A102871 a(n) = a(n-3) - 5*a(n-2) + 5*a(n-1), a(0) = 1, a(1) = 3, a(2) = 10.

1, 3, 10, 36, 133, 495, 1846, 6888, 25705, 95931, 358018, 1336140, 4986541, 18610023, 69453550, 259204176, 967363153, 3610248435, 13473630586, 50284273908, 187663465045, 700369586271, 2613814880038, 9754889933880, 36405744855481, 135868089488043, 507066613096690

Offset: 0

Author

Creighton Dement, Mar 01 2005

Keywords

• nonn
• easy

Comments

A floretion-generated sequence resulting from a particular transform of the periodic sequence (-1,1).

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.

Also indices of the centered triangular numbers which are triangular numbers - Richard Choulet, Oct 09 2007

Place 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

Examples

			For n=5, a(5)=495; b(5)=286; binomial(495,2) = 122265 = 3*binomial(286,2). - _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).

Crossrefs

Cf. A001075 (first differences), A001834, A082841, A101265.

Programs

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 *)

Formula

2*a(n) - A001834(n) = (-1)^(n+1); a(n) = 4*a(n-1) - a(n) - 1;

G.f.: (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); 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) - 1/2 + (1/2)*(12*a(n)^2 - 12*a(n) + 9)^(1/2). - Richard Choulet, Oct 09 2007

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

E.g.f.: (exp(2*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) + cosh(x) + sinh(x))/2. - Stefano Spezia, Sep 19 2023