A368066 -id:A368066 - cp's OEIS frontend

A368065 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5ij + j^2).

1, 7, 44100, 3210672937500, 12804360424787610000000000, 8591751256288909159255104643281750000000000, 2333034616280404811605303958158227652934766912996000000000000000000

Offset: 0

Vaclav Kotesovec, Dec 10 2023

nonn

Cf. A367543, A324403, A367542, A079478^2, A368067, A368064, A368066.

Cf. A368068, A368069.

Table[Product[i^2 + 5*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

a(n) ~ c * 7^(7*n*(n+1)/2) * ((5-sqrt(21))/2)^(sqrt(21)*n*(n+1)/2) * n^(2*n^2 - 4/3) / exp(3*n^2), where c = A368069.

A368064 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4ij + j^2).

Original entry on oeis.org

1, 6, 24336, 870746557824, 1311726482483997806493696, 256433546267136937832915286844640487014400, 15678550451426175377500759401206644047210595564950427820202393600

Offset: 0

Vaclav Kotesovec, Dec 10 2023

nonn

Cf. A367543, A324403, A367542, A079478^2, A368067, A368065, A368066.

Table[Product[i^2 + 4*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

a(n) ~ 2^((3+sqrt(3))*n*(n+1) + (sqrt(3)-1)/6) * 3^(3*n*(n+1) + 13/24) * n^(2*n^2 - 7/6) / (Gamma(1/3)^(1/2) * Gamma(1/4)^(1/3) * Pi^(7/12) * (1 + sqrt(3))^((6*n*(n+1) + 1)/sqrt(3) - 1/2) * exp(3*n^2)).

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3ij + j^2).

Original entry on oeis.org

1, 5, 12100, 188898484500, 91554454518735288960000, 4263420404009649597344435073399120000000, 46073465749493255153019723901007197815549903333795840000000000

Offset: 0

Vaclav Kotesovec, Dec 10 2023

nonn

Cf. A367543, A324403, A367542, A079478^2, A368064, A368065, A368066.

Table[Product[i^2 + 3*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

a(n) ~ 5^(5*n*(n+1)/2 + 1/2) * n^(2*n^2 - 1) / (2 * Pi * exp(3*n^2) * phi^(sqrt(5)*(n*(n+1) + 1/6) - 1/2)), where phi = A001622 is the golden ratio.

A368069 Decimal expansion of a constant related to the asymptotics of A368065.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 10 2023

			0.374988675248948775420442838991310915249099682697586346616093763186183412...

Cf. A368065, A368066.

Equals limit_{n->oo} A368065(n) / (7^(7*n*(n+1)/2) * ((5-sqrt(21))/2)^(sqrt(21)*n*(n+1)/2) * n^(2*n^2 - 4/3) / exp(3*n^2)).

cp's OEIS Frontend

A368065 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5ij + j^2).

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Formula

A368064 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4ij + j^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3ij + j^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A368069 Decimal expansion of a constant related to the asymptotics of A368065.

Views

Author

Keywords

Examples

Crossrefs

Formula

cp's OEIS Frontend

A368065 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5*i*j + j^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A368064 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4*i*j + j^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3*i*j + j^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A368069 Decimal expansion of a constant related to the asymptotics of A368065.

Views

Author

Keywords

Examples

Crossrefs

Formula

A368065 a(n) = Product_{i=1..n, j=1..n} (i^2 + 5ij + j^2).

A368064 a(n) = Product_{i=1..n, j=1..n} (i^2 + 4ij + j^2).

A368067 a(n) = Product_{i=1..n, j=1..n} (i^2 + 3ij + j^2).