Consider all "integer" points in the 1st Quadrant, i.e. NorthEast part of the coordinates system.
How many lattice paths from (0,0) to (b,a) exist if only east (1,0), north (0,1), and northeast (1,1) steps are allowed?
Provide a general recurrence formula, supported by few samples, say all (a,b) points between (0,0) and (6,6).
What can be said about the numbers thus obtained?
Try to formulate a direct formula for the integer points on the y=x line, i.e. D(m,m)=...
(In reply to
solution by Charlie)
The numbers are also appear to be the centered orthoplicial polytopic numbers, and can be given by the recurrence equation:
_{c}P^{(d)}(2d,n) = _{c}P^{(d)}(2d,n1)
+ _{c}P^{(d1)}(2(d1),n)
+ _{c}P^{(d1)}(2(d1),n1),
with initial conditions
_{c}P^{(d)}(2d,0) = 1
_{c}P^{(1)}(2·1,n) = 2n + 1
_{c}P^{(1)}(2·1,n) = (2n+1)
Centered square gnomom (1orthoplex) numbers:
{1,3,5,7,9,11,13,...}
_{c}P^{(2)}(2·2,n) = 2n(n+1)+1
Centered square (2orthoplex) Tetragonal numbers:
{1,5,13,25,41,61,85,...}
_{c}P^{(3)}(2·3,n) = (2n+1)(2n^{2}+2n+3)/3
Centered octahedral (3orthoplex) Octahedral numbers:
{1,7,25,63,129,231,377,...}
_{c}P^{(4)}(2·4,n) = (3+8n+10n^{2}+4n^{3}+2n^{4})/3
Centered tetracross (4orthoplex) numbers:
{1,9,41,129,321,681,1280,...}
_{c}P^{(5)}(2·5,n) = (15+46n+50n^{2}+40n^{3}+10n^{4}+4n^{5})/15
Centered pentacross (5orthoplex) numbers:
{1,11,61,231,681,1683,3653,...}
_{c}P^{(6)}(2·6,n) =(45+138n+196n^{2}+120n^{3}+70n^{4}+12n^{5}+4n^{6})/45
Centered hexacross (6orthoplex) numbers:
{1,13,85,377,1289,3653,8989,...}

Posted by Dej Mar
on 20130322 06:06:00 