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.

A052338 a(n) = A050443(n-th prime)/(n-th prime).

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 2, 4, 11, 16, 43, 87, 123, 250, 736, 2189, 3152, 9501, 19909, 28861, 88250, 186540, 575703, 2605063, 5556059, 8118935, 17356074, 25389718, 54388168, 789812633, 1700415401, 5380882786, 7903301746, 54185536387, 79678647536, 253610377475
Offset: 1

Views

Author

Christian G. Bower, Dec 23 1999

Keywords

Links

Crossrefs

Cf. A000040, A001608, A050443.

Programs

  • PARI
    a(n) = my(p = prime(n)); polcoef((4-x^3)/(1-x^3-x^4) + O(x^(p+1)), p)/p; \\ Michel Marcus, Mar 04 2019

Extensions

Corrected by Seiichi Manyama, Mar 04 2019

A306785 Primes p such that p^2 divides A050443(p).

Original entry on oeis.org

2, 5, 1051
Offset: 1

Views

Author

Seiichi Manyama, Mar 09 2019

Keywords

Comments

A050443(p) is divisible by p for p prime, so sequence looks for primes p such that p^2 divides A050443(p).
No more terms < 10^11. - Lucas A. Brown, Jan 27 2021

Examples

			A050443(2) = 0 is divisible by 2^2.
A050443(5) = 0 is divisible by 5^2.
A050443(1051) is divisible by 1051^2.

Crossrefs

Cf. A000040, A050443, A173656, A306786.

Programs

  • PARI
    M = [0,1,0,0; 0,0,1,0; 0,0,0,1; 1,1,0,0];
b(n) = lift( ( Mod(M,n^2)^n * [4,0,0,3]~)[1] ); \\ A050443(n) mod n^2
forprime(n=2, 10^10, if( b(n)==0, print1(n,", ") ) ); \\ Joerg Arndt, Mar 11 2019

A306646 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. (k+1-x^k)/(1-x^k-x^(k+1)).

Original entry on oeis.org

2, 3, 1, 4, 0, 3, 5, 0, 2, 4, 6, 0, 0, 3, 7, 7, 0, 0, 3, 2, 11, 8, 0, 0, 0, 4, 5, 18, 9, 0, 0, 0, 4, 0, 5, 29, 10, 0, 0, 0, 0, 5, 3, 7, 47, 11, 0, 0, 0, 0, 5, 0, 7, 10, 76, 12, 0, 0, 0, 0, 0, 6, 0, 4, 12, 123, 13, 0, 0, 0, 0, 0, 6, 0, 4, 3, 17, 199
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2019

Keywords

Examples

			A(6,1) = 6*Sum_{j=1..6} binomial(j,6-j)/j = 6*(1/3+3/2+1+1/6) = 18.
A(6,2) = 6*Sum_{j=1..3} binomial(j,6-2*j)/j = 6*(1/2+1/3) = 5.
Square array begins:
    2,  3, 4, 5, 6, 7, 8, 9, 10, 11, ...
    1,  0, 0, 0, 0, 0, 0, 0,  0,  0, ...
    3,  2, 0, 0, 0, 0, 0, 0,  0,  0, ...
    4,  3, 3, 0, 0, 0, 0, 0,  0,  0, ...
    7,  2, 4, 4, 0, 0, 0, 0,  0,  0, ...
   11,  5, 0, 5, 5, 0, 0, 0,  0,  0, ...
   18,  5, 3, 0, 6, 6, 0, 0,  0,  0, ...
   29,  7, 7, 0, 0, 7, 7, 0,  0,  0, ...
   47, 10, 4, 4, 0, 0, 8, 8,  0,  0, ...
   76, 12, 3, 9, 0, 0, 0, 9,  9,  0, ...

Links

Crossrefs

Columns 1-9 give A000032, A001608, A050443, A087935, A087936, A306755, A306756, A306757, A306758.
Cf. A306713, A306735.

Programs

  • Mathematica
    T[0, k_] := k + 1; T[n_, k_] := n *Sum[Binomial[j, n - k*j]/j, {j, 1, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

Formula

A(0,k) = k+1 and A(n,k) = n*Sum_{j=1..floor(n/k)} binomial(j,n-k*j)/j for n > 0.
A(n,k) = (k+1)*A306713(n,k) - A306713(n-k,k) for n >= k.

A087935 Perrin sequence of order 5.

Original entry on oeis.org

5, 0, 0, 0, 4, 5, 0, 0, 4, 9, 5, 0, 4, 13, 14, 5, 4, 17, 27, 19, 9, 21, 44, 46, 28, 30, 65, 90, 74, 58, 95, 155, 164, 132, 153, 250, 319, 296, 285, 403, 569, 615, 581, 688, 972, 1184, 1196, 1269, 1660, 2156, 2380, 2465, 2929, 3816, 4536, 4845, 5394, 6745, 8352, 9381
Offset: 0

Views

Author

Benoit Cloitre, Oct 27 2003

Keywords

Comments

If p is prime, p divides a(p).

Links

Crossrefs

Column 4 of A306646.
Cf. A001608, A050443.
Cf. A087936.

Programs

  • GAP
    a:=[5,0,0,0,4];; for n in [6..60] do a[n]:=a[n-4]+a[n-5]; od; Print(a); # Muniru A Asiru, Mar 06 2019
  • Magma
    I:=[5,0,0,0,4]; [n le 5 select I[n] else Self(n-4) +Self(n-5): n in [1..60]]; // G. C. Greubel, Mar 06 2019
  • Maple
    seq(coeff(series((x^4-5)/(x^5+x^4-1),x,n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Mar 06 2019
  • Mathematica
    LinearRecurrence[{0,0,0,1,1},{5,0,0,0,4},60] (* Harvey P. Dale, Oct 03 2016 *)
  • PARI
    my(x='x+O('x^60)); Vec((5-x^4)/(1-x^4-x^5)) \\ G. C. Greubel, Mar 06 2019
  • PARI
    polsym(x^5-x-1,66) \\ Joerg Arndt, Mar 10 2019
  • Sage
    ((5-x^4)/(1-x^4-x^5)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Mar 06 2019

Formula

a(n) = a(n-4) + a(n-5), with a(0)=5, a(1)=a(2)=a(3)=0.
a(n) = (x_1)^n + (x_2)^n + (x_3)^n + (x_4)^n + (x_5)^n where (x_i) 1 <= i <= 5 are the roots of x^5=x+1.
G.f.: (5 - x^4)/(1 -x^4 -x^5). - Colin Barker, Jun 16 2013
a(0) = 5 and a(n) = n*Sum_{k=1..floor(n/4)} binomial(k,n-4*k)/k for n > 0. - Seiichi Manyama, Mar 04 2019
From Aleksander Bosek, Mar 06 2019: (Start)
a((s+5)*n + m) = Sum_{j=0..n} binomial(n-j,j)*a(s*n+j+m) for all s > 0, m > 0.
a(m) = Sum_{j=0..n} (-1)^(n-j)*binomial(n-j,j)*a(m+n+4*j) for all m > 0. (End)

A087936 Perrin sequence of order 6.

Original entry on oeis.org

6, 0, 0, 0, 0, 5, 6, 0, 0, 0, 5, 11, 6, 0, 0, 5, 16, 17, 6, 0, 5, 21, 33, 23, 6, 5, 26, 54, 56, 29, 11, 31, 80, 110, 85, 40, 42, 111, 190, 195, 125, 82, 153, 301, 385, 320, 207, 235, 454, 686, 705, 527, 442, 689, 1140, 1391, 1232, 969, 1131, 1829, 2531, 2623, 2201, 2100
Offset: 0

Views

Author

Benoit Cloitre, Oct 27 2003

Keywords

Comments

If p is prime, p divides a(p).

Links

Crossrefs

Column 5 of A306646.
Cf. A001608, A050443.
Cf. A087935.

Programs

  • GAP
    Concatenation([6],List([1..65],n->n*Sum([1..Int(n/5)],k->Binomial(k,n-5*k)/k))); # Muniru A Asiru, Mar 09 2019
  • Maple
    a:=n->n*add(binomial(k,n-5*k)/k,k=1..floor(n/5)): 6,seq(a(n),n=1..65); # Muniru A Asiru, Mar 09 2019
  • PARI
    polsym(x^6-x-1,66) \\ Joerg Arndt, Mar 10 2019

Formula

a(n) = a(n-5) + a(n-6) with a(0)=6, a(1)=a(2)=a(3)=a(4)=0, a(5)=5.
a(n) = Sum_{i=1..6} (x_i)^n where x_i are the roots of x^6 = x+1.
G.f.: (x^5-6) / (x^6+x^5-1). - Colin Barker, Jun 16 2013
a(0) = 6 and a(n) = n*Sum_{k=1..floor(n/5)} binomial(k,n-5*k)/k for n > 0. - Seiichi Manyama, Mar 04 2019
From Aleksander Bosek, Mar 06 2019: (Start)
a((s+6)*n+m) = Sum_{l=0..n} binomial(n-l,l)*a(s*n+l+m) for all s > 0, m > 0.
a(m) = Sum_{l=0..n}(-1)^{n-l} binomial(n-l,l)*a(m+n+5*l)for all m > 0. (End)
Showing 1-5 of 5 results.

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