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A102871
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a(n) = a(n-3) - 5*a(n-2) + 5*a(n-1), a(0) = 1, a(1) = 3, a(2) = 10.
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6
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1, 3, 10, 36, 133, 495, 1846, 6888, 25705, 95931, 358018, 1336140, 4986541, 18610023, 69453550, 259204176, 967363153, 3610248435, 13473630586, 50284273908, 187663465045, 700369586271, 2613814880038
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| A floretion-generated sequence resulting from a particular transform of the periodic sequence (-1,1).
Also indices of the centered triangular numbers which are triangular numbers - R. Choulet (richardchoulet(AT)yahoo.fr), Oct 09 2007
a(n) red and b(n) blue balls in an urn; draw 2 balls without return. Probability(2 red balls) = 3*Probability(2 blue balls); b(n)=A101265(n). - Paul Weisenhorn (weisenhorn-f.p(AT)online.de), Aug 02 2010
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LINKS
| Harvey P. Dale, Table of n, a(n) for n = 0..1000
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FORMULA
| 2*a(n) - A001834(n) = (-1)^(n+1); a(n) = 4*a(n-1) - a(n) - 1; G.f. x*(2*x-1)/((x-1)*(x^2-4*x+1)). Superseeker results: a(n+2) - 2a(n+1) + a(n) = A001834(n+1) (from this and the first relation involving A001834 it follows that 4a(n+1) - a(n+2) - a(n) = (-1)^n as well as the recurrence relation given for A001834 ); a(n+1) - a(n) = A001075(n+1) (Chebyshev's T(n, x) polynomials evaluated at x=2); a(n+2) - a(n) = A082841(n+1).
a[j+3]-3*a[j+2]-3*a[j+1]+a[j] = -2 for all j.
a(n+1) = 2*a(n)-0.5+0.5*(12*a(n)^2-12*a(n)+9)^0.5. - R. Choulet (richardchoulet(AT)yahoo.fr), Oct 09 2007
a(n) = 1/2-(1/4)*sqrt(3)*(2-sqrt(3))^n+(1/4)*sqrt(3)*(2+sqrt(3))^n+(1/4)*(2-sqrt(3))^n+(1/4)*(2+sqrt(3))^n, with n>=0. [From Paolo P. Lava (paoloplava(AT)gmail.com), Oct 03 2008]
Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Aug 02 2010: (Start)
a(n) = (sqrt(12*b(n)*(b(n)-1)+1)+1)/2; b(n)=A101265(n).
(End)
a(n) = A001571(n)+1. - Johannes Boot, Jun 17 2011
G.f.: (2*x-1)/(x^3-5*x^2+5*x-1) [From Harvey P. Dale, Oct 04 2011]
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EXAMPLE
| Contribution from Weisenhorn Paul (weisenhorn-f.p(AT)online.de), Aug 02 2010: (Start)
For n=5 a(5)=495; b(5)=286; binomial(495,2)=122265=3*binomial(286,2)
(End)
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MAPLE
| a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=4*a[n-1]-a[n-2]-1 od: seq(a[n], n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 08 2008
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MATHEMATICA
| LinearRecurrence[{5, -5, 1}, {1, 3, 10}, 30] (* From Harvey P. Dale, Oct 04 2011 *)
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PROG
| Floretion Algebra Multiplication Program, FAMP Code: .5em[J* ]forseq[ .25( 'i + 'j + 'k + i' + j' + k' + 'ii' + 'jj' + 'kk' + 'ij' + 'ik' + 'ji' + 'jk' + 'ki' + 'kj' + e ) ], em[J]forseq = A001834, vesforseq = (1, -1, 1, -1). ForType 1A. Identity used: em[J]forseq + em[J* ]forseq = vesforseq.
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CROSSREFS
| Cf. A001834, A001075, A082841.
Sequence in context: A047107 A149040 A055989 * A119374 A126188 A081909
Adjacent sequences: A102868 A102869 A102870 * A102872 A102873 A102874
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KEYWORD
| nonn,easy
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AUTHOR
| Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Mar 01 2005
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EXTENSIONS
| More terms from R. Choulet (richardchoulet(AT)yahoo.fr), Oct 09 2007
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