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A171898
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Forward van Eck transform of A181391.
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11
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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, 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, 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
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OFFSET
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1,2
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COMMENTS
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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.
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
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
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LINKS
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FORMULA
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A181391(i+a(i)+1) = a(i) for any i, a(i)>0.
Conversely, a(j-A(181391(j)-1) = A181391(j) for any j, A181391(j)>0. (End)
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MAPLE
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ECKf:=proc(a) local b, i, m, n;
if whattype(a) <> list then RETURN([]); fi:
b:=[];
for n from 1 to nops(a)-1 do
# does a(n) appear again?
m:=0;
for i from n+1 to nops(a) do
if (a[i]=a[n]) then m:=i-n; break; fi
od:
b:=[op(b), m];
od:
b:=[op(b), 0];
RETURN(b);
end:
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MATHEMATICA
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terms = 100;
m = 14 terms; (* Increase m until no zero appears in the output *)
ClearAll[b, last]; b[_] = 0; last[_] = -1; last[0] = 2; nxt = 1;
Do[hist = last[nxt]; b[n] = nxt; last[nxt] = n; nxt = 0; If[hist > 0, nxt = n - hist], {n, 3, m}];
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]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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