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 1-10 of 11 results. Next

A333198 Decimal expansion of a constant related to the asymptotics of A306734 and A333179.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 11 2020

Keywords

Examples

			1.86429525435844065924747599856112246877295244507368421574403...

Crossrefs

Cf. A306734, A333179, A333180, A333181, A357471, A376621.

Programs

  • Mathematica
    RealDigits[E^Sqrt[4*Log[r]^2/3 + 4*PolyLog[2, 1-r] - Pi^2/3] /. r -> (2 - 5*(2/(-11 + 3*Sqrt[69]))^(1/3) + ((-11 + 3*Sqrt[69])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Formula

Equals limit_{n->infinity} A306734(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A333179(n)^(1/sqrt(n)).
Equals exp(sqrt(4*log(r)^2/3 + 4*polylog(2, 1-r) - Pi^2/3)), where r = 1 - A357471 = 0.4301597090019467340886... is the real root of the equation r^2 = (1-r)^3. - Vaclav Kotesovec, Oct 07 2024

Extensions

More digits from Vaclav Kotesovec, Oct 07 2024

A306734 Expansion of Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 8, 9, 8, 8, 9, 9, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 14, 15, 15, 15, 15, 15, 15, 16
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2019

Keywords

Links

Crossrefs

Cf. A003114, A053256, A053258, A053261, A333179, A333180.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[Sum[x^(k^2) Product[(1 + x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Mar 10 2020 *)

Formula

a(n) ~ c * A333198^sqrt(n) / sqrt(n), where c = 0.424889520435345887204307524... = sqrt((23 + (10051/2 - (1173*sqrt(69))/2)^(1/3) + ((23/2)*(437 + 51*sqrt(69)))^(1/3))/69)/2, c = sqrt(s)/2, where s is the real root of the equation -1 + 6*s - 23*s^2 + 23*s^3 = 0. - Vaclav Kotesovec, Mar 11 2020
Limit_{n->infinity} a(n) / A333179(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885... - Vaclav Kotesovec, Mar 11 2020

A376542 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j))^2.

Original entry on oeis.org

1, 1, 0, 2, 1, 1, 2, 0, 3, 1, 4, 2, 3, 3, 2, 6, 2, 7, 2, 8, 3, 10, 6, 8, 9, 8, 12, 8, 16, 6, 20, 8, 22, 10, 24, 14, 27, 20, 26, 26, 25, 34, 26, 42, 25, 51, 26, 58, 31, 66, 36, 72, 43, 76, 56, 82, 70, 82, 86, 84, 106, 87, 124, 90, 145, 95, 168, 102, 187, 115, 206
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 28 2024

Keywords

Links

Crossrefs

Cf. A216222, A306734, A333179, A340647, A369557, A376530.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ A369557(n) / 4.

A333374 G.f.: Sum_{k>=1} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)/(1 - x^j)).

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 3, 4, 6, 8, 10, 12, 15, 18, 22, 28, 34, 42, 52, 62, 75, 90, 106, 126, 150, 176, 208, 246, 288, 338, 397, 462, 538, 626, 724, 838, 968, 1114, 1282, 1474, 1690, 1936, 2217, 2532, 2890, 3296, 3750, 4264, 4844, 5492, 6222, 7042, 7958, 8986, 10138
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 17 2020

Keywords

Comments

The g.f. Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j) = Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j + 2*x^j) == Sum_{k >= 1} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

Links

Crossrefs

Cf. A001935, A001936, A003106, A053261, A066447, A003114, A306734, A333179.
Cf. A192918, A376841.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[(1+x^k)/(1-x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).
Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).
a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.10511708841962944170826735560432... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 - sinh(arcsinh(sqrt(11)/4)/3) / (12*sqrt(11))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - Vaclav Kotesovec, Mar 17 2020, updated Oct 10 2024

A376631 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j)).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 2, 0, 3, 0, 2, 1, 3, 1, 3, 1, 2, 3, 2, 3, 2, 4, 1, 5, 2, 5, 2, 6, 1, 7, 2, 7, 3, 6, 4, 7, 5, 6, 7, 6, 7, 7, 9, 5, 11, 5, 12, 6, 14, 5, 15, 6, 16, 7, 17, 7, 18, 9, 18, 11, 19, 12, 20, 14, 19, 17, 19, 19, 20, 23, 18, 27, 18, 29, 20, 32, 19
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A000009, A053258, A053261, A264905, A333179, A376580, A376632, A376660.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j)).
a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.39098976711379944962936707496887239986756106886318...
a(n) ~ A376580(n) * (A376660/A376621)^sqrt(n).

A376630 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j-1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 2, 2, 0, 1, 2, 2, 1, 1, 2, 3, 2, 1, 3, 2, 2, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 5, 5, 5, 4, 4, 7, 7, 5, 6, 8, 7, 7, 6, 8, 10, 8, 8, 10, 11, 9, 10, 12, 12, 11, 12, 14, 14, 13, 13, 16, 17, 15, 17, 18, 18, 19, 19, 20, 21, 22, 22, 24, 24, 25, 26, 27, 28, 29, 30
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A000009, A053258, A053261, A333179, A376629, A376660.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k - 1))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j-1)).
a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = sqrt(cosh(arccosh(sqrt(31)/2) / 3))/31^(1/4) = 0.456748282933947534736955792823221857...

A376813 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)^2.

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 3, 2, 2, 2, 3, 6, 7, 8, 10, 8, 8, 8, 6, 8, 10, 12, 16, 20, 22, 24, 27, 26, 25, 26, 25, 26, 29, 32, 37, 44, 52, 58, 66, 72, 76, 82, 82, 84, 87, 88, 91, 96, 103, 112, 126, 138, 154, 174, 190, 208, 225, 238, 253, 268, 275, 284, 296, 304
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 05 2024

Keywords

Links

Crossrefs

Cf. A333179, A376623, A376812, A216222.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[1+x^k, {k, 1, n}]^2, {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^k)*(1 + x^k)*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(2*j))^2.
a(n) ~ phi^(1/2) * exp(Pi*sqrt(2*n/15)) / (4 * 5^(1/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A376943 G.f.: Sum_{k>=0} 2^k * x^(k*(k+1)) * Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

1, 0, 2, 2, 0, 0, 4, 4, 4, 4, 0, 0, 8, 8, 8, 16, 8, 8, 8, 0, 16, 16, 16, 32, 32, 32, 32, 32, 16, 16, 48, 32, 32, 64, 64, 96, 96, 96, 96, 96, 96, 64, 128, 96, 96, 160, 128, 192, 256, 256, 256, 320, 320, 320, 320, 256, 384, 384, 320, 384, 384, 448, 576, 704, 640, 768, 896
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 10 2024

Keywords

Comments

In general, if d >= 1, b > 0 and g.f. = Sum_{k>=0} d^k * x^(b*k^2 + c*k) * Product_{j=1..k} (1 + x^j), then a(n) ~ r^c * (1+r) * exp(sqrt((2*log(d)^2 + 8*b*log(d)*log(r) + 4*b*(2*b+1)*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((r + 2*b*(1+r))*n)), where r is the smallest positive real root of the equation d*r^(2*b)*(1+r) = 1.

Links

Crossrefs

Cf. A273065, A333179, A376944, A376945, A376947.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Sum[2^k * x^(k*(k+1)) * Product[1+x^j, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 80; p = 1; s = 1; Do[p = Normal[Series[2*p*(1 + x^k) * x^(2*k), {x, 0, nmax}]]; s += p; , {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

a(n) ~ r * (1+r) * exp(sqrt((2*log(2)^2 + 8*log(2)*log(r) + 12*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((3*r + 2)*n)), where r = ((46 - 6*sqrt(57))^(1/3) + (46 + 6*sqrt(57))^(1/3) - 2)/6 is the real root of the equation 2*r^2*(1+r) = 1 (A273065).

A333181 G.f.: Sum_{k>=1} (k * x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 0, 0, 3, 3, 3, 6, 3, 3, 3, 0, 4, 4, 4, 8, 8, 8, 8, 8, 4, 4, 9, 5, 5, 10, 10, 15, 15, 15, 15, 15, 15, 10, 16, 11, 11, 17, 12, 18, 24, 24, 24, 30, 30, 30, 30, 24, 31, 31, 25, 26, 26, 27, 34, 41, 35, 42, 49, 49, 56, 56, 56, 56, 64
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 10 2020

Keywords

Links

Crossrefs

Cf. A333153, A333179, A333180.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[n*x^(n*(n+1))*Product[1+x^k, {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 0; Do[p = Expand[p*(1 + x^k)*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += k*p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

a(n) ~ c * A333198^sqrt(n), where c = 0.2895947615240435716456...
Limit_{n->infinity} A333180(n) / a(n) = A060006 = (1/2 + sqrt(23/3)/6)^(1/3) + (1/2 - sqrt(23/3)/6)^(1/3) = 1.32471795724474602596090885...

A376628 G.f.: Sum_{k>=0} x^(k*(k+1)) / Product_{j=1..k} (1 - x^(2*j-1)).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 8, 10, 10, 12, 14, 15, 17, 19, 21, 23, 27, 29, 31, 37, 39, 43, 49, 52, 58, 64, 70, 76, 84, 92, 99, 111, 119, 129, 143, 153, 167, 183, 197, 213, 233, 251, 271, 295, 317, 343, 372, 400, 430, 466, 500, 538, 582, 622, 670
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A003106, A053251, A053254, A053258, A053282, A333179.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Sum[x^(k*(k+1))/Product[1-x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} x^(2*j)/(1 - x^(2*j-1)).
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/2) * sqrt(n)).
Conjectural g.f.: (1 + q * nu(-q))/(1 + q) = 1 + Sum_{k >= 0} q^(k+2)*Product_{j = 1..k} 1 + q^(2*j+1), where nu(q) is the g.f. of A053254. - Peter Bala, Jan 03 2025
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