A147704 Diagonal sums of Riordan array ((1-2x)/(1 - 3x + x^2),x(1-x)/(1 - 3x + x^2)).

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

Author

Paul Barry, Nov 10 2008

Keywords

• easy
• nonn

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.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Crossrefs

Cf. A000045, A001622, A130880.

Programs

Mathematica

LinearRecurrence[{3,0,-1},{1,1,3},30] (* Harvey P. Dale, May 24 2016 *)

PARI

Vec((1-x^2)/(1-3*x+x^3) + O(x^20)) \\ Michel Marcus, Feb 16 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