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A102871 a(n) = a(n-3) - 5*a(n-2) + 5*a(n-1), a(0) = 1, a(1) = 3, a(2) = 10. 6
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; text; internal format)
OFFSET

0,2

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 - Richard Choulet, Oct 09 2007

a(n) red and b(n) blue balls in an urn; draw 2 balls without replacement. Probability(2 red balls) = 3*Probability(2 blue balls); b(n)=A101265(n). - Paul Weisenhorn, Aug 02 2010

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

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. - Richard Choulet, 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. - Paolo P. Lava, Oct 03 2008

a(n) = (sqrt(12*b(n)*(b(n)-1) + 1) + 1)/2; b(n) = A101265(n). - Paul Weisenhorn, Aug 02 2010

a(n) = A001571(n) + 1. - Johannes Boot, Jun 17 2011

G.f.: (2*x - 1)/(x^3 - 5*x^2 + 5*x - 1). - Harvey P. Dale, Oct 04 2011

EXAMPLE

For n=5, a(5)=495; b(5)=286; binomial(495,2) = 122265 = 3*binomial(286,2). - Paul Weisenhorn, Aug 02 2010

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, Mar 08 2008

MATHEMATICA

LinearRecurrence[{5, -5, 1}, {1, 3, 10}, 30] (* Harvey P. Dale, Oct 04 2011 *)

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.

CROSSREFS

Cf. A001075, A001834, A082841.

Sequence in context: A047107 A149040 A055989 * A277287 A119374 A272686

Adjacent sequences:  A102868 A102869 A102870 * A102872 A102873 A102874

KEYWORD

nonn,easy

AUTHOR

Creighton Dement, Mar 01 2005

STATUS

approved

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Last modified November 21 17:44 EST 2017. Contains 295004 sequences.