A131040 a(n) = (1/2+(1/2)*i*sqrt(11))^(n+1) + (1/2-(1/2)*i*sqrt(11))^(n+1), where i=sqrt(-1).

1, -5, -8, 7, 31, 10, -83, -113, 136, 475, 67, -1358, -1559, 2515, 7192, -353, -21929, -20870, 44917, 107527, -27224, -349805, -268133, 781282, 1585681, -758165, -5515208, -3240713, 13304911, 23027050, -16887683, -85968833, -35305784

Offset

0,2

Comments

Generating floretion is 1.5i' + .5j' + .5k' + .5e whereas in A131039 it is 'i + .5i' + .5j' + .5k' + .5e

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[ 1.5i' + .5j' + .5k' + .5e]

Essentially the Lucas sequence V(1,3). - Peter Bala, Jun 23 2015

Links

Table of n, a(n) for n=0..32.

Wikipedia, Lucas sequence

Index entries for linear recurrences with constant coefficients, signature (1,-3).

Formula

a(n) = a(n-1) - 3*a(n-2).

G.f.: (1 - 6*x)/(1 - x + 3*x^2).

a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x - 11*x^2))/2 )^n. - Peter Bala, Jun 23 2015

Mathematica

LinearRecurrence[{1, -3}, {1, -5}, 50] (* Paolo Xausa, Mar 12 2026 *)

Prog

(SageMath) [lucas_number2(n, 1, 3) for n in range(1, 34)] # Zerinvary Lajos, May 14 2009

Crossrefs

Cf. A131039, A131041, A131042, A002316, A002531.

Sequence in context: A314570 A039678 A259234 * A231786 A007450 A303816

Adjacent sequences: A131037 A131038 A131039 * A131041 A131042 A131043

Keyword

easy,sign

Author

Creighton Dement, Jun 11 2007

Extensions

Name corrected by Sean A. Irvine, Mar 17 2026

Status

approved