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

0,3

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

Table of n, a(n) for n=0..29.

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).

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

Cf. A000045, A001622, A130880.

Sequence in context: A078054 A018043 A291037 * A116410 A199841 A106606

Adjacent sequences: A147701 A147702 A147703 * A147705 A147706 A147707

Keyword

easy,nonn

Author

Paul Barry, Nov 10 2008

Status

approved