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A147704
Diagonal sums of Riordan array ((1-2x)/(1 - 3x + x^2),x(1-x)/(1 - 3x + x^2)).
4
1, 1, 3, 8, 23, 66, 190, 547, 1575, 4535, 13058, 37599, 108262, 311728, 897585, 2584493, 7441751, 21427668, 61698511, 177653782, 511533678, 1472902523, 4241053787, 12211627683, 35161980526, 101244887791, 291523035690, 839407126544, 2416976491841, 6959406439833
OFFSET
0,3
COMMENTS
Diagonal sums of A147703.
Hankel transform is := 1,2,3,0,0,0,0,0,0,0,... - Philippe Deléham, Dec 15 2008
For n -> infinity, a(n+1)/a(n) -> 2.87938... = 1/A130880 = the largest diagonal of a nonagon (9-gon) with side 1 (see Redondo & Huylebrouck); compare to the F(n+1)/F(n) -> 1.618... = A001622 = the golden section or diagonal of a pentagon with side 1, where F is the Fibonacci sequence A000045. - Dirk Huylebrouck, Feb 15 2015
LINKS
A. Redondo Buitrago and D. Huylebrouck, Nonagons in the Hagia Sophia and the Selimiye Mosque, Nexus Network Journal on Architecture and Mathematics, January 2015.
FORMULA
G.f.: (1-x^2)/(1 - 3x + x^3).
a(n) = 3*a(n-1) - a(n-3), n>2 ; a(0)=1, a(1)=1, a(2)=3. - Philippe Deléham, Dec 15 2008
a(n) = (floor(A^n)+1)/3 for n>=1 where A = 2.8793... is the largest root of x^3-3x^2+1. - Stephen Bartell, Aug 15 2024
MATHEMATICA
LinearRecurrence[{3, 0, -1}, {1, 1, 3}, 30] (* Harvey P. Dale, May 24 2016 *)
PROG
(PARI) Vec((1-x^2)/(1-3*x+x^3) + O(x^20)) \\ Michel Marcus, Feb 16 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Nov 10 2008
STATUS
approved