrt(2): a := n -> (w/4)*((2 - w)^n - (2 + w)^n):
seq(simplify(a(n)), n = 0..20);  # (after Taras Goy), Peter Luschny, Jan 03 2025

Mathematica

LinearRecurrence[{4,-6}, {0,1}, 50]

PARI

a(n)=([0,1;0,0]*[0,-6;1,4]^n)[1,1] \\ Charles R Greathouse IV, May 31 2011

SageMath

A190965=BinaryRecurrenceSequence(4,-6,0,1)
[A190965(n) for n in range(41)] # G. C. Greubel, Jan 10 2024

Formula

G.f.: x/(1-4*x+6*x^2). - Philippe Deléham, Oct 12 2011

2*a(n)^2 + A266046(n)^2 = 6^n. - Stanislav Sykora, Dec 20 2015

From G. C. Greubel, Jan 10 2024: (Start)

a(n) = 6^((n-1)/2)*ChebyshevU(n-1, sqrt(2/3)).

E.g.f.: (1/sqrt(2))*exp(2*x)*sin(sqrt(2)*x). (End)

a(n) = (i*sqrt(2)/4)*((2 - i*sqrt(2))^n - (2 + i*sqrt(2))^n), where i = sqrt(-1). - Taras Goy, Jan 03 2025