A227345 -id:A227345 - cp's OEIS frontend

A227566 Number of partitions of n into distinct parts with boundary size 9.

1, 2, 3, 6, 10, 15, 24, 35, 50, 71, 99, 134, 184, 246, 321, 424, 547, 699, 891, 1123, 1400, 1751, 2158, 2648, 3239, 3938, 4751, 5732, 6857, 8174, 9721, 11501, 13535, 15924, 18622, 21728, 25278, 29320, 33873, 39102, 44939, 51542, 58995, 67348, 76654, 87161

Offset: 61

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 61..1000

Column k=9 of A227345, A227551.

A227567 Number of partitions of n into distinct parts with boundary size 10.

Original entry on oeis.org

1, 1, 3, 5, 9, 14, 23, 32, 49, 69, 98, 134, 186, 247, 334, 440, 574, 742, 962, 1218, 1549, 1943, 2430, 3011, 3728, 4564, 5590, 6795, 8227, 9909, 11914, 14223, 16954, 20124, 23795, 28044, 32974, 38592, 45093, 52530, 60991, 70640, 81667, 94095, 108214, 124177

Offset: 75

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 75..1000

Column k=10 of A227345, A227551.

A227568 Largest k such that a partition of n into distinct parts with boundary size k exists.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Alois P. Heinz, Jul 16 2013

nonn

The boundary size is the number of parts having fewer than two neighbors.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Where records occur: A077043.

Cf. A227345, A227551.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, 1, 0),
      `if`(i<1, 0, max(`if`(t>1, 1, 0)+b(n, i-1, iquo(t, 2)),
      `if`(i>n, 0, `if`(t=2, 1, 0)+b(n-i, i-1, iquo(t, 2)+2)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t > 1, 1, 0], If[i < 1, 0, Max[If[t > 1, 1, 0] + b[n, i - 1, Quotient[t, 2]], If[i > n, 0, If[t == 2, 1, 0] + b[n - i, i - 1, Quotient[t, 2] + 2]]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

a(n) = max { k : A227345(n,k) > 0 } = max { k : A227551(n,k) > 0 }.

a(n) = floor(2*sqrt(n/3)).

cp's OEIS Frontend

A227566 Number of partitions of n into distinct parts with boundary size 9.

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A227567 Number of partitions of n into distinct parts with boundary size 10.

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A227568 Largest k such that a partition of n into distinct parts with boundary size k exists.

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