Tribonacci sequence(order 3)
f(0)=0
f(1)=0
f(2)=1
f(i)= f(i-3)+f(i-2)+f(i-1)
for i=3,4,5, . . .
Tetranacci sequence(order 4)
f(0)=0
f(1)=0
f(2)=0
f(3)=1
f(i)= f(i-4)+f(i-3)+f(i-2)+f(i-1)
for i=4,5,6, . . .
n-nacci sequence(order n)
f(0)=0
f(1)=0
f(2)=0
.
.
.
f(n-2)=0
f(n-1)=1
f(i)= f(i-n)+f(i-(n-1))+. . .
+f(i-3)+f(i-2)+f(i-1) for i=n,n+1,n+2, . . .
f(0)=x[0]
f(1)=x[1]
f(2)=x[2]
.
.
.
f(n-2)=x[n-2]
f(n-1)=x[n-1]
f(i)= c[n]*f(i-n)+c[n-1]*f(i-(n-1))+. . .
+c[3]*f(i-3)+c[2]*f(i-2)+c[1]*f(i-1) for
i=n,n+1,n+2, . . .
examples:
Padovan sequence x={1,1,1} and c={0,1,1}
n-nacci sequence x={0,0, . . . ,0,1} and c={1,1, . . . ,1,1}
An application: multiple recursive generator
x[i]=(a1*x[i-1]+a2*x[i-2]+a3*x[i-3]+a4*x[i-4]+a5*x[i-5]) mod m.
for a1=107374182, a2=0, a3=0, a4=0, a5=104480, and m=2^31-1.
where initial ate set by a certain rule or at random.