A192646 -id:A192646 - cp's OEIS frontend

A192645 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2 - y^2 > 0 then x^2 - y^2 is in S, and 1 and 2 are in S.

1, 2, 3, 5, 8, 16, 21, 24, 39, 55, 60, 63, 135, 185, 192, 231, 247, 252, 255, 320, 369, 377, 416, 432, 437, 440, 512, 551, 567, 572, 575, 944, 945, 1080, 1265, 1457, 1496, 1504, 1512, 1517, 1520, 1521, 1889, 2079, 2448, 2449, 2495, 2584, 2631, 2639

Offset: 1

Clark Kimberling, Jul 06 2011

nonn

See A192476 for a general discussion.  Related sequences:
A192645:  f(x,y) = x^2 - y^2 > 0, start={1,2};
A192647:  f(x,y) = x^2 - y^2 > 0, start={1,3};
A192648:  f(x,y) = x^2 - y^2 > 0, start={2,3};
A192649:  f(x,y) = x^2 - y^2 > 0, start={1,2,4}.

			2^2 - 1^2 = 3;
3^2 - 2^2 = 5, 3^2 - 1^2 = 8;
5^2 - 3^2 = 16, 5^2 - 2^2 = 21, 5^2 - 1^2 = 24.
Taking the generating procedure in the order just indicated results in the monotonic ordering of the sequence and also suggests a triangular format for the generated terms:
    3;
    5,   8;
   16,  21,  24;
   39,  55,  60,  63;
  135, 185, 192, 231, 247;
  ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A192476, A192646 (first differences).

start = {1, 2};
f[x_, y_] := If[MemberQ[Range[1, 5000], x^2 - y^2], x^2 - y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      5000 &]];
t = FixedPoint[b, start]  (* A192645 *)
Differences[t] (* A192646 *)

A192649 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2-y^2>0 then x^2-y^2 is in S, and 1, 2, and 4 are in S.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 12, 15, 16, 17, 21, 24, 31, 32, 33, 39, 40, 45, 48, 55, 56, 60, 63, 64, 65, 72, 77, 79, 80, 81, 95, 111, 112, 119, 127, 128, 129, 135, 140, 143, 144, 145, 152, 159, 161, 175, 176, 185, 192, 200, 207, 208, 209, 216, 221, 223, 224, 225, 231

Offset: 1

Clark Kimberling, Jul 06 2011

nonn

See A192645.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A192476, A192646, A192650.

start = {1, 2, 4};
f[x_, y_] := If[MemberQ[Range[1, 700], x^2 - y^2], x^2 - y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      700 &]];
t = FixedPoint[b, start]  (* A192649 *)
Differences[t] (* A192650 *)

A192647 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2 - y^2 > 0 then x^2 - y^2 is in S, and 1 and 3 are in S.

Original entry on oeis.org

1, 3, 8, 55, 63, 944, 2961, 3016, 3024, 3905, 3960, 3968, 48320, 63424, 328735, 377055, 432575, 495999, 887167, 888111, 891072, 891127, 891135, 6104449, 6152769, 6481504, 6537024, 6585344, 6600448, 6648768, 6914079, 6977503, 7876385, 8205120

Offset: 1

Clark Kimberling, Jul 06 2011

nonn

See A192645.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A192476, A192646.

start = {1, 3};
f[x_, y_] := If[MemberQ[Range[1, 500000], x^2 - y^2], x^2 - y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      500000 &]];
t = FixedPoint[b, start]  (* A192647 *)

A192648 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2-y^2>0 then x^2-y^2 is in S, and 2 and 3 are in S.

Original entry on oeis.org

2, 3, 5, 16, 21, 185, 231, 247, 252, 416, 432, 437, 2495, 4345, 7648, 10143, 13568, 17913, 19136, 26784, 29279, 33784, 33969, 34200, 34216, 34221, 52920, 53105, 53336, 53352, 53357, 60568, 60753, 60984, 61000, 61005, 63063, 63248, 63479

Offset: 1

Clark Kimberling, Jul 06 2011

nonn

See A192645.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A192476, A192646.

start = {2, 3};
f[x_, y_] := If[MemberQ[Range[1, 150000], x^2 - y^2], x^2 - y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      150000 &]];
t = FixedPoint[b, start]  (* A192648 *)

A192650 First differences of A192649.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 3, 7, 1, 1, 6, 1, 5, 3, 7, 1, 4, 3, 1, 1, 7, 5, 2, 1, 1, 14, 16, 1, 7, 8, 1, 1, 6, 5, 3, 1, 1, 7, 7, 2, 14, 1, 9, 7, 8, 7, 1, 1, 7, 5, 2, 1, 1, 6, 9, 7, 5, 3, 1, 1, 7, 9, 6, 1, 5, 2, 1, 1, 8, 15, 8, 31, 9, 9, 8, 8, 7, 23, 1, 1, 8, 7, 5, 3, 7, 1, 1, 15, 7, 24, 1

Offset: 1

Clark Kimberling, Jul 06 2011

nonn

Are there infinitely many pairs 1,1?

Cf. A192649, A192646.

start = {1, 2, 4};
f[x_, y_] := If[MemberQ[Range[1, 700], x^2 - y^2], x^2 - y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      700 &]];
t = FixedPoint[b, start]  (* A192649 *)
Differences[t] (* A192650 *)

cp's OEIS Frontend

A192645 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2 - y^2 > 0 then x^2 - y^2 is in S, and 1 and 2 are in S.

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A192649 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2-y^2>0 then x^2-y^2 is in S, and 1, 2, and 4 are in S.

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A192647 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2 - y^2 > 0 then x^2 - y^2 is in S, and 1 and 3 are in S.

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A192648 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2-y^2>0 then x^2-y^2 is in S, and 2 and 3 are in S.

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A192650 First differences of A192649.

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