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A108051
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a(n+1) = 4*(a(n)+a(n-1)) for n>1, a(1)=1, a(2)=6.
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1
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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; internal format)
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OFFSET
| 0,3
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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.
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LINKS
| Robert Munafo, Sequences Related to Floretions
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,4).
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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 (pbarry(AT)wit.ie), Jun 07 2005
a(n)=((3+sqrt18)(2+sqrt8)^n+(3-sqrt18)(2-sqrt8)^n)/6 offset 0. [From Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009]
a(n) = 2^(n-1) * A001333(n), n>0. [Ralf Stephan, Dec 02 2010]
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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'
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CROSSREFS
| Cf. A057087, A001108, A002315, A096980.
Sequence in context: A171859 A084778 A155588 * A199315 A001599 A074247
Adjacent sequences: A108048 A108049 A108050 * A108052 A108053 A108054
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KEYWORD
| easy,nonn
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AUTHOR
| Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jun 01 2005
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