Heptanacci Number

Author

Eric W. Weisstein
April 27, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/IntegerSequences/HeptanacciNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HeptanacciNumber.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Initialization

FibonacciNStep::usage="FibonacciNStep[n,k] gives the nth k-step Fibonacci number."

(*Fibonacci N-Step*)

FibonacciNStep[n_Integer?Negative,k_]:=0
FibonacciNStep[0,k_]:=0
FibonacciNStep[1,k_]:=1
FibonacciNStep[2,k_]:=1
FibonacciNStep[n_,k_]:=Module[{i},
FibonacciNStep[n,k]=Sum[FibonacciNStep[n-i,k],{i,k}]
]

Enumeration

FibonacciNStep[#,7]&/@Range[0,20]

Recurrence

RSolve[{H[0]==0,H[1]==1,H[2]==1,H[3]==2,H[4]==4,H[5]==8,H[6]==16,H[n]==H[n-1]+H[n-2]+H[n-3]+H[n-4]+H[n-5]+H[n-6]+H[n-7]},H[n],n]

Table[RootSum[Function[x,x^7-x^6-x^5-x^4-x^3-x^2-x-1],(((#)^n)/(-#^6+#^4+2#^3+3#^2+12#-1))&],{n,20}]

Ratio of Terms

Reduce[x^7-x^6-x^5-x^4-x^3-x^2-x-1==0&&x>0,x,Reals]

N[%,20]