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.

Previous Showing 11-14 of 14 results.

A353246 Integer part of e[n]e, where [n] indicates hyper-n and e = 2.718281828... (using H. Kneser's proposal for n > 3).

Original entry on oeis.org

4, 5, 7, 15, 2075
Offset: 0

Views

Author

Marco Ripà, Apr 08 2022

Keywords

Comments

The common hyperoperation sequence is defined as follows: hyper-0 = zeration, hyper-1 = addition, hyper-2 = multiplication, hyper-3 = exponentiation, hyper-4 = tetration, and so on...
Thus e[0]e = e + 2, e[1]e = 2*e, e[2]e = e^2, e[3]e = e^e, and so on.
The fifth term of the twin sequence of the present one, floor(Pi[4]Pi), is much larger than 2075 and it is harder to calculate, while the integer part of e[4]Pi is 37149801960 (17.9 million times bigger than a(4)).

Examples

			For n = 3, a(3) = floor(e[3]e) = floor(e^e) = 15.

Links

Crossrefs

Cf. A001113, A019762, A072334, A073226, A351727, A352396.

Formula

a(n) = floor(e[n]e).

A371831 a(n) = numerator(Sum_{k=1..n} k^2/k!).

Original entry on oeis.org

0, 1, 3, 9, 31, 43, 217, 3913, 9133, 73067, 1972819, 6576067, 24112247, 372017527, 1612075951, 157983443203, 7109254944151, 37916026368811, 644572448269793, 34806912206568841, 2422459091299663, 7775794614048301, 277759159408419360043, 2036900502328408640323, 46848711553553398727437
Offset: 0

Views

Author

Stefano Spezia, Apr 07 2024

Keywords

Links

Crossrefs

Cf. A019762, A030297, A371832.
Cf. A014973, A092043.

Programs

  • Mathematica
    a[n_]:=Numerator[(2(E*Gamma[n+1,1]-1)-n)/n!]; Array[a,25,0]
  • PARI
    a(n) = numerator(sum(k=1, n, k^2/k!)); \\ Michel Marcus, Apr 07 2024

Formula

a(n) = numerator((2*(e*Gamma(n+1, 1) - 1) - n)/n!).
a(n) = numerator(A030297(n)/n!).
Limit_{n->oo} a(n)/A371832(n) = 2*e = A019762.

A371832 a(n) = denominator(Sum_{k=1..n} k^2/k!).

Original entry on oeis.org

1, 1, 1, 2, 6, 8, 40, 720, 1680, 13440, 362880, 1209600, 4435200, 68428800, 296524800, 29059430400, 1307674368000, 6974263296000, 118562476032000, 6402373705728000, 445586448384000, 1430277488640000, 51090942171709440000, 374666909259202560000, 8617338912961658880000
Offset: 0

Views

Author

Stefano Spezia, Apr 07 2024

Keywords

Links

Crossrefs

Cf. A019762, A030297, A371831.
Cf. A014973, A092043.

Programs

  • Mathematica
    a[n_]:=Denominator[(2(E*Gamma[n+1,1]-1)-n)/n!]; Array[a,25,0]
  • PARI
    a(n) = denominator(sum(k=1, n, k^2/k!)); \\ Michel Marcus, Apr 07 2024

Formula

a(n) = denominator((2*(e*Gamma(n+1, 1) - 1) - n)/n!).
a(n) = denominator(A030297(n)/n!).
Limit_{n->oo} A371831(n)/a(n) = 2*e = A019762.

A252848 Decimal expansion of Sum_{n>0} Sum_{k=0..n} exp(k)/n! = e*(e^e - 1)/(e - 1).

Original entry on oeis.org

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

Views

Author

Richard R. Forberg, Dec 22 2014

Keywords

Comments

Terms in the sum begin: 1 + (1 + e)/1 + (1 + e + e^2)/2 + (1 + e + e^2 + e^3)/6 + ... .
The largest term in the sum is at n = 2, where that term is 5.5536... .
The double sum converges to a similar algebraic form using any base for exponentiation. For instance, using Pi as the base shows the general closed form:
Pi*(e^Pi - e)/(Pi - 1), which equals 29.9584963... .
As the base approaches 1, the ratio converges to 2e = Sum_{n>0} Sum_{k=0..n} 1/n! = 5.43656... . See A019762.

Examples

			22.391713168940217114413... .

Programs

  • Mathematica
    Sum[N[Sum[Exp[k]/n!, {k, 0, n}], 100], {n, 0, Infinity}]

Formula

e*(e^e - 1)/(e - 1).
Previous Showing 11-14 of 14 results.

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