again numbers

are the answers to problem on numbers is correct? i am confused. i am self taught but how do i become sure? i have none to certify me and i am long ago decided that i need no certificates. the year was 1974 february when i left school. my brother did not leave school nor he did anything to get any approval or certificate. completeness or self sufficiency and license to do the same.

i have of course received many certificates of insanity and i receive them everyday without exception.

i am talking of numbers.

there is staircase of 10 stairs. I have limited vision or ability to climb the stairs one at a time or skip one jump to the next. how many ways can i reach the top of the staircase? the answer is 10th Fibonacci number. 

f(n)= f(n-1)+f(n-2)for n>=2; f(m)=0 for m<0;

f(0)=1; f(1)=1; this is recursive definition.

f(2)= f(1)+f(0); ... f(10)=f(9)+f(8)

n is number of stairs to climb.

now, i have greater ability and can jump up to 3 steps at a time. how many ways can i reach top of the stairs. 

g(0)=1; g(1)=1; g(2)=2; g(n)=g(n-2)+g(n-1)+g(n-3) for n>=3; g(m)=0 for m<0

g(3)=4; g(5)=7; g(6)=13; g(7)=24; g(8)=44; g(9)=81; g(10)=149;

how many ways can i go to 10th floor from floor 0 in an elevator?

there is a 2x10 rectangle board; dominoes are we know of dimension 1x2 rectangle. how many ways can we fill the board with dominoes. it is again Fibonacci number f(10).

suppose the board is yx10 and where y>2; can staircase problem has an equivalent rectangle filling generalization for dominoes of any size 1, 2, 3 ... i don't know. i could not write a recursive definition.

not all problems can be solved recursively. all recursions can be written as iteration but not  all iterations is recursive.