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.

User: Andrew Nelson

Andrew Nelson's wiki page.

Andrew Nelson has authored 4 sequences.

A332635 a(n) = n!! mod prime(n).

Original entry on oeis.org

1, 2, 3, 1, 4, 9, 3, 4, 2, 12, 10, 15, 40, 34, 9, 11, 3, 28, 50, 55, 15, 24, 31, 80, 8, 16, 86, 65, 54, 40, 71, 54, 62, 85, 122, 114, 1, 40, 4, 87, 45, 126, 172, 53, 93, 109, 139, 28, 167, 78, 19, 222, 182, 136, 230, 231, 110, 163, 264, 45, 92, 134, 177, 241
Offset: 1

Views

Author

Andrew Nelson, Feb 17 2020

Keywords

Comments

a(n) > 0, as n!! cannot be divisible by prime(n): n < prime(n) for all n, so the prime factorization of n!! never includes prime(n).
a(n) = 1 for n = 1, 4, 37, 2721, ... .
a(n) = n for n = 1, 2, 3, 86, 122, ... .

Examples

			For n = 4, a(4) = 4!! mod prime(4) = 8 mod 7 = 1.

Links

Crossrefs

Cf. A000040 (primes), A006882 (double factorials), A091858 (n! mod prime(n)).

Programs

  • Mathematica
    Table[Mod[n!!, Prime[n]], {n, 100}]
  • PARI
    a(n) = my(p=prime(n)); lift(prod(i=0, (n-1)\2, Mod(n-2*i, p))); \\ Michel Marcus, Feb 25 2020

Formula

a(n) = n!! mod prime(n), where n!! denotes the double factorial of n.

A332409 a(n) = n!! mod Fibonacci(n).

Original entry on oeis.org

0, 0, 1, 2, 0, 0, 1, 6, 27, 45, 71, 0, 228, 73, 605, 861, 956, 2376, 1199, 5235, 7137, 5017, 21617, 40320, 49250, 72900, 94129, 253071, 125204, 188760, 786046, 1041600, 3306329, 2717231, 8692580, 4869072, 10661888, 33618132, 14333453, 66880275, 110783982
Offset: 1

Views

Author

Andrew Nelson, Feb 17 2020

Keywords

Comments

a(n) = 0 for n = 1, 2, 5, 6, 12 (a(n) < 500).

Examples

			For n = 1, a(1) = 1!! mod Fibonacci(1) = 1 mod 1 = 0.
For n = 4, a(4) = 4!! mod Fibonacci(4) = 8 mod 3 = 2.

Crossrefs

Cf. A000045, A006882, A182213.

Programs

  • Mathematica
    Table[Mod[n!!,Fibonacci[n]],{n,50}] (* Harvey P. Dale, Sep 08 2020 *)
  • PARI
    a0(n) = my(f=fibonacci(n)); prod(i=0, (n-1)\2, n - 2*i) % f; \\ Michel Marcus, Mar 17 2020

Formula

a(n) = n!! mod Fibonacci(n).
where n!! denotes the double factorial of n (n!! = n*a(n-2) for n > 1, a(0) = a(1) = 1), and Fibonacci(n) denotes the n-th Fibonacci number.

A333250 a(n) = concatenate(n-1, a(n-1)) mod n, with initial condition a(0) = 0.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 2, 6, 4, 3, 3, 4, 6, 9, 13, 3, 9, 16, 6, 15, 15, 20, 8, 21, 17, 17, 21, 2, 20, 7, 27, 20, 16, 15, 17, 22, 30, 4, 32, 10, 30, 12, 38, 24, 12, 2, 38, 32, 28, 26, 26, 28, 32, 38, 46, 1, 47, 4, 52, 11, 31, 53, 15, 41, 5, 60, 26, 60, 28, 66, 36, 7, 69, 42, 16, 66, 42, 19, 75, 54, 34
Offset: 0

Views

Author

Andrew Nelson, Mar 13 2020

Keywords

Comments

a(n) = 0 for n = 0, 1, 2, 4, 5, 359, 841, 5129, 5180, 5292, 22255, 50109, 85763 (a(n) < 10^6).
a(n) merges with a similar sequence with an odd initial condition (e.g., a(0) = 1) at term a(250).

Examples

			a(2) = concatenate(1, a(1)) mod 2 = concatenate(1, 0) mod 2 = 10 mod 2 = 0.
a(20) = concatenate(19, a(19)) mod 20 = concatenate(19, 15) mod 20 = 1915 mod 20 = 15.

Links

Programs

  • PARI
    lista(nn) = {my(a = 0, list = List()); listput(list, a); for (n=1, nn, a = eval(concat(Str(n-1), a)) % n; listput(list, a);); Vec(list);} \\ Michel Marcus, Mar 17 2020

A333125 a(n) = binomial(Fibonacci(n),n).

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 28, 1716, 203490, 52451256, 29248649430, 36519676207704, 103619293824707388, 681222021538453426360, 10526080837282875691177000, 387340445158332035685509830240, 34306348668342682111244774082795555, 7379087300345635546662027722168990277849
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 08 2020, following a suggestion of Andrew Nelson

Keywords

Links

Crossrefs

Cf. A000045, A014070.

Programs

  • Mathematica
    Table[Binomial[Fibonacci[n], n], {n, 0, 20}]
  • PARI
    a(n) = binomial(fibonacci(n), n); \\ Michel Marcus, Mar 10 2020

Formula

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

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