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A257904
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Sequence (d(n)) generated by Algorithm (in Comments) with a(1) = 0 and d(1) = 2.
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3
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3, 1, 2, -1, 4, -2, 5, -4, 6, -3, 7, -8, 9, -6, 8, -5, 10, -11, 12, -10, 11, 13, -21, 14, -12, 15, -14, 16, -15, 18, -17, 19, -13, 17, -19, 20, -16, 21, -24, 22, -20, 23, -9, 24, -35, 25, -22, 26, -28, 27, -23, 28, -25, 29, -27, 30, -18, -7, 31, -34, 32, -26
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OFFSET
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1,1
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COMMENTS
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Algorithm: For k >= 1, let A(k) = {a(1), …, a(k)} and D(k) = {d(1), …, d(k)}. Begin with k = 1 and nonnegative integers a(1) and d(1). Let h be the least integer > -a(k) such that h is not in D(k) and a(k) + h is not in A(k). Let a(k+1) = a(k) + h and d(k+1) = h. Replace k by k+1 and repeat inductively.
Conjecture: if a(1) is an nonnegative integer and d(1) is an integer, then (a(n)) is a permutation of the nonnegative integers (if a(1) = 0) or a permutation of the positive integers (if a(1) > 0). Moreover, (d(n)) is a permutation of the integers if d(1) = 0, or of the nonzero integers if d(1) > 0.
See A257883 for a guide to related sequences.
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LINKS
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FORMULA
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a(k+1) - a(k) = d(k+1) for k >= 1.
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EXAMPLE
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a(1) = 0, d(1) = 2;
a(2) = 1, d(2) = 1;
a(3) = 4, d(3) = 3;
a(4) = 2, d(4) = -2.
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MATHEMATICA
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a[1] = 0; d[1] = 3; k = 1; z = 10000; zz = 120;
A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];
c[k_] := Complement[Range[-z, z], diff[k]];
T[k_] := -a[k] + Complement[Range[z], A[k]]
Table[{h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h,
d[k + 1] = h, k = k + 1}, {i, 1, zz}];
u = Table[a[k], {k, 1, zz}] (* A257903 *)
Table[d[k], {k, 1, zz}] (* A257904 *)
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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