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

0,5

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

References

Email from James Propp, Nov 28 2000.

Links

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

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

Example

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 ...

Crossrefs

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

Main diagonal gives A099933.

Sequence in context: A090234 A286380 A275866 * A144866 A058732 A060082

Adjacent sequences: A007751 A007752 A007753 * A007755 A007756 A007757

Keyword

nonn,easy,nice,tabl

Author

N. J. A. Sloane, Nov 28 2000

Extensions

More terms from Christian G. Bower, Dec 02 2000

Status

approved