A340257 -id:A340257 - cp's OEIS frontend

A340298 a(n) = a(floor(n/2)) + a(ceiling(n/2)) + n*floor(log_2(n)) for n >= 2, a(n) = n for n <= 1.

0, 1, 4, 8, 16, 22, 28, 38, 56, 65, 74, 83, 92, 105, 118, 139, 176, 189, 202, 215, 228, 241, 254, 267, 280, 297, 314, 331, 348, 373, 398, 439, 512, 530, 548, 566, 584, 602, 620, 638, 656, 674, 692, 710, 728, 746, 764, 782, 800, 822, 844, 866, 888, 910, 932, 954

Offset: 0

Alois P. Heinz, Jan 03 2021

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A033156, A340257, A340301.

a:= proc(n) option remember; `if`(n<2, n, (h->
      a(h)+a(n-h)+n*ilog2(n))(iquo(n, 2)))
    end:
seq(a(n), n=0..55);

a(2^n) = A340257(n).

A369324 Array read by ascending antidiagonals: A(n,k) is the number of words of length n on an alphabet [k], avoiding 120 and 210, and sortable by a stack of depth 2, where k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 16, 25, 16, 5, 1, 0, 1, 32, 65, 56, 25, 6, 1, 0, 1, 64, 161, 176, 105, 36, 7, 1, 0, 1, 128, 385, 512, 385, 176, 49, 8, 1, 0, 1, 256, 897, 1408, 1281, 736, 273, 64, 9, 1, 0, 1, 512, 2049, 3712, 3969, 2752, 1281, 400, 81, 10, 1

Offset: 0

Stefano Spezia, Jan 20 2024

			The array begins:
  0, 1,  1,   1,   1,    1, ...
  0, 1,  2,   3,   4,    5, ...
  0, 1,  4,   9,  16,   25, ...
  0, 1,  8,  25,  56,  105, ...
  0, 1, 16,  65, 176,  385, ...
  0, 1, 32, 161, 512, 1281, ...
  ...

Toufik Mansour, Howard Skogman, and Rebecca Smith, Sorting inversion sequences, arXiv:2401.06662 [math.CO], 2024. See Theorem 3.18 at page 10.

Cf. A000004 (k=0), A000012 (k=1), A000079 (k=2), A002064 (k=3), A340257 (k=4).
Cf. A000290 (n=2), A001477 (n=1), A057427 (n=0), A131423 (n=3), A164039.
Cf. A000035, A369325 (main diagonal), A369326.

A[n_,k_]:=(1-(-1)^k)/2+2^n Sum[Binomial[n+k-3-2i,n-1],{i,0,Floor[(k-2)/2]}]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

A(n,k) = A000035(k) + 2^n*Sum_{i=0..floor((k-2)/2)} binomial(n + k - 3 - 2*i, n - 1).

Sum_{k=0..n} A(n-k,k) = A164039(n-1).

cp's OEIS Frontend

A340298 a(n) = a(floor(n/2)) + a(ceiling(n/2)) + n*floor(log_2(n)) for n >= 2, a(n) = n for n <= 1.

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