OFFSET
0,2
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
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.: (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
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Creighton Dement, Mar 01 2005
STATUS
approved