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A113405
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G.f.: x^3/(1-2x+x^3-2x^4) = x^3/( (1-2x)(1+x)(1-x+x^2) ).
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20
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0, 0, 0, 1, 2, 4, 7, 14, 28, 57, 114, 228, 455, 910, 1820, 3641, 7282, 14564, 29127, 58254, 116508, 233017, 466034, 932068, 1864135, 3728270, 7456540, 14913081, 29826162, 59652324, 119304647, 238609294, 477218588, 954437177, 1908874354, 3817748708
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OFFSET
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0,5
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COMMENTS
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A transform of the Jacobsthal numbers. A059633 is the equivalent transform of the Fibonacci numbers.
Paul Curtz, Aug 05 2007, observes that the inverse binomial transform of 0,0,0,1,2,4,7,14,28,57,114,228,455,910,1820,... gives the same sequence up to signs. That is, the extended sequence is an eigen-sequence for the inverse binomial transform.
Contribution from Ross Drewe (rd(AT)labyrinth.net.au), Sep 03 2009: (Start)
The round() function enables the closed (non-recurrence) formula to take a very simple form: see Formula section. This can be generalised without loss of simplicity to a(n) = round(b^n/c), where b and c are very small, incommensurate integers (c may also be an integer fraction). Particular choices of small integers for b and c produce a number of well-known sequences which are usually defined by a recurrence - see Cross Reference. (End)
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to linear recurrences with constant coefficients, signature (2,0,-1,2).
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
N. J. A. Sloane, Transforms
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FORMULA
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a(n)=2a(n-1)-a(n-3)+2a(n-4); a(n)=sum{k=0..floor(n/2), C(n-k, k)A001045(k)}; a(n)=sum{k=0..n, C((n+k)/2, k)A001045((n-k)/2)(1+(-1)^(n-k))/2}.
a(3n) = A015565(n), a(3n+1) = 2*A015565(n), a(3n+2) = 4*A015565(n). - Paul Curtz, Nov 30 2007
a(n+1)-2a(n)= A131531(n). a(n)+a(n+3)=2^n. - Paul Curtz, Dec 16 2007
a(n)=round(2^n/9) [From Ross Drewe (rd(AT)labyrinth.net.au), Sep 03 2009]
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MAPLE
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A010892 := proc(n) op((n mod 6)+1, [1, 1, 0, -1, -1, 0]) ; end proc:
A113405 := proc(n) (2^n-(-1)^n)/9 -A010892(n-1)/3; end proc: # R. J. Mathar, Dec 17 2010
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MATHEMATICA
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CoefficientList[Series[x^3/(1-2x+x^3-2x^4), {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 0, -1, 2}, {0, 0, 0, 1}, 40] (* From Harvey P. Dale, Apr 30 2011 *)
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PROG
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(PARI) a(n)=2^n\/9 \\ Charles R Greathouse IV, Jun 05, 2011
(MAGMA) [Round(2^n/9): n in [0..40]]; // Vincenzo Librandi, Aug 11 2011
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CROSSREFS
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Contribution from Ross Drewe (rd(AT)labyrinth.net.au), Sep 03 2009: (Start)
Other sequences a(n) = round(b^n / c), where b and c are very small integers:
A001045 b = 2; c = 3
A007910 b = 2; c = 5
A016029 b = 2; c = 5/3
A077947 b = 2; c = 7
abs(A078043) b = 2; c = 7/3
A007051 b = 3; c = 2
A015518 b = 3; c = 4
A034478 b = 5; c = 2
A003463 b = 5; c = 4
A015531 b = 5; c = 6
(End)
Sequence in context: A068060 A057744 A210210 * A119340 A119341 A119342
Adjacent sequences: A113402 A113403 A113404 * A113406 A113407 A113408
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KEYWORD
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easy,nonn,changed
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AUTHOR
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Paul Barry, Oct 28 2005
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EXTENSIONS
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Edited by N. J. A. Sloane, Dec 13 2007
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STATUS
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approved
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