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A084214
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Inverse binomial transform of a math magic problem.
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10
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1, 1, 4, 6, 14, 26, 54, 106, 214, 426, 854, 1706, 3414, 6826, 13654, 27306, 54614, 109226, 218454, 436906, 873814, 1747626, 3495254, 6990506, 13981014, 27962026, 55924054, 111848106, 223696214, 447392426, 894784854, 1789569706, 3579139414
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Inverse binomial transform of A060816.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (1,2)
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FORMULA
| a(n)=(5*2^n-3*0^n+4*(-1)^n)/6.
G.f.: (1+x^2)/((1+x)*(1-2*x)).
E.g.f.: (5*exp(2*x)-3*exp(0)+4*exp(-x))/6.
The binomial transform of A084214(n+1) is A020989(n). a(n)=A001045(n-1)+A001045(n+1)-0^n/2. - Paul Barry (pbarry(AT)wit.ie), May 04 2004
a(n)=sum{k=0..n, A001045(n+1)C(1, k/2)(1+(-1)^k)/2} - Paul Barry (pbarry(AT)wit.ie), Oct 15 2004
a(n) = a(n-1)+2*a(n-2) for n > 2. [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Dec 01 2009]
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MAPLE
| a[0]:=1:a[1]:=4:for n from 2 to 50 do a[n]:=a[n-1]+2*a[n-2]od: seq(a[n], n=-1..31); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 15 2008]
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MATHEMATICA
| f[n_]:=2/(n+1); x=3; Table[x=f[x]; Numerator[x], {n, 0, 5!}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Mar 12 2010]
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PROG
| (MAGMA) [(5*2^n-3*0^n+4*(-1)^n)/6: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011
(Haskell)
a084214 n = a084214_list !! n
a084214_list = 1 : xs where
xs = 1 : 4 : zipWith (+) (map (* 2) xs) (tail xs)
-- Reinhard Zumkeller, Aug 01 2011
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CROSSREFS
| Cf. A048654.
Sequence in context: A200186 A192782 A188576 * A030138 A009849 A103419
Adjacent sequences: A084211 A084212 A084213 * A084215 A084216 A084217
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), May 19 2003
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