A074742 -id:A074742 - cp's OEIS frontend

A027965 T(n, 2*n-3), T given by A027960.

3, 7, 15, 28, 47, 73, 107, 150, 203, 267, 343, 432, 535, 653, 787, 938, 1107, 1295, 1503, 1732, 1983, 2257, 2555, 2878, 3227, 3603, 4007, 4440, 4903, 5397, 5923, 6482, 7075, 7703, 8367, 9068, 9807, 10585, 11403, 12262, 13163, 14107, 15095, 16128, 17207, 18333, 19507, 20730, 22003

Offset: 2

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

A column of triangle A027011.

List([2..50], n-> (18-10*n+3*n^2+n^3)/6) # G. C. Greubel, Jun 30 2019

[(18-10*n+3*n^2+n^3)/6: n in [2..50]]; // G. C. Greubel, Jun 30 2019

LinearRecurrence[{4,-6,4,-1}, {3,7,15,28}, 50] (* G. C. Greubel, Jun 30 2019 *)

vector(50, n, n++; (18-10*n+3*n^2+n^3)/6) \\ G. C. Greubel, Jun 30 2019

[(18-10*n+3*n^2+n^3)/6 for n in (2..50)] # G. C. Greubel, Jun 30 2019

a(n+2) = A074742(n-1) = A008778(n) + 2 = A000297(n-1) + 3.
From Ralf Stephan, Feb 07 2004: (Start)
G.f.: x^2*(3 - 2*x)*(1 - x + x^2)/(1-x)^4.
Differences of A027966. (End)
From G. C. Greubel, Jun 30 2019: (Start)
a(n) = (18 - 10*n + 3*n^2 + n^3)/6.
E.g.f.: (-18 - 12*x + (18 - 6*x + 6*x^2 + x^3)*exp(x))/6. (End)

Terms a(32) onward added by G. C. Greubel, Jun 30 2019

A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 4, 7, 11, 14, 5, 9, 15, 21, 25, 6, 11, 19, 28, 36, 41, 7, 13, 23, 35, 47, 57, 63, 8, 15, 27, 42, 58, 73, 85, 92, 9, 17, 31, 49, 69, 89, 107, 121, 129, 10, 19, 35, 56, 80, 105, 129, 150, 166, 175

Offset: 1

Gary W. Adamson, Sep 19 2008

Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

			The array is formed by A004736 * A144328 (transform) where A004736 = the natural number decrescendo triangle and A144328 = a crescendo triangle. First few rows of the array =
  1, 1,  1,  1,  1,  1, ...
  2, 3,  3,  3,  3,  3, ...
  3, 5,  7,  7,  7,  7, ...
  4, 7, 11, 14, 14, 14, ...
  5, 9, 15, 21, 25, 25, ...
  ...
Triangle begins as:
   1;
   2,  3;
   3,  5,  7;
   4,  7, 11, 14;
   5,  9, 15, 21, 25;
   6, 11, 19, 28, 36,  41;
   7, 13, 23, 35, 47,  57,  63;
   8, 15, 27, 42, 58,  73,  85,  92;
   9, 17, 31, 49, 69,  89, 107, 121, 129;
  10, 19, 35, 56, 80, 105, 129, 150, 166, 175;
  ...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A004006, A006008, A027965, A050407, A074742, A144328.

T[n_, k_]:= (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6;
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 18 2021 *)

def A144680(n,k): return (3*(k^2-k+2)*n - k*(k-1)*(2*k-1))/6
flatten([[A144680(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 18 2021

Sum_{k=1..n} T(n, k) = A006008(n).
From G. C. Greubel, Oct 18 2021: (Start)
T(n, k) = (1/6)*( 3*(k^2 - k + 2)*n - k*(k-1)*(2*k-1) ).
T(n, n) = A004006(n).
T(n, n-1) = A050407(n+2).
T(n, n-2) = A027965(n-1) = A074742(n-2). (End)

A338760 Subword complexity of the infinite word Prod_{i>=1} Prod_{j=1..i} a^(i-j+1) b^j.

Original entry on oeis.org

1, 2, 4, 8, 15, 28, 47, 73, 107, 150, 203, 267, 343, 432, 535, 653, 787, 938, 1107, 1295, 1503, 1732, 1983, 2257, 2555, 2878, 3227, 3603, 4007, 4440, 4903, 5397, 5923, 6482, 7075, 7703, 8367, 9068, 9807, 10585, 11403, 12262, 13163, 14107, 15095, 16128, 17207

Offset: 0

Jeffrey Shallit, Nov 07 2020

nonn

The infinite word is (ab)(aab.abb)(aaab.aabb.abbb)(aaaab.aaabb.aabbb.abbbb)... . Subword complexity is the number of distinct length-n blocks appearing in the sequence.

			For n=4 the only word omitted is baba.

Luke Schaeffer and Kaiyo Wu, Two Infinite Words with Cubic Subword Complexity, J. Integer Sequences 23 (2020), Article 20.10.8.

Cf. A074742, A338761.

Equal to 2^n for n <= 3, and n^3/6+n^2/2-5n/3+3 = A074742(n-1) for n >= 4.

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A027965 T(n, 2*n-3), T given by A027960.

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A144680 Triangle read by rows, lower half of an array formed by A004736 * A144328 (transform).

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Mathematica

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A338760 Subword complexity of the infinite word Prod_{i>=1} Prod_{j=1..i} a^(i-j+1) b^j.

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