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 1-6 of 6 results.

A072714 Duplicate of A051786.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 43, 452, 45351, 125920291, 60027819184831
Offset: 1

Views

Author

Keywords

A072713 a(1)=a(2)=a(3)=a(4)=a(5)=1; for n>5, a(n)*a(n-5) = a(n-1)*a(n-2)*a(n-3)*a(n-4)+1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 7, 43, 1807, 815861, 147917502976, 1339566593057489572791, 6793440021984612817314824762112917427331, 607759339422199886496126580428414916308278553796099069562650354036190535151
Offset: 1

Views

Author

Benoit Cloitre, Aug 07 2002

Keywords

Links

Crossrefs

Cf. A005246, A051786.

Programs

  • Mathematica
    nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,(b*c*d*e+1)/a}; Join[{1,1,1,1}, Transpose[ NestList[nxt,{1,1,1,1,1},15]][[5]]] (* Harvey P. Dale, Oct 03 2012 *)

A276453 a(n) = (a(n-1)+1)*(a(n-2)+1)*(a(n-3)+1)/a(n-4) with a(0) = a(1) = 1, a(2) = 2, a(3) = 6.

Original entry on oeis.org

1, 1, 2, 6, 42, 903, 136052, 881442036, 2581196224947732, 342795531574625708871288171, 5732512385084161208637718426682572229606557631, 5754497648510061274107897581706624823818534711463525598519384262130236399970112
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2016

Keywords

Comments

From Antoine de Saint Germain, Dec 30 2024: (Start)
Sequence consists of integers, see Math StackExchange link.
Values of a unitary Y-frieze pattern associated to the linearly oriented quiver K4 (i.e., the quiver whose underlying graph is the complete graph on the vertices {1,2,3,4}, oriented such that i -> j whenever i < j). (End)

Links

Crossrefs

Cf. A051786, A101879, A276175.

Programs

  • Magma
    I:=[1, 1, 2, 6]; [n le 4 select I[n] else (Self(n-1)+1)*(Self(n-2)+1)*(Self(n-3)+1)/Self(n-4): n in [1..13]]; // Vincenzo Librandi, Dec 30 2024
  • Mathematica
    RecurrenceTable[{a[n] == (a[n - 1] + 1) (a[n - 2] + 1) (a[n - 3] + 1)/a[n - 4], a[0] == 1, a[1] == 1, a[2] == 2, a[3] == 6}, a, {n, 0, 11}] (* Michael De Vlieger, Sep 03 2016 *)
  • Ruby
    def A276453(n)
  a = [1, 1, 2, 6]
  ary = [1]
  while ary.size < n + 1
    i = a[1..-1].inject(1){|s, i| s * (i + 1)}
    break if i % a[0] > 0
    a = *a[1..-1], i / a[0]
    ary << a[0]
  end
  ary
end

Formula

a(n) = A051786(n)*A051786(n+1)*A051786(n+2).

A086889 Real part of Propp's cubic recurrence starting with i (i^2=-1).

Original entry on oeis.org

0, 0, 0, 0, -1, 1, -3, -3, -41, 140, -93456, 280356252, -471833669834248, 40261350925439315783335171, 20143180524957577201436416071451979623101429, 9048277144756224572757714728840507662214586396123974675011643142128755383165
Offset: 1

Views

Author

Benoit Cloitre, Sep 17 2003

Keywords

Comments

b(n)=x(n)+i*y(n) where x(n) and y(n) are integer sequences.

Crossrefs

Cf. A051786, A086890.

Formula

a(n)=Re(b(n)) where b(1)=b(2)=b(3)=b(4)=i and b(n)=(1+b(n-1)*b(n-2)*b(n-3))/b(n-4)

A086890 Imaginary part of Propp's cubic recurrence starting with i (i^2=-1).

Original entry on oeis.org

1, 1, 1, 1, -1, -2, 0, -10, 29, -689, 79779, -306741785, -537059115735445, 32782820784889194174016393, 123899509182081496572342154385115870319926534, 6620488215208352146569523585853411114659592927465468100410366988520167986216
Offset: 1

Views

Author

Benoit Cloitre, Sep 17 2003

Keywords

Comments

b(n)=x(n)+i*y(n) where x(n) and y(n) are integer sequences.

Crossrefs

Cf. A051786, A086889.

Formula

a(n)=Im(b(n)) where b(1)=b(2)=b(3)=b(4)=i and b(n)=(1+b(n-1)*b(n-2)*b(n-3))/b(n-4)

A376085 a(0..5) = 1 and a(n) = 1 - a(n-1) - a(n-2) + a(n-1)*a(n-2)*a(n-3)/a(n-4) + a(n-2)*a(n-3)*a(n-4)/a(n-5) + a(n-3)*a(n-4)*a(n-5)/a(n-6), for n > 5.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 4, 17, 68, 2312, 668169, 6179226912, 140378107463180352, 250687119058419133437352005889, 325446213917387462112884613611747886778483963398144, 1853431255195849256571682148793108515162996950284389365029788837512893363822697947303936
Offset: 0

Views

Author

Thomas Scheuerle, Sep 09 2024

Keywords

Comments

Will this recurrence result into integer values for all n? Yes, because with the start condition a(0..5) = [x, x, x^2, y, y*x^2, y*x^3], we will obtain a sequence of polynomials:
a(6) = (y^3 + y^2 - y)*x^3 + 1.
a(7) = (y^4 + y^3 - y^2)*x^8 + y*x^5.
a(8) = (y^7 + 2*y^6 - y^5 - 2*y^4 + y^3)*x^12 + (2*y^4 + 2*y^3 - 2*y^2)*x^9 + y*x^6 + (-y^2 + y)*x^3.
a(9) = (y^13 + 4*y^12 + 2*y^11 - 8*y^10 - 5*y^9 + 8*y^8 + 2*y^7 - 4*y^6 + y^5)*x^20 + (4*y^10 + 12*y^9 - 20*y^7 + 12*y^5 - 4*y^4)*x^17 + (6*y^7 + 12*y^6 - 6*y^5 - 12*y^4 + 6*y^3)*x^14 + (-y^8 - y^7 + 3*y^6 + y^5 + y^4 + 5*y^3 - 4*y^2)*x^11 + (-2*y^5 + 4*y^3 - 2*y^2 + y)*x^8 + (-y^2 + y)*x^5 + (y^2 - y)*x^3 + 1.
...
For this polynomials a(3*n) divides a(3*n+1).
With the start condition [x, x, x^2, y, y*x^2, y^2*x^3], a(3*n) divides a(3*n+2) too. These polynomials are also more elegant:
a(6) = y^4*x^3 + 1.
a(7) = y^6*x^8 + y^2*x^5.
a(8) = y^11*x^12 + 2*y^7*x^9 + y^3*x^6.
a(9) = y^19*x^20 + 4*y^15*x^17 + 6*y^11*x^14 + 4*y^7*x^11 + y^3*x^8 + 1.
...

Crossrefs

Cf. A051786.

Programs

  • PARI
    a(n) = if( n<0, n = 6-n); if( n<6, 1, 1-a(n-1)-a(n-2)+a(n-1)*a(n-2)*a(n-3)/a(n-4)+a(n-2)*a(n-3)*a(n-4)/a(n-5)+a(n-3)*a(n-4)*a(n-5)/a(n-6))

Formula

a(n) = 1 - a(n-1) - a(n-2) + (a(n-5)^2*a(n-4)^2*a(n-3) + a(n-6)*a(n-4)^2*a(n-3)*a(n-2) + a(n-6)*a(n-5)*a(n-3)*a(n-2)*a(n-1))/(a(n-6)*a(n-5)*a(n-4)).
a(3*n) divides a(3*n+1) and a(3*n+2) too.
a(3*n-1)*a(3*n) divides a(3*n+1) and a(3*n+2).
if the prime p divides a(3*n+1) or a(3*n+2), then it will also divide a(3*n-1)*a(3*n), new prime factors appear the first time in a(3*n) only.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.