A121621 Real part of (2 + 3i)^n.

1, 2, -5, -46, -119, 122, 2035, 6554, -239, -86158, -341525, -246046, 3455641, 17021162, 23161315, -128629846, -815616479, -1590277918, 4241902555, 37641223154, 95420159401, -107655263398, -1671083125805, -5284814079046, 584824319281

Offset: 0

Author

Gary W. Adamson and Nick Williams, Aug 10 2006

Keywords

• sign
• easy

Comments

A121622 is the companion sequence generated from (3 + 2i).

Examples

			a(5) = 122 since (2 + 3i)^5 = (122 - 597i).
a(5) = 122 = 4*(-119) - 13*(-46) = 4*a(4) - 13*a(3).
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..500
• Beata Bajorska-Harapińska, Barbara Smoleń, 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,-13)

Crossrefs

Cf. A121622.

Programs

Maple

A121621:=proc(n)global a:if(n=0)then a[0]:=1:elif(n=1)then a[1]:=2:else a[n]:=4*a[n-1]-13*a[n-2]:fi:return a[n]:end:
seq(A121621(n),n=0..20); # Nathaniel Johnston, Apr 15 2011

Mathematica

f[n_] := Re[(2 + 3I)^n]; Table[f[n], {n, 0, 24}] (* Robert G. Wilson v *)

Formula

Re(2 + 3i)^n = a(n) = 4*a(n-1) - 13*a(n-2).

G.f.: ( 1-2*x ) / ( 1 - 4*x + 13*x^2 ). - R. J. Mathar, Mar 03 2013

Extensions

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