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# User:Jean-François Alcover

Retired engineer, born 1947, Alma mater: Ecole Centrale Paris 1970.

Name: Jean-François Alcover

Location: Paris, France

## Mathematica scripts for autosequences

firstKindQ[T_List] :=

Module[{ta, tb},

tb = Table[Sum[(-1)^(n-k)*Binomial[n, k]*Part[T,k+1],

{k, 0, Length[T] - 1}], {n, 0, Length[T]-1}];

ta = Table[(-1)^(n + 1) Part[T,n + 1], {n, 0, Length[T]-1}] ;

First[T] == 0 && ta == tb];

secondKindQ[T_List] :=

Module[{ta, tb},

tb = Table[Sum[(-1)^(n - k)*Binomial[n, k]*Part[T,k+1],

{k, 0, Length[T]-1}], {n, 0, Length[T]-1}];

ta = Table[(-1)^n Part[T,n+1], {n, 0, Length[T]-1}] ; ta == tb];

toSecondKind[T_?firstKindQ] :=

Table[2 Part[T,n+1] - Part[T,n], {n, 1, Length[T]-1}];

toFirstKind[T_?secondKindQ] :=

Table[1/2^n Sum[2^k Part[T,k+1], {k, 0, n-1}], {n, 0, Length[T]}];

autosequenceQ[T_List] := Which[

firstKindQ[T], Print["first kind, its second kind companion is ", toSecondKind[T]]; True,

secondKindQ[T], Print["second kind, its first kind companion is ", toFirstKind[T]]; True,

True, Print["not an autosequence"]; False];

Example:

autosequenceQ[Table[Fibonacci[n], {n, 0, 10}]]

first kind, its second kind companion is {2,1,3,4,7,11,18,29,47,76}