A121622 Real part of (3 + 2i)^n.

1, 3, 5, -9, -119, -597, -2035, -4449, -239, 56403, 341525, 1315911, 3455641, 3627003, -23161315, -186118929, -815616479, -2474152797, -4241902555, 6712571031, 95420159401, 485257533003, 1671083125805, 3718150825791, 584824319281

Offset: 0

Author

Gary W. Adamson and Nick Williams, Aug 10 2006

Keywords

• sign
• easy

Comments

Companion sequence A121621 is real((2 + 3i)^n).

Examples

			a(5) = -597 since (3 + 2i)^5 = (-597 + 122i).
a(5) = -597 = 6*(-119) - 13*(-9) = 6*a(5) -13*a(4).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..500
• 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 (6,-13).

Crossrefs

Cf. A121621.

Cf. A193410, A066771.

Programs

Mathematica

f[n_] := Re[(3 + 2I)^n]; Table[f[n], {n, 0, 24}] (* Robert G. Wilson v, Aug 17 2006 *)
LinearRecurrence[{6,-13},{1,3},30] (* Harvey P. Dale, Apr 24 2017 *)

PARI

a(n) = real((3 + 2*I)^n); \\ Michel Marcus, Jun 12 2021

Formula

a(n) = real((3 + 2i)^n).

a(n) = 6*a(n-1) - 13*a(n-2).

G.f.: ( 1-3*x ) / ( 1-6*x+13*x^2 ). - R. J. Mathar, Aug 12 2012

E.g.f.: exp(3*x)*cos(2*x). - Sergei N. Gladkovskii, Jan 20 2014

Extensions

More terms from Robert G. Wilson v, Aug 17 2006