A274190 -id:A274190 - cp's OEIS frontend

A274199 Limiting reverse row of the array A274190.

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 67, 101, 152, 228, 342, 511, 763, 1138, 1695, 2523, 3752, 5578, 8287, 12307, 18272, 27119, 40241, 59700, 88556, 131340, 194772, 288815, 428229, 634900, 941263, 1395397, 2068560, 3066372, 4545387, 6737633, 9987026, 14803303

Offset: 0

Clark Kimberling, Jun 13 2016

The triangular array (g(n,k)) at A274190 is defined as follows:  g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,2k) for n > 0, k > 1.
From Gus Wiseman, Mar 12 2021: (Start)
Also (apparently) the number of compositions of n where all adjacent parts (x, y), satisfy x < 2y. For example, the a(1) = 1 through a(6) = 12 compositions are:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (111)  (22)    (23)     (24)
                    (112)   (32)     (33)
                    (1111)  (113)    (114)
                            (122)    (123)
                            (1112)   (132)
                            (11111)  (222)
                                     (1113)
                                     (1122)
                                     (11112)
                                     (111111)
(End)

			Row (g(14,k)):  1, 51, 73, 69, 55, 40, 28, 19, 12, 8, 5, 3, 2, 1, 1; the reversal is 1 1 2 3 5 8 12 19 28 ..., which agrees with A274199 up to 19.

Daniel Gabric and Jeffrey Shallit, Smallest and Largest Block Palindrome Factorizations, arXiv:2302.13147 [math.CO], 2023.

Cf. A274190, A274200, A274201.

Cf. A000929, A003242, A154402, A224957, A342094, A342095, A342096, A342097, A342098, A342191, A342330-A342342.

g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 2 k]];
z = 300; u = Reverse[Table[g[z, k], {k, 0, z}]];
z = 301; v = Reverse[Table[g[z, k], {k, 0, z}]];
w = Join[{1}, Intersection[u, v]] (* A274199 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]<2*#[[i-1]],{i,2,Length[#]}]&]],{n,15}] (* Gus Wiseman, Mar 12 2021 *)

A274192 Decimal expansion of limiting ratio described in Comments.

Original entry on oeis.org

1, 4, 8, 2, 1, 4, 6, 2, 2, 1, 0, 4, 5, 7, 9, 6, 4, 7, 3, 9, 5, 1, 0, 9, 4, 5, 0, 5, 0, 8, 9, 2, 9, 2, 1, 8, 8, 1, 0, 0, 7, 2, 2, 0, 9, 9, 2, 0, 0, 8, 2, 7, 9, 6, 3, 7, 8, 9, 8, 7, 8, 3, 7, 8, 8, 6, 2, 3, 2, 4, 8, 7, 2, 9, 2, 5, 0, 1, 6, 9, 3, 4, 8, 5, 8, 6

Offset: 1

Clark Kimberling, Jun 13 2016

As in A274190, define g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,2k) for n > 0, k > 1. The sum of numbers in the n-th row of the array {g(n,k)} is given by A274184; viz., this sum is also the number of numbers in the n-th row of the array in A274183. In other words, if we put h(0) = (0) and for n > 0 define h(n) inductively as the concatenation of h(n-1) and the numbers k/2 as k ranges through the even numbers k in h(n-1), and then let H(n) be the number of numbers in h(n), then H(n)/H(n-1) approaches 1.48214622...

This constant appears on p. 439 of Tangora's paper cited in Links.

			Limiting ratio = 1.48214622104579647395109450508929...

M. C. Tangora, Level number sequences of trees and the lambda algebra, European J. Combinatorics 12 (1991), 433-443.

Cf. A274190, A274184, A274195, A274198, A274209 (reciprocal).

z = 1500; g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 2 k]];
t = Table[g[n, k], {n, 0, z}, {k, 0, n}];
w = Map[Total, t];   (* A274184 *)
u = N[w[[z]]/w[[z - 1]], 100]
RealDigits[u][[1]] (* A274192 *)

A274193 Triangular array read by rows: g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,3k) for n > 0, k > 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 3, 2, 2, 1, 1, 1, 1, 3, 4, 3, 2, 2, 1, 1, 1, 1, 4, 4, 4, 3, 2, 2, 1, 1, 1, 1, 5, 6, 5, 4, 3, 2, 2, 1, 1, 1, 1, 6, 7, 7, 5, 4, 3, 2, 2, 1, 1, 1, 1, 8, 9, 8, 7, 5, 4, 3

Offset: 0

Clark Kimberling, Jun 14 2016

			First  10 rows:
1
1   1
1   1   1
1   1   1   1
1   2   1   1   1
1   2   2   1   1   1
1   2   2   2   1   1   1
1   3   3   2   2   1   1   1
1   3   4   3   2   2   1   1   1
1   4   4   4   3   2   2   1   1   1

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A274194 (row sums), A274195, A274200 (limiting reverse row), A274190, A274196.

g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 3 k]];
t = Table[g[n, k], {n, 0, 14}, {k, 0, n}]
TableForm[t] (* A274193 array *)
u = Flatten[t] (* A274193 sequence *)

A274196 Triangular array read by rows: g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,4k) for n > 0, k > 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 3, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 3, 4, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 4, 4, 4, 4, 3, 2, 2

Offset: 0

Clark Kimberling, Jun 16 2016

			First  10 rows:
1
1   1
1   1   1
1   1   1   1
1   1   1   1   1
1   2   1   1   1   1
1   2   2   1   1   1   1
1   2   2   2   1   1   1   1
1   2   2   2   2   1   1   1   1
1   3   3   2   2   2   1   1   1   1

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A274197 (row sums), A274201 (limiting reverse row), A274190.

g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 4 k]];
t = Table[g[n, k], {n, 0, 14}, {k, 0, n}]
TableForm[t] (* A274196 array *)
u = Flatten[t] (* A274196 sequence *)

A274209 Decimal expansion of the reciprocal of the constant in A274192; see Comments.

Original entry on oeis.org

6, 7, 4, 6, 9, 7, 2, 6, 3, 8, 7, 3, 4, 6, 8, 5, 5, 7, 2, 7, 6, 8, 0, 8, 6, 2, 9, 7, 5, 4, 9, 5, 0, 1, 5, 2, 7, 5, 4, 3, 1, 2, 9, 5, 6, 3, 0, 9, 6, 5, 1, 7, 0, 3, 9, 5, 5, 2, 2, 1, 9, 2, 8, 8, 1, 5, 1, 6, 2, 9, 1, 9, 3, 7, 9, 7, 5, 3, 2, 6, 7, 7, 6, 2, 7, 7

Offset: 0

Clark Kimberling, Jun 16 2016

As in A274190, define g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,2k) for n > 0, k > 1. The sum of numbers in the n-th row of the array {g(n,k)} is given by A274184; viz., this sum is also the number of numbers in the n-th row of the array in A274183. In other words, if we put h(0) = (0) and for n > 0 define h(n) inductively as the concatenation of h(n-1) and the numbers k/2 as k ranges through the even numbers k in h(n-1), and then let H(n) be the number of numbers in h(n), then H(n)/H(n+1) approaches 0.67469726387...

This constant appears on p. 439 of Tangora's paper cited in Links.

			Limiting ratio = 0.6746972638734685572768086297549501...

M. C. Tangora, Level number sequences of trees and the lambda algebra, European J. Combinatorics 12 (1991), 433-443.

Cf. A274192, A274183, A274184, A274210, A274211.

z = 1600; g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 2 k]];
t = Table[g[n, k], {n, 0, z}, {k, 0, n}];
w = Map[Total, t]; (*A274184*)
u = N[w[[z]]/w[[z + 1]], 100]
d = RealDigits[u][[1]] (*A274209*)

cp's OEIS Frontend

A274199 Limiting reverse row of the array A274190.

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A274192 Decimal expansion of limiting ratio described in Comments.

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A274193 Triangular array read by rows: g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,3k) for n > 0, k > 1.

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A274196 Triangular array read by rows: g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,4k) for n > 0, k > 1.

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A274209 Decimal expansion of the reciprocal of the constant in A274192; see Comments.

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