A257047 -id:A257047 - cp's OEIS frontend

A257905 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 0 and d(1) = 0.

0, 1, 3, 2, 5, 11, 4, 9, 6, 13, 7, 15, 10, 8, 17, 35, 12, 25, 14, 29, 16, 33, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 30, 26, 53, 24, 49, 40, 28, 57, 27, 55, 31, 63, 32, 65, 38, 42, 34, 69, 36, 73, 48, 97, 44, 89, 46, 93, 51, 103, 52, 105, 50, 101

Offset: 1

Clark Kimberling, May 16 2015

Rule 3 follows.  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).
Step 1:   If there is an integer h such that 1 - a(k) < h < 0 and h is not in D(k) and a(k) + h is not in A(k), let d(k+1) be the least such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.
Step 2:  Let h be the least positive integer not in D(k) such that 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 do Step 1.
Conjecture:  suppose that a(1) is an nonnegative integer and d(1) is an integer.
If a(1) = 0 and d(1) != 1, then (a(n)) is a permutation of the nonnegative integers;
if a(1) = 0 and d(1) = 1, then (a(n)) is a permutation of the nonnegative integers excluding 1;
if a(1) = 1, then (a(n)) is a permutation of the positive integers;
if a(1) > 1, then (a(n)) is a permutation of the integers >1;
if d(1) = 0, then (d(n)) is a permutation of the integers;
if d(1) !=0, then (d(n)) is a permutation of the nonzero integers.
Guide to related sequences:
a(1)  d(1)      (a(n))       (d(n))
0       0      A257905      A258047
0       1      A257906      A257907
0       2      A257908      A257909
0       3      A257910      A257980
1       0      A258046      A258047
1       1      A257981      A257982
1       2      A257983      A257909
2       0      A257985      A257047
2       1      A257986      A257982
2       2      A257987      A257909

			a(1) = 0, d(1) = 0;
a(2) = 1, d(2) = 1;
a(3) = 3, d(3) = 2;
a(4) = 2, d(4) = -1.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A258047, A257705, A257883, A175498.

Cf. A256283 (putative inverse).

import Data.List ((\\))
a257905 n = a257905_list !! (n-1)
a257905_list = 0 : f [0] [0] where
   f xs@(x:_) ds = g [2 - x .. -1] where
     g [] = y : f (y:xs) (h:ds) where
                  y = x + h
                  (h:_) = [z | z <- [1..] \\ ds, x - z `notElem` xs]
     g (h:hs) | h `notElem` ds && y `notElem` xs = y : f (y:xs) (h:ds)
              | otherwise = g hs
              where y = x + h
-- Reinhard Zumkeller, Jun 03 2015

{a, f} = {{0}, {0}}; Do[tmp = {#, # - Last[a]} &[Min[Complement[#, Intersection[a, #]]&[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f, #] && ! MemberQ[a, Last[a] - #]) &]]]; AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {120}]; {a, f} (* Peter J. C. Moses, May 14 2015 *)

a(n) = A258046(n) - 1 for n >= 1.

A256914 Trace of the enhanced squares representation of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 2, 3, 4, 1, 2, 16, 1, 2, 3, 4, 1, 2, 3, 1, 25, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 36, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 2, 3, 49, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 2, 3, 4, 1, 64, 1, 2, 3, 4, 1, 2, 3, 1, 9, 1, 2, 3, 4, 1, 2, 16, 81, 1, 2

Offset: 0

Clark Kimberling, Apr 14 2015

See A256913 for definitions.

a(A257046(n)) = 1; a(A257047(n)) != 1. - Reinhard Zumkeller, Apr 15 2015

			R(0) = 0, so trace = 0.
R(1) = 1, so trace = 1.
R(8) = 4 + 3 + 1, so trace = 1.
R(43) = 36 + 4 + 3, so trace = 3.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000290, A256913, A256915 (number of terms).

Cf. A257046, A257047, A257070.

a256914 = last . a256913_row  -- Reinhard Zumkeller, Apr 15 2015

b[n_] := n^2; bb = Insert[Table[b[n], {n, 0, 100}]  , 2, 3];
s[n_] := Table[b[n], {k, 1, 2 n + 1}];
h[1] = {0, 1, 2, 3}; h[n_] := Join[h[n - 1], s[n]];
g = h[100]; Take[g, 100]
r[0] = {0}; r[1] = {1}; r[2] = {2}; r[3] = {3}; r[8] = {4, 3, 1};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
t = Table[r[n], {n, 0, 120}] (* A256913, before concatenation *)
Flatten[t]  (* A256913 *)
Table[Last[r[n]], {n, 0, 120}]    (* A256914 *)
Table[Length[r[n]], {n, 0, 200}]  (* A256915 *)

A257046 Numbers having trace 1 in their enhanced squares representation, see A256913.

Original entry on oeis.org

1, 5, 8, 10, 14, 17, 21, 24, 26, 30, 33, 35, 37, 41, 44, 46, 50, 54, 57, 59, 63, 65, 69, 72, 74, 78, 82, 86, 89, 91, 95, 98, 101, 105, 108, 110, 114, 117, 122, 126, 129, 131, 135, 138, 142, 145, 149, 152, 154, 158, 161, 165, 168, 170, 174, 177, 179, 183, 186

Offset: 1

Reinhard Zumkeller, Apr 15 2015

nonn

A256914(a(n)) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A256913, A256914, A257047 (complement).

a257046 n = a257046_list !! (n-1)
a257046_list = filter ((== 1) . a256914) [0..]

cp's OEIS Frontend

A257905 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 0 and d(1) = 0.

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A256914 Trace of the enhanced squares representation of n.

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Haskell

Mathematica

A257046 Numbers having trace 1 in their enhanced squares representation, see A256913.

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Haskell