A379914 -id:A379914 - cp's OEIS frontend

A379998 Irregular triangle read by rows: T(n,k) is number of sequences of length k over {0,1,...,n-1} containing no two consecutive blocks with the same average, n >= 1, 0 <= k <= A379914(n).

1, 1, 1, 2, 2, 2, 1, 3, 6, 8, 1, 4, 12, 28, 38, 50, 24, 6, 1, 5, 20, 64, 148, 316, 370, 340, 152, 38, 1, 6, 30, 126, 406, 1142, 2142, 3380, 4022, 3910, 2794, 2048, 988, 496, 234, 82, 14, 10, 4, 2, 1, 7, 42, 216, 898, 3314, 9014, 21760, 41026, 63898, 78204, 87820, 71434, 53984, 34232, 16716, 6400, 2346, 644, 148, 12

Offset: 1

Pontus von Brömssen, Jan 09 2025

See A379914 for details.

A sequence, its reversal, and its complement (where all terms x are replaced by n-1-x) are all counted.

			Triangle begins:
  1, 1;
  1, 2,  2,  2;
  1, 3,  6,  8;
  1, 4, 12, 28,  38,  50,  24,   6;
  1, 5, 20, 64, 148, 316, 370, 340, 152, 38;
  ...

Pontus von Brömssen, Table of n, a(n) for n = 1..139 (rows 1..9)

Cf. A245996, A379914, A379999, A380000.

T(n,0) = 1.
T(n,1) = n.
T(n,2) = n*(n-1) for n >= 2.
T(n,3) = A245996(n-1) for n >= 2.
Empirically: T(n,4) = T(n-1,4) + T(n-2,4) - T(n-5,4) - T(n-6,4) - T(n-7,4) + T(n-8,4) + T(n-9,4) + T(n-10,4) - T(n-13,4) - T(n-14,4) + T(n-15,4) for n >= 19.

A379999 Number of longest sequences over {0,1,...,n-1} containing no two consecutive blocks with the same average.

Original entry on oeis.org

1, 2, 8, 6, 38, 2, 12, 8, 2

Offset: 1

Pontus von Brömssen, Jan 09 2025

a(n) is the last element of row n in A379998.
The sequences counted by a(n) have length A379914(n).
a(n) is even for all n >= 2, because each term x in a sequence can be replaced by n-1-x, giving another sequence of maximum length.

			For 1 <= n <= 4, the following sequences are counted:
  n | longest sequences
  --+-----------------------------------------------------
  1 | 0
  2 | 010, 101
  3 | 010, 012, 020, 101, 121, 202, 210, 212
  4 | 0203202, 1310131, 1310313, 2023020, 2023202, 3130131

Cf. A379914, A379998, A380000.

A380000 Number of sequences over {0,1,...,n-1} containing no two consecutive blocks with the same average.

Original entry on oeis.org

2, 7, 18, 163, 1454, 21837, 492116, 23699853, 1507394232

Offset: 1

Pontus von Brömssen, Jan 09 2025

See A379914 for details.

a(n) is even if and only if n is odd; for each sequence can be paired up with its complement (where all terms x are replaced by n-1-x), and the only self-complementary sequences (that also satisfy the consecutive blocks condition) are the empty sequence and those with a single term (n-1)/2 for odd n.

			For n = 3, the a(3) = 18 sequences are: (), 0, 1, 2, 01, 02, 10, 12, 20, 21, 010, 012, 020, 101, 121, 202, 210, 212.

Row sums of A379998.

Cf. A379914, A379999.

cp's OEIS Frontend

A379998 Irregular triangle read by rows: T(n,k) is number of sequences of length k over {0,1,...,n-1} containing no two consecutive blocks with the same average, n >= 1, 0 <= k <= A379914(n).

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A379999 Number of longest sequences over {0,1,...,n-1} containing no two consecutive blocks with the same average.

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A380000 Number of sequences over {0,1,...,n-1} containing no two consecutive blocks with the same average.

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