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A171898 Forward van Eck transform of A181391.

%I A171898 #37 Feb 07 2025 05:48:44
%S A171898 1,2,6,2,2,5,1,6,42,5,2,4,5,9,14,3,9,3,15,2,4,6,17,3,6,32,56,5,3,131,
%T A171898 5,11,5,3,20,6,2,8,15,31,170,3,31,18,3,3,33,5,1,11,46,56,4,37,152,307,
%U A171898 3,7,92,4,7,62,52,3,42,3,6,2,19,6,8,3,9,3,650,2,23,8,223,7,206,3,21,25,5,8
%N A171898 Forward van Eck transform of A181391.
%C A171898 Given a sequence a, the forward van Eck transform b is defined as follows: If a(n) also appears again in a later position, let a(m) be the next occurrence, and set b(n)=m-n; otherwise b(n)=0.
%C A171898 This is a permutation of the positive terms in A181391, where each term m > 0 from that sequence is shifted backwards m+1 positions. - _Jan Ritsema van Eck_, Aug 16 2019
%C A171898 The backwards van Eck transform searches backwards for a repeated value: if a(n) also has appeared in earlier positions, a(m)=a(n) with m<n, then b(n) is the minimum n-m. - _R. J. Mathar_, Jun 24 2021
%H A171898 Alois P. Heinz, <a href="/A171898/b171898.txt">Table of n, a(n) for n = 1..10000</a>
%F A171898 From _Jan Ritsema van Eck_, Aug 16 2019: (Start)
%F A171898 A181391(i+a(i)+1) = a(i) for any i, a(i)>0.
%F A171898 Conversely, a(j-A181391(j)-1) = A181391(j) for any j, A181391(j)>0. (End)
%p A171898 ECKf:=proc(a) local b,i,m,n;
%p A171898 if whattype(a) <> list then RETURN([]); fi:
%p A171898 b:=[];
%p A171898 for n from 1 to nops(a)-1 do
%p A171898 # does a(n) appear again?
%p A171898 m:=0;
%p A171898 for i from n+1 to nops(a) do
%p A171898 if (a[i]=a[n]) then m:=i-n; break; fi
%p A171898 od:
%p A171898 b:=[op(b),m];
%p A171898 od:
%p A171898 b:=[op(b),0];
%p A171898 RETURN(b);
%p A171898 end:
%t A171898 terms = 100;
%t A171898 m = 14 terms; (* Increase m until no zero appears in the output *)
%t A171898 ClearAll[b, last]; b[_] = 0; last[_] = -1; last[0] = 2; nxt = 1;
%t A171898 Do[hist = last[nxt]; b[n] = nxt; last[nxt] = n; nxt = 0; If[hist > 0, nxt = n - hist], {n, 3, m}];
%t A171898 A181391 = Array[b, m];
%t A171898 ECKf[a_List] := Module[{b = {}, i, m, n}, For[n = 1, n <= Length[a]-1, n++, m = 0; For[i = n+1, i <= Length[a], i++, If[a[[i]] == a[[n]], m = i-n; Break[]]]; b = Append[b, m]]; b = Append[b, 0]; Return[b]];
%t A171898 ECKf[A181391][[;; terms]] (* _Jean-François Alcover_, Oct 30 2020, after Maple *)
%Y A171898 Cf. A181391 (van Eck's sequence), A171899, A171942.
%K A171898 nonn
%O A171898 1,2
%A A171898 _N. J. A. Sloane_, Oct 22 2010