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A108051 a(n+1) = 4*(a(n)+a(n-1)) for n>1, a(1)=1, a(2)=6. 4
0, 1, 6, 28, 136, 656, 3168, 15296, 73856, 356608, 1721856, 8313856, 40142848, 193826816, 935878656, 4518821888, 21818802176, 105350496256, 508677193728, 2456110759936, 11859151814656, 57261050298368, 276480808452096 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Let (a_n) be the sequence and (a_(n+1)) the sequence beginning at 1. Let B and iB be the binomial and inverse binomial transforms, respectively. Then B((a_n)) = A001108(n) (a(n)-th triangular number is a square); B((a_(n+1))) = A002315(n) (NSW Numbers); iB((a_(n+1))) = A096980(n). Note: a 2nd sequence generated by the same floretion is A057087 (Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci sequence.). As is often the case with two sequences corresponding to a single floretion, both satisfy the same recurrence relation.

LINKS

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

Martin Burtscher, Igor Szczyrba, RafaƂ Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

Robert Munafo, Sequences Related to Floretions

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

FORMULA

a(n+1) = -(1/2)*(2-2*2^(1/2))^n*(-1+2^(1/2))-(1/2)*(2+2*2^(1/2))^n(-1-2^(1/2)); G.f.: x*(1+2*x)/(1-4*x-4*x^2).

a(n) = sum{k=0..n, (-1)^k*C(n-1, k)*(Pell(2n-2k)-Pell(2n-2k-1))}, n>0, where Pell(n) = A000129(n). - Paul Barry, Jun 07 2005

a(n+1) = ((3+sqrt18)(2+sqrt8)^n+(3-sqrt18)(2-sqrt8)^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009, index corrected Jul 11 2012

a(n) = 2^(n-1) * A001333(n), n>0. - Ralf Stephan, Dec 02 2010

a(n) = A057087(n-1) + 2*A057087(n-2). - R. J. Mathar, Jul 11 2012

MATHEMATICA

CoefficientList[Series[x*(1+2*x)/(1-4*x-4*x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 26 2012 *)

PROG

Floretion Algebra Multiplication Program, FAMP Code: (a_n) = 2ibasekseq[A*B] (with initial term zero), (a_(n+1)) = 1tesseq[A*B], A = + .5'i - .5'j + .5'k + .5i' - .5j' + .5k' - .5'ij' - .5'ik' - .5'ji' - .5'ki'; B = - .5'i + .5'j + .5'k - .5i' + .5j' + .5k' - .5'ik' - .5'jk' - .5'ki' - .5'kj'

(MAGMA) I:=[0, 1, 6]; [n le 3 select I[n] else 4*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 26 2012

CROSSREFS

Cf. A057087, A001108, A002315, A096980.

Sequence in context: A084778 A155588 A208439 * A199315 A001599 A074247

Adjacent sequences:  A108048 A108049 A108050 * A108052 A108053 A108054

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Jun 01 2005

STATUS

approved

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Last modified March 23 22:30 EDT 2017. Contains 283985 sequences.