A322071 Triangle read by rows: T(n, k) is the largest integer m such that m*k^k <= 2*n^k.

2, 4, 2, 6, 4, 2, 8, 8, 4, 2, 10, 12, 9, 4, 2, 12, 18, 16, 10, 4, 2, 14, 24, 25, 18, 10, 5, 2, 16, 32, 37, 32, 20, 11, 5, 2, 18, 40, 54, 51, 37, 22, 11, 5, 2, 20, 50, 74, 78, 64, 42, 24, 11, 5, 2, 22, 60, 98, 114, 103, 75, 47, 25, 12, 5, 2, 24, 72, 128, 162, 159, 128, 87, 51, 26, 12, 5, 2

Offset

1,1

Comments

For k >= 5 and n >= k, T(n, k) is the maximum number of induced copies of a cycle of length k in a n-vertex graph (see Král, Norin and Volec Theorem 1).

Links

Table of n, a(n) for n=1..78.

Daniel Král, Sergey Norin and Jan Volec, A bound on the inducibility of cycles, arXiv:1801.01556 [math.CO], 2018; Journal of Combinatorial Theory, Series A, 161 (2019) 359-363.

Formula

T(n, k) = floor(2*n^k/k^k).

T(n, k) = floor(A005843(n^k/A000312(k))).

Example

  n\k|  1   2   3   4   5   6   7   8   9  10
  ---+----------------------------------------
   1 |  2
   2 |  4   2
   3 |  6   4   2
   4 |  8   8   4   2
   5 | 10  12   9   4   2
   6 | 12  18  16  10   4   2
   7 | 14  24  25  18  10   5   2
   8 | 16  32  37  32  20  11   5   2
   9 | 18  40  54  51  37  22  11   5   2
  10 | 20  50  74  78  64  42  24  11   5   2
  ...

Maple

a := (n, k) -> floor(2*n^k/k^k): seq(seq(a(n, k), k = 1 .. n), n = 1 .. 20)

Mathematica

Flatten[Table[Floor[2*n^k/k^k], {n, 1, 15}, {k, 1, n}]]

Prog

(Magma) [[Floor(2*n^k/k^k): k in [1..n]]: n in [1..10]]; // triangle output
(Maxima) sjoin(v, j) := apply(sconcat, rest(join(makelist(j, length(v)), v)))$ display_triangle(n) := for i from 1 thru n do disp(sjoin(makelist(floor(2*i^j/j^j), j, 1, i), " ")); display_triangle(10);
(PARI)
T(n, k) = floor(2*n^k/k^k);
tabl(nn) = for(i=1, nn, for(j=1, i, print1(T(i, j), ", ")); print);
tabl(10) \\ triangle output
(GAP) Flat(List([1..15], n->List([1..n], k->Int(2*n^k/k^k)))); # Muniru A Asiru, Nov 25 2018

Crossrefs

Cf. A000312, A005843, A322072 (row sums).

Sequence in context: A215841 A272327 A212012 * A176342 A340148 A049200

Adjacent sequences: A322068 A322069 A322070 * A322072 A322073 A322074

Keyword

nonn,tabl

Author

Stefano Spezia, Nov 25 2018

Extensions

a(71)-a(78) from Muniru A Asiru, Nov 25 2018

Status

approved