cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

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Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Examples

			0.374988675248948775420442838991310915249099682697586346616093763186183412...

Crossrefs

Cf. A368065, A368066.

Formula

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)).

A368066 a(n) = Product_{i=1..n, j=1..n} (i^2 + 6*i*j + j^2).

Original entry on oeis.org

1, 8, 73984, 10027173445632, 93867986947606492024406016, 185865459466664040069739311383413462872883200, 186896871826703385639703785281909582209471190408233074664996759142400
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Comments

In general, for d >= -1, Product_{i=1..n, j=1..n} (i^2 + d*i*j + j^2) ~ c(d) * (d+2)^((d+2)*n*(n+1)/2) * n^(2*n^2 - 1/2 - d/6) / ((d/2 + sqrt(d^2/4 - 1))^(sqrt(d^2 - 4)*n*(n+1)/2) * exp(3*n^2)), where c(d) is a constant (dependent only on d).
c(-1) = 3^(1/6) * exp(Pi/(6*sqrt(3))) * Gamma(1/3)^2 / (2*Pi)^(5/3).
c(0) = exp(Pi/12) * Gamma(1/4) / (2*Pi)^(5/4).
c(1) = 3^(5/12) * exp(Pi/(12*sqrt(3))) * Gamma(1/3) / (2*Pi)^(4/3).
c(2) = A^2 / (2^(1/6) * exp(1/6) * Pi), where A = A074962.
c(3) = 2^((sqrt(5) - 9)/6) * sqrt(5) * (1 + sqrt(5))^(1/2 - sqrt(5)/6) / Pi.
c(4) = 2^((sqrt(3) - 1)/6) * 3^(13/24) * (1 + sqrt(3))^(1/2 - 1/sqrt(3)) / (Pi^(7/12) * Gamma(1/4)^(1/3) * Gamma(1/3)^(1/2)).
c(5) = A368069.
c(6) = 2^(25/8) * (1 + sqrt(2))^(3/4 - 2*sqrt(2)/3) / (Pi^(1/4) * Gamma(1/8) * Gamma(1/4)^(1/2)).
Special (non-integer) case: Product_{i=1..n, j=1..n} (i^2 + (d + 1/d)*i*j + j^2) ~ A^(2/d) * (Product_{j=1..d} Gamma(j/d)^(2*j/d)) * (d+1)^((d/2 + 1 + 1/(2*d))*2*n*(n+1) + (d+1)^2/(6*d) + 1/6) * n^(2*n^2 - d/6 - 1/2 - 1/(6*d)) / ((2*Pi)^((d+1)/2) * exp(3*n^2 + 1/(6*d)) * d^((d+1)*n*(n+1) - 1/(6*d))), where A = A074962 is the Glaisher-Kinkelin constant.

Crossrefs

Cf. A367543 (d=-1), A324403 (d=0), A367542 (d=1), A079478^2 (d=2), A368067 (d=3), A368064 (d=4), A368065 (d=5).
Cf. A368069, A368068.

Programs

  • Mathematica
    Table[Product[i^2 + 6*i*j + j^2, {i, 1, n}, {j, 1, n}], {n, 0, 7}]

Formula

a(n) ~ 2^(12*n*(n+1) + 25/8) * n^(2*n^2 - 3/2) / (Pi^(1/4) * Gamma(1/4)^(1/2) * Gamma(1/8) * (1 + sqrt(2))^(2*sqrt(2)*(6*n*(n+1) + 1)/3 - 3/4) * exp(3*n^2)).

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

Original entry on oeis.org

1, 6, 24336, 870746557824, 1311726482483997806493696, 256433546267136937832915286844640487014400, 15678550451426175377500759401206644047210595564950427820202393600
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

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

Programs

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

Formula

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 + 3*i*j + j^2).

Original entry on oeis.org

1, 5, 12100, 188898484500, 91554454518735288960000, 4263420404009649597344435073399120000000, 46073465749493255153019723901007197815549903333795840000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

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

Programs

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

Formula

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.

A368068 a(n) = Product_{i=1..n, j=1..n} (2*i^2 + 5*i*j + 2*j^2).

Original entry on oeis.org

1, 9, 129600, 40327580160000, 1311346674278439321600000000, 13821139470331790817454891043295068160000000000, 114180111981355345833797461507302737916551512227408406118400000000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 10 2023

Keywords

Crossrefs

Cf. A324402, A368065.

Programs

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

Formula

a(n) = Product_{i=1..n, j=1..n} (i + 2*j) * (2*i + j).
a(n) = A324402(n)^2.
a(n) ~ A * 3^(9*n*(n+1)/2 + 11/12) * n^(2*n^2 - 11/12) / (Pi * 2^(2*n^2 + 3*n + 17/12) * exp(3*n^2 + 1/12)), where A is the Glaisher-Kinkelin constant A074962.
Showing 1-5 of 5 results.

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