A035621 -id:A035621 - cp's OEIS frontend

A035620 Number of partitions of n into parts 3k+1 and 3k+2 with at least one part of each type.

0, 0, 1, 1, 2, 4, 5, 7, 11, 14, 19, 26, 33, 43, 55, 70, 88, 111, 137, 170, 208, 256, 311, 378, 456, 551, 658, 790, 940, 1119, 1325, 1570, 1847, 2179, 2554, 2996, 3499, 4088, 4753, 5533, 6414, 7436, 8593, 9931, 11439, 13180, 15140, 17391, 19926, 22827

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 80 terms from Robert Price)

Cf. A035441-A035468, A035618-A035619, A035621-A035699.

nmax = 50; s1 = Range[0, nmax/3]*3 + 1; s2 = Range[0, nmax/3]*3 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 50; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(3 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(3 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(3 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(3 k + 2))). - Robert Price, Aug 16 2020

A035622 Number of partitions of n into parts 4k and 4k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 4, 0, 10, 0, 11, 0, 22, 0, 25, 0, 44, 0, 51, 0, 83, 0, 98, 0, 149, 0, 177, 0, 259, 0, 309, 0, 436, 0, 521, 0, 716, 0, 857, 0, 1151, 0, 1376, 0, 1816, 0, 2170, 0, 2818, 0, 3361, 0, 4309, 0, 5132, 0, 6502, 0, 7728, 0, 9695, 0, 11501, 0, 14298

Offset: 0

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 101 terms from Robert Price)

Bisections give: A006477 (even part), A000004 (odd part).

Cf. A035441-A035468, A035618-A035621, A035623-A035699.

nmax = 70; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 0, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 70; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

G.f.: (-1 + 1/Product_{k>=1} (1 - x^(4 k)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 2))). - Robert Price, Aug 16 2020

A035673 Number of partitions of n into parts 8k and 8k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 4, 0, 4, 0, 4, 0, 10, 0, 11, 0, 11, 0, 11, 0, 22, 0, 25, 0, 26, 0, 26, 0, 44, 0, 51, 0, 54, 0, 55, 0, 84, 0, 98, 0, 105, 0, 108, 0, 153, 0, 178, 0, 193, 0, 200, 0, 269, 0, 313, 0, 341, 0, 356, 0, 459, 0, 531, 0, 582, 0, 611, 0

Offset: 1

Olivier Gérard

nonn

Robert Price, Table of n, a(n) for n = 1..1000

Cf. A035441-A035468, A035618-A035672, A035674-A035699.

Bisections give: A035621 (even part), A000004 (odd part).

nmax = 81; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 2)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 12 2020

cp's OEIS Frontend

A035620 Number of partitions of n into parts 3k+1 and 3k+2 with at least one part of each type.

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A035622 Number of partitions of n into parts 4k and 4k+2 with at least one part of each type.

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A035673 Number of partitions of n into parts 8k and 8k+2 with at least one part of each type.

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