%I #33 Jun 06 2023 05:05:58

%S 1,1,1,1,2,1,1,3,5,2,1,4,11,18,7,1,5,19,52,85,33,1,6,29,110,301,492,

%T 191,1,7,41,198,751,2055,3359,1304,1,8,55,322,1555,5898,16139,26380,

%U 10241,1,9,71,488,2857,13797,52331,143196,234061,90865,1,10,89,702

%N Array (a frieze pattern) defined by a(n,k) = (a(n-1,k)*a(n-1,k+1) - 1) / a(n-2,k+1), read by antidiagonals.

%C Let u be a sequence with u(0)=p, u(1)=q, and u(i)^(i+k) = u(i-1)*u(i+1). Then u(n)= q^a(n-1,k)/p^a(n-2,k+1). - Example for k=1, u(5)=q^7/p^18 and for k=2, u(5)=q^85/p^52. - _Olivier Gérard_, Sep 19 2016

%D Email from _James Propp_, Nov 28 2000.

%H Seiichi Manyama, <a href="/A007754/b007754.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n, k) = (n+k)*a(n-1, k)-a(n-2, k) with a(0, k)=1 and a(-1, k)=0. - _Henry Bottomley_, Feb 28 2001

%F a(n, k) = Pi*(BesselJ(n+k+1, 2)*BesselY(k, 2) - BesselY(n+k+1, 2)*BesselJ(k, 2)). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 21 2005

%F Column asymptotics (i.e. for fixed k and n -> infinity): a(n, k) ~ BesselJ(k, 2)*(n+k)!. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 21 2005

%e Array begins:

%e 1 1 1 1 1 1 1 1 ...

%e 1 2 3 4 5 6 7 ...

%e 1 5 11 19 29 41 ...

%e 2 18 52 110 198 ...

%e 7 85 301 751 ...

%Y Row 0-3: A000012, A000027(n+1), A028387, A058794-A058796. Columns 0-2: A058797-A058799.

%Y Main diagonal gives A099933.

%K nonn,easy,nice,tabl

%O 0,5

%A _N. J. A. Sloane_, Nov 28 2000

%E More terms from _Christian G. Bower_, Dec 02 2000