A147704 - OEIS

%I #40 Aug 16 2024 10:10:35

%S 1,1,3,8,23,66,190,547,1575,4535,13058,37599,108262,311728,897585,

%T 2584493,7441751,21427668,61698511,177653782,511533678,1472902523,

%U 4241053787,12211627683,35161980526,101244887791,291523035690,839407126544,2416976491841,6959406439833

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

%C Diagonal sums of A147703.

%C Hankel transform is := 1,2,3,0,0,0,0,0,0,0,... - _Philippe Deléham_, Dec 15 2008

%C 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

%H A. Redondo Buitrago and D. Huylebrouck, <a href="http://dx.doi.org/10.1007/s00004-015-0235-y">Nonagons in the Hagia Sophia and the Selimiye Mosque</a>, Nexus Network Journal on Architecture and Mathematics, January 2015.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-1).

%F G.f.: (1-x^2)/(1 - 3x + x^3).

%F 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

%F 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

%t LinearRecurrence[{3,0,-1},{1,1,3},30] (* _Harvey P. Dale_, May 24 2016 *)

%o (PARI) Vec((1-x^2)/(1-3*x+x^3) + O(x^20)) \\ _Michel Marcus_, Feb 16 2015

%Y Cf. A000045, A001622, A130880.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Nov 10 2008