A139030 Real part of (4 + 3i)^n.

4, 7, -44, -527, -3116, -11753, -16124, 164833, 1721764, 9653287, 34182196, 32125393, -597551756, -5583548873, -29729597084, -98248054847, -42744511676, 2114245277767, 17982575014036, 91004468168113, 278471369994004, -47340744250793, -7340510203856444

Offset

1,1

Comments

sqrt (a(n)^2 + (A139031(n))^2) = 5^n. Example: a(3) = -44, A139031(3) = 117. Sqrt (-44^2 + 117^2) = 5^3.

Links

Table of n, a(n) for n=1..23.

Index entries for linear recurrences with constant coefficients, signature (8,-25).

Formula

Real part of (4 + 3i)^n. Term (1,1) of [4,-3; 3,4]^n a(n), n>2 = 8*a(n-1) - 25*a(n-2), given a(1) = 4, a(2) = 7. Odd-indexed terms of A066776 interleaved with even-indexed terms of A066771, irrespective of sign.

G.f.: -x*(-4+25*x) / ( 1-8*x+25*x^2 ). - R. J. Mathar, Feb 05 2011

a(n) = 8*a(n-1) - 25*a(n-2); a(1)=4, a(2)=7. - Harvey P. Dale, Nov 09 2011

Example

a(5) = -3116 since (4 + 3i)^5 = (-3116 - 237i) where -237 = A139031(5).

Mathematica

Re[(4+3I)^Range[40]] (* or *) LinearRecurrence[{8, -25}, {4, 7}, 40] (* Harvey P. Dale, Nov 09 2011 *)

Crossrefs

Cf. A139031, A066771, A066776.

Sequence in context: A093102 A199404 A284971 * A115439 A348625 A192168

Adjacent sequences: A139027 A139028 A139029 * A139031 A139032 A139033

Keyword

sign

Author

Gary W. Adamson, Apr 06 2008

Extensions

More terms from Harvey P. Dale, Nov 09 2011

Status

approved