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 1-3 of 3 results.

A084174 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).

Original entry on oeis.org

1, 1, 3, 6, 14, 29, 61, 124, 252, 507, 1019, 2042, 4090, 8185, 16377, 32760, 65528, 131063, 262135, 524278, 1048566, 2097141, 4194293, 8388596, 16777204, 33554419, 67108851, 134217714, 268435442, 536870897, 1073741809, 2147483632
Offset: 0

Views

Author

Paul Barry, May 18 2003

Keywords

Comments

Original name was: Generalized Jacobsthal numbers.
Row sums of triangle A131087. - Gary W. Adamson, Jun 14 2007

Links

Crossrefs

Cf. A131087.

Programs

  • Magma
    [2^n+(-1)^n/4-(2*n+1)/4: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011
  • Maple
    seq(floor(2^n-n(n+1)/2),n=0..31); # Zerinvary Lajos, Apr 29 2007
  • Mathematica
    LinearRecurrence[{3,-1,-3,2},{1,1,3,6},40] (* Harvey P. Dale, Feb 17 2021 *)

Formula

a(n) = 2^n + (-1)^n/4 - (2*n+1)/4.
a(n+2) = a(n+1) + 2*a(n) + n, a(0)=1, a(1)=1.
a(n) = A084172(n) - A001045(n).
a(n) = floor(2^n - n*(n+1)/2). - Zerinvary Lajos, Apr 29 2007
G.f.: (1 - 2*x + x^2 + x^3)/(1 - 3*x + x^2 + 3*x^3 - 2*x^4). - Colin Barker, Jan 16 2012

A109225 Triangle read by rows: T(n,0) = T(n,n) = 1 and for 0 < k < n: T(n,k) = T(n-1,k-1) + 1 - T(n-1,k-1) mod 2 + T(n-1,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 6, 1, 1, 5, 12, 13, 8, 1, 1, 6, 17, 26, 21, 10, 1, 1, 7, 24, 43, 48, 31, 12, 1, 1, 8, 31, 68, 91, 80, 43, 14, 1, 1, 9, 40, 99, 160, 171, 124, 57, 16, 1, 1, 10, 49, 140, 259, 332, 295, 182, 73, 18, 1, 1, 11, 60, 189, 400, 591, 628, 477
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 23 2005

Keywords

Comments

Row sums give A084172, the generalized Jacobsthal numbers;
T(n,1) = n for n>0;
T(n,2) = A074148(n) for n>1.

Links

Crossrefs

Cf. A007318.

A084173 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).

Original entry on oeis.org

1, 3, 5, 13, 27, 59, 121, 249, 503, 1015, 2037, 4085, 8179, 16371, 32753, 65521, 131055, 262127, 524269, 1048557, 2097131, 4194283, 8388585, 16777193, 33554407, 67108839, 134217701, 268435429, 536870883, 1073741795, 2147483617
Offset: 0

Views

Author

Paul Barry, May 18 2003

Keywords

Comments

Original name was: A sum of generalized Jacobsthal numbers.

Links

Programs

  • Magma
    [2^(n+1)-(2*n+1)/2-(-1)^n/2: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011
  • Mathematica
    a[n_]:=2^(n+1) - (2*n+1)/2 - (-1)^n/2; Array[a, 50, 0] (* or *)
CoefficientList[Series[(4*x^3 - 3*x^2 + 1)/(-2*x^4 + 3*x^3 + x^2 - 3*x + 1), {x, 0, 50}], x] (* or *)
LinearRecurrence[{3, -1, -3, 2}, {1, 3, 5, 13}, 50] (* Stefano Spezia, Oct 08 2018 *)
  • PARI
    x='x+O('x^33); Vec((4*x^3-3*x^2+1)/(-2*x^4+3*x^3+x^2-3*x+1)) \\ Altug Alkan, Oct 08 2018

Formula

a(n) = 2^(n+1) - (2*n+1)/2 - (-1)^n/2.
G.f.: (4*x^3 - 3*x^2 + 1)/(-2*x^4 + 3*x^3 + x^2 - 3*x + 1).
a(n) = A084172(n) + A083579(n).
Showing 1-3 of 3 results.

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