A144434 Triangle read by rows formed from golden-mean-like generalized factorials: t(n,m)=Round(phi*a(n)/(a(m)*a(n-m))), where phi=-(1 - Sqrt[5])/2, b(n)=b(n-1)+phi, a(n)=b(n)*a(n-1).

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 4, 7, 9, 7, 4, 1, 1, 4, 9, 14, 14, 9, 4, 1, 1, 4, 12, 21, 25, 21, 12, 4, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 6, 18, 41, 66, 77, 66, 41, 18, 6, 1

Offset

1,5

Comments

Row sums are {1, 2, 4, 6, 9, 18, 33, 56, 101, 186, 341, ...}.

The actual rational numbers are:

{1},

{1, 1},

{1, 3/2, 1},

{1, 2, 2, 1},

{1, 5/2, 10/3, 5/2, 1},

{1, 3, 5, 5, 3, 1},

{1, 7/2, 7, 35/4, 7, 7/2, 1},

{1, 4, 28/3, 14, 14, 28/3, 4, 1},

{1, 9/2, 12, 21, 126/5, 21, 12, 9/2, 1},

{1, 5, 15, 30, 42, 42, 30, 15, 5, 1},

{1, 11/2, 55/3, 165/4, 66, 77, 66, 165/4, 55/3, 11/2, 1}

...

Links

Table of n, a(n) for n=1..66.

Formula

Phi=-(1 - Sqrt[5])/2; b(n)=b(n-1)+phi; a(n)=b(n)*a(n-1); t(n,m) = Round(Phi*a(n)/(a(m)*a(n-m))). - corrected by Joshua Swanson, Sep 16 2016

Empirically, t(n,m)=Round[(1/(m+1))*Binomial[n+1, m]]. - Joshua Swanson, Sep 16 2016

Example

Triangle begins:
{1},
{1, 1},
{1, 2, 1},
{1, 2, 2, 1},
{1, 2, 3, 2, 1},
{1, 3, 5, 5, 3, 1},
{1, 4, 7, 9, 7, 4, 1},
{1, 4, 9, 14, 14, 9, 4, 1},
{1, 4, 12, 21, 25, 21, 12, 4, 1},
{1, 5, 15, 30, 42, 42, 30, 15, 5, 1},
{1, 6, 18, 41, 66, 77, 66, 41, 18, 6, 1}
...

Mathematica

Clear[a, n, b, c] (* generalized Beta integer factorial of the golden mean*);
b[0] = -(1 - Sqrt[5])/2; b[n_] := b[n] = b[n - 1] - (1 - Sqrt[5])/2;
a[0] = -(1 - Sqrt[5])/2; a[n_] := a[n] = b[n]*a[n - 1]; (* combinations based on the gen-beta -factorials*)
c = Table[Table[FullSimplify[ExpandAll[((-1 + Sqrt[5])/2)*a[n]/(a[m]*a[n - m])]], {m, 0, n}], {n, 0, 10}];
Round[Flatten[c]]
(* edited by Joshua Swanson, Sep 16 2016 *)

Crossrefs

Sequence in context: A300119 A323211 A110537 * A322057 A323767 A357824

Adjacent sequences: A144431 A144432 A144433 * A144435 A144436 A144437

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 04 2008

Extensions

Name corrected by Joshua Swanson, Sep 16 2016

Status

approved