A171898 Forward van Eck transform of A181391.

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

Offset: 1

Author

N. J. A. Sloane, Oct 22 2010

Keywords

• nonn

Comments

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 mR. J. Mathar, Jun 24 2021

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A181391 (van Eck's sequence), A171899, A171942.

Programs

Maple

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:

Mathematica

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}];
A181391 = Array[b, 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]];
ECKf[A181391][[;; terms]] (* Jean-François Alcover, Oct 30 2020, after Maple *)

Formula

From Jan Ritsema van Eck, Aug 16 2019: (Start)

A181391(i+a(i)+1) = a(i) for any i, a(i)>0.

Conversely, a(j-A181391(j)-1) = A181391(j) for any j, A181391(j)>0. (End)