A242448 -id:A242448 - cp's OEIS frontend

A242364 Irregular triangular array of all the integers ordered as in Comments.

1, 0, 2, -1, 4, -3, -2, 3, 8, -7, -6, -4, 5, 6, 16, -15, -14, -12, -8, -5, 7, 9, 10, 12, 32, -31, -30, -28, -24, -16, -11, -10, -9, 13, 14, 15, 17, 18, 20, 24, 64, -63, -62, -60, -56, -48, -32, -23, -22, -20, -19, -18, -17, -13, 11, 25, 26, 28, 29, 30, 31

Offset: 1

Clark Kimberling, Jun 11 2014

Let f1(x) = 2x, f2(x) = 1-x, f3(x) = 2-x, g(1) = (1), and g(n) = union(f1(g(n-1)), f2(g(n-1)),f3(g(n-1))) for n >1. Let T be the array whose n-th row consists of the numbers in g(n) arranged in increasing order. It is easy to prove that every integer occurs exactly once in T. Conjectures: (1) |g(n)| = 2*F(n-1) for n >=2, where F = A000045 (the Fibonacci numbers), and exactly half of the numbers in g(n) are positive; (2) the number of even numbers in g(n) is 2*F(n-2) and the number of odd numbers is 2*F(n-3).

			First 6 rows of the array:
1
0 .... 2
-1 ... 4
-3 ... -2 .... 3 .... 8
-7 ... -6 ... -4 .... 5 .... 6 .... 16
-15 .. -14 .. -12 .. -8 .... -5 ... 7 ... 9 ... 10 ... 12 ... 32

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A242365, A243735, A243736, A242448, A000045.

z = 12; g[1] = {1}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]], f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, 9}]
u1 = Flatten[u]  (* A242364 *)
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 1, z}]
v1 = Flatten[v]  (* A242365 *)
w1 = Table[Apply[Plus, g[n]], {n, 1, 20}]   (* A243735 *)
w2 = Table[Apply[Plus, v[[n]]], {n, 1, 10}] (* A243736 *)

A242365 Irregular triangular array of the positive integers ordered as in Comments.

Original entry on oeis.org

1, 2, 4, 8, 3, 16, 6, 5, 32, 12, 10, 9, 7, 64, 24, 20, 18, 17, 15, 14, 13, 128, 48, 40, 36, 34, 33, 31, 30, 29, 28, 26, 25, 11, 256, 96, 80, 72, 68, 66, 65, 63, 62, 61, 60, 58, 57, 56, 52, 50, 49, 23, 22, 21, 19, 512, 192, 160, 144, 136, 132, 130, 129, 127

Offset: 1

Clark Kimberling, Jun 11 2014

As in A242364, let f1(x) = 2x, f2(x) = 1-x, f3(x) = 2-x, g(1) = (1), and g(n) = union(f1(g(n-1)), f2(g(n-1)),f3(g(n-1))) for n >1. Let T be the array whose n-th row consists of the positive numbers in g(n) arranged in increasing order. It is easy to prove that every positive integer occurs exactly once in T.

Conjectures: (1) |g(n)| = F(n-1) for n >=2, where F = A000045 (the Fibonacci numbers); (2) the number of even numbers in g(n) is F(n-2) and the number of odd numbers is F(n-3).

			First 7 rows of the array:
1
2
4
8 ... 3
16 .. 6 ... 5
32 .. 12 .. 10 .. 9 ... 7
64 .. 24 .. 20 .. 18 .. 17 .. 15 .. 14 .. 13

Clark Kimberling, Table of n, a(n) for n = 1..1500

Cf. A242364, A243735, A243736, A242448, A000045.

z = 12; g[1] = {1}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]], f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, 9}]
u1 = Flatten[u]  (* A242364 *)
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 1, z}]
v1 = Flatten[v]  (* A242365 *)
w1 = Table[Apply[Plus, g[n]], {n, 1, 20}]   (* A243735 *)
w2 = Table[Apply[Plus, v[[n]]], {n, 1, 10}] (* A243736 *)

A243735 Sum of the numbers in row n of the array at A242364.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 26, 46, 88, 174, 342, 660, 1258, 2398, 4616, 8998, 17710, 35028, 69378, 137430, 272344, 540334, 1073798, 2137396, 4259770, 8496366, 16954536, 33843606, 67575358, 134965204

Offset: 1

Clark Kimberling, Jun 11 2014

			First 6 rows of the array at A242364:
1
0 .... 2
-1 ... 4
-3 ... -2 .... 3 .... 8
-7 ... -6 ... -4 .... 5 .... 6 .... 16
-15 .. -14 .. -12 .. -8 .... -5 ... 7 ... 9 ... 10 ... 12 ... 32;
the row sums are 1, 2, 3, 6, 10,16,...

Cf. A242364, A242365, A243736, A242448, A000045.

z = 12; g[1] = {1}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]], f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, 9}]
u1 = Flatten[u]  (* A242364 *)
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 1, z}]
v1 = Flatten[v]  (* A242365 *)
w1 = Table[Apply[Plus, g[n]], {n, 1, 20}]   (* A243735 *)
w2 = Table[Apply[Plus, v[[n]]], {n, 1, 10}] (* A243736 *)

A243736 Sum of the numbers in row n of the array at A242365.

Original entry on oeis.org

1, 2, 4, 11, 27, 70, 185, 499, 1356, 3695, 10075, 27482, 74997, 204751, 559172, 1527379, 4172439, 11398634, 31140481, 85075307, 232426476, 634995031, 1734829379, 4739627674, 12948881373, 35376966263, 96651610708, 264057013451, 721417015071, 1970947675114

Offset: 1

Clark Kimberling, Jun 11 2014

Cf. A242364, A242365, A243735, A242448, A000045.

z = 12; g[1] = {1}; f1[x_] := 2 x; f2[x_] := 1 - x; f3[x_] := 2 - x; h[1] = g[1]; b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]], f3[g[n - 1]]]]; h[n_] := h[n] = Union[h[n - 1], g[n - 1]]; g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]; u = Table[g[n], {n, 1, 9}]
u1 = Flatten[u]  (* A242364 *)
v = Table[Reverse[Drop[g[n], Fibonacci[n - 1]]], {n, 1, z}]
v1 = Flatten[v]  (* A242365 *)
w1 = Table[Apply[Plus, g[n]], {n, 1, 20}]   (* A243735 *)
w2 = Table[Apply[Plus, v[[n]]], {n, 1, 10}] (* A243736 *)

First 7 rows of the array at A242365:
1
2
4
8 ... 3
16 .. 6 ... 5
32 .. 12 .. 10 .. 9 ... 7
64 .. 24 .. 20 .. 18 .. 17 .. 15 .. 14 .. 13;
the row sums are 1, 2, 4, 11, 27, 70 185,...

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A242364 Irregular triangular array of all the integers ordered as in Comments.

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A242365 Irregular triangular array of the positive integers ordered as in Comments.

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A243735 Sum of the numbers in row n of the array at A242364.

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A243736 Sum of the numbers in row n of the array at A242365.

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