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 1-4 of 4 results.

A260006 a(n) = f(1,n,n), where f is the Sudan function defined in A260002.

Original entry on oeis.org

0, 3, 12, 35, 90, 217, 504, 1143, 2550, 5621, 12276, 26611, 57330, 122865, 262128, 557039, 1179630, 2490349, 5242860, 11010027, 23068650, 48234473, 100663272, 209715175, 436207590, 905969637, 1879048164, 3892314083, 8053063650, 16642998241, 34359738336
Offset: 0

Views

Author

Natan Arie Consigli, Jul 23 2015

Keywords

Comments

f(1,n,n) = 2^n*(n+2) - (n+2) = (2^n - 1)*(n+2).
To evaluate the Sudan function see A260002 and A260003.
The numbers are alternately even and odd because for even n (2^n-1)*(n+2) is even and (2^(n+1)-1)*(n+1+2) is odd.
From Enrique Navarrete, Oct 02 2021: (Start)
a(n-2) is the number of ways we can write [n] as the union of 2 sets of sizes i, j which intersect in exactly 1 element (1 < i, j < n; i = j allowed), n>=2.
For n = 4, a(n-2) = 12 since [4] can be written as the unions:
{1,2} U {1,3,4}; {2,3} U {1,2,4}; {1,2} U {2,3,4}; {2,3} U {1,3,4};
{1,3} U {1,2,4}; {2,4} U {1,2,3}; {1,3} U {2,3,4}; {2,4} U {1,3,4};
{1,4} U {1,2,3}; {3,4} U {1,2,3}; {1,4} U {2,3,4}; {3,4} U {1,2,4}. (End)

Examples

			a(4) = (2^4 - 1)*(4 + 2) = 90.

Links

Crossrefs

Cf. A000295 (f(1,0,n)), A000325 (f(1,2,n)), A005408 (f(1,n,1) = 2n+1), A001787 (n*2^(n-1)), A079583 (f(1,1,n)), A123720 (f(1,4,n)), A133124 (f(1,3,n)).
Cf. A260002, A260003, A260004, A260005.
Cf. A000984, A002697, A303602.

Programs

  • Magma
    [(2^n-1)*(n+2): n in [0..30]]; // Vincenzo Librandi, Aug 22 2015
  • Mathematica
    Table[(2^n -1)(n+2), {n, 0, 30}] (* Michael De Vlieger, Aug 22 2015 *)
CoefficientList[Series[x(3 -6x +2x^2)/((1-x)^2 (1-2x)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 22 2015 *)
LinearRecurrence[{6,-13,12,-4},{0,3,12,35},40] (* Harvey P. Dale, Mar 04 2023 *)
  • PARI
    vector(40, n, n--; (2^n-1)*(n+2)) \\ Michel Marcus, Jul 29 2015
  • PARI
    concat(0, Vec(x*(3-6*x+2*x^2)/((1-x)^2*(1-2*x)^2) + O(x^40))) \\ Colin Barker, Jul 29 2015
  • Sage
    [(n+2)*(2^n -1) for n in (0..30)] # G. C. Greubel, Dec 30 2021

Formula

a(n) = (2^n -1)*(n+2).
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4) for n>3. - Colin Barker, Jul 29 2015
G.f.: x*(3 - 6*x + 2*x^2) / ((1-x)^2*(1-2*x)^2). - Colin Barker, Jul 29 2015
E.g.f.: 2*(x+1)*exp(2*x) - (x+2)*exp(x). - Robert Israel, Aug 23 2015
From Enrique Navarrete, Oct 02 2021: (Start)
a(n-2) = Sum_{j=2..n/2} binomial(n,j)*j, n even > 2.
a(n-2) = (Sum_{j=2..floor(n/2)} binomial(n,j)*j) + (1/2)*binomial(n, ceiling(n/2))*ceiling(n/2), n odd > 1. (End)
From Wolfdieter Lang, Nov 12 2021: (Start)
The previous bisection becomes for a(n):
a(2*k) = 2*(A002697(k+1) - (k+1)), and a(2*k+1) = A303602(k+1) - (2*k+3)*(2 - A000984(k+1))/2 = (2*k+3)*(4^(k+1) - 2)/2, for k >= 0. (End)

A260003 Values f(1,x,y) with x>=0, y>0, in increasing order, where f is the Sudan function defined in A260002.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75
Offset: 1

Views

Author

Natan Arie Consigli, Jul 23 2015

Keywords

Comments

Equivalently, numbers of the form 2^y(x+2)-y-2.
Using f(1,x,y) = f(0, f(1,x,y-1), f(1,x,y-1)+y) = 2*f(1,x,y-1) + y
f(1,x,y) + y + 2 = 2*(f(1,x,y-1)+y-1+2) let g(y) = f(1,x,y) + y + 2 then g(y) = 2*g(y-1). This means g(y)=2^y*g(0) and f(1,x,y) + y + 2 = 2^y(f(1,x,0)+2) but f(1,x,0) = x so f(1,x,y) = 2^y(x+2) - y - 2.
In this list we suppose that y>0. If we include y=0, every natural number would be in the sequence.

Examples

			19 is listed because f(1,1,3) = 2^3*(1+2) - 3 - 2 = 19.

Crossrefs

Cf. A000325 (f(1,2,n)), A005408 (f(1,n,1)=2n+1), A048493 (f(1,n,2)), A079583 (f(1,1,n)), A123720 (f(1,4,n)), A133124(f(1,3,n)), A260002, A260004, A260005 (f(2,n,2)), A260006.

A133804 Triangle read by rows: A007318 * A133080 * A133566.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 7, 4, 1, 5, 14, 10, 5, 1, 6, 25, 20, 16, 6, 1, 7, 41, 35, 41, 21, 7, 1, 8, 63, 56, 91, 56, 29, 8, 1, 9, 92, 84, 182, 126, 92, 36, 9, 1, 10, 129, 120, 336, 252, 246, 120, 46, 10, 1, 11, 175, 165, 582, 462, 582, 330, 175, 55, 11, 1, 12, 231, 220, 957, 792, 1254, 792, 550, 220, 67, 12, 1
Offset: 1

Views

Author

Gary W. Adamson, Sep 23 2007

Keywords

Comments

Row sums = A133124: (1, 3, 7, 16, 35, 74, 153, ...).
A133805 = binomial transform of (A133566 * A133080).

Examples

			First few rows of the triangle:
  1;
  2,  1;
  3,  3,  1;
  4,  7,  4,  1;
  5, 14, 10,  5,  1;
  6, 25, 20, 16,  6,  1;
  7, 41, 35, 41, 21,  7,  1;
  ...

Crossrefs

Cf. A133080, A133124, A133566.

Formula

Binomial transform of (A133080 * A133566), where A133080 * A133566 = an infinite lower triangular matrix with (1,1,1,...) in the main and subdiagonals and (0,1,0,1,0,...) in the subsubdiagonal.

Extensions

a(21) = 1 inserted and more terms from Georg Fischer, Jun 08 2023

A133938 A007318 * (A129686 + A133080 - I), where I is the identity matrix.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 4, 4, 1, 11, 8, 11, 4, 1, 16, 15, 25, 11, 6, 1, 22, 26, 50, 26, 22, 6, 1, 29, 42, 91, 56, 63, 22, 8, 1, 37, 64, 154, 112, 154, 64, 37, 8, 1, 46, 93, 24, 210, 336, 162, 129, 37, 10, 1
Offset: 1

Views

Author

Gary W. Adamson, Sep 29 2007

Keywords

Comments

Left column = A000124: (1, 2, 4, 7, 11, ...).
Row sums = A133124: (1, 3, 7, 16, 35, 74, ...).

Examples

			First few rows of the triangle:
   1;
   2,  1;
   4,  2,  1;
   7,  4,  4,  1;
  11,  8, 11,  4,  1;
  16, 15, 25, 11,  6,  1;
  22, 26, 50, 26, 22,  6,  1;
  ...

Crossrefs

Cf. A000124, A133124, A129686, A133080.

Formula

Binomial transform of matrix M, where M = a tridiagonal matrix with (1,1,1,...) in the main diagonal, (1,0,1,0,...) in the subdiagonal and (1,1,1,...) in the subsubdiagonal. M = (A129686 + A133080 - I), I = Identity matrix.
Showing 1-4 of 4 results.

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