A100215 Expansion of (4 - 7*x + 2*x^2)/((1-2*x)*(1 - 2*x + 2*x^2)).

4, 9, 14, 18, 24, 44, 104, 248, 544, 1104, 2144, 4128, 8064, 16064, 32384, 65408, 131584, 263424, 525824, 1049088, 2095104, 4189184, 8382464, 16775168, 33562624, 67129344, 134242304, 268443648, 536838144

Offset

0,1

Comments

a(n) = (-1)^n*A009116(n+3) + A100216(n) + A038503(n+1), where A009116, A100216 and A038503 can be generated by the operators jes, les and tes of the Floretion algebra, which is a product factor space Q x Q /{(1,1), (-1,-1)}.

Binomial transform of the sequence 4,5,0,-1 (repeated with period length 4). - R. J. Mathar, Apr 18 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Creighton Dement, Floretion Online Multiplier.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4).

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3).

a(n) = (-1)^n*A009116(n+3) + A100216(n) + A038503(n+1).

a(n) = vesseq(.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e), where ves sums over all floretion basis vector coefficients.

a(n) = 2^(n+1) + 2*A099087(n) + A099087(n-1). - R. J. Mathar, Apr 18 2009

Example

a(2) = 14 because (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)^3 = 1'j + 1'k + 1j' + 1k' + 3'ii' + 2'jj' + 2'kk' + 1'jk' + 1'kj' + 1e and the sum of these coefficients is 1 + 1 + 1 + 1 + 3 + 2 + 2 + 1 + 1 + 1 = 14 (see comment).

Mathematica

LinearRecurrence[{4, -6, 4}, {4, 9, 14}, 40] (* Vincenzo Librandi, Jun 25 2012 *)

Prog

(Magma) I:=[4, 9, 14]; [n le 3 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Jun 25 2012
(SageMath)
A099087=BinaryRecurrenceSequence(2, -2, 1, 2)
def A100215(n): return 2^(n+1) + 2*A099087(n) + A099087(n-1)
[A100215(n) for n in range(41)] # G. C. Greubel, Mar 29 2024

Crossrefs

Cf. A009116, A038503, A099087, A100213, A100214, A100216.

Sequence in context: A313077 A313078 A313079 * A313080 A313081 A100213

Adjacent sequences: A100212 A100213 A100214 * A100216 A100217 A100218

Keyword

nonn,easy

Author

Creighton Dement, Nov 11 2004

Extensions

Definition replaced with the more precise g.f. by R. J. Mathar, Nov 17 2010

Status

approved