A007754 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.

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, 191, 1, 7, 41, 198, 751, 2055, 3359, 1304, 1, 8, 55, 322, 1555, 5898, 16139, 26380, 10241, 1, 9, 71, 488, 2857, 13797, 52331, 143196, 234061, 90865, 1, 10, 89, 702

Offset: 0

Author

N. J. A. Sloane, Nov 28 2000

Keywords

• nonn
• easy
• nice
• tabl

Comments

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

Examples

			Array begins:
  1   1   1   1   1    1    1   1 ...
    1   2   3   4   5    6    7 ...
      1   5  11  19   29   41 ...
        2  18  52  110  198 ...
          7  85  301  751 ...
		

References

• Email from James Propp, Nov 28 2000.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

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

Main diagonal gives A099933.

Formula

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

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

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

Extensions

More terms from Christian G. Bower, Dec 02 2000