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.

A060923 Bisection of Lucas triangle A060922: even-indexed members of column sequences of A060922 (not counting leading zeros).

Original entry on oeis.org

1, 4, 1, 11, 17, 1, 29, 80, 39, 1, 76, 303, 315, 70, 1, 199, 1039, 1687, 905, 110, 1, 521, 3364, 7470, 6666, 2120, 159, 1, 1364, 10493, 29634, 37580, 20965, 4311, 217, 1, 3571, 31885, 109421, 181074, 148545
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Examples

			Triangle begins:
  {1};
  {4,1};
  {11,17,1};
  {29,80,39,1};
  ...
pLe(2,x) = 1+11*x-11*x^2+4*x^3.

Crossrefs

Row sums give A060926.
Column sequences (without leading zeros) are, for m=0..3: A002878, A060934-A060936.
Companion triangle A060924 (odd part).
Cf. A060922.

Formula

a(n, m) = A060922(2*n-m, m).
a(n, m) = ((2*(n-m)+1)*A060924(n-1, m-1) + 2*(4*n-3*m)*a(n-1, m-1) + 4*(2*n-m-1)*A060924(n-2, m-1))/(5*m), m >= n >= 1; a(n, 0)= A002878(n); else 0.
G.f. for column m >= 0: x^m*pLe(m+1, x)/(1-3*x+x^2)^(m+1), where pLe(n, x) := Sum_{m=0..n+floor(n/2)} A061186(n, m)*x^m are the row polynomials of the (signed) staircase A061186.
T(n,k) = 3*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2) + 4*T(n-3,k-2), T(0,0) = 1, T(1,0) = 4, T(1,1) = 1, T(2,0) = 11, T(2,1) = 17, T(2,2) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 21 2014

A060928 Expansion of 1/(1 - 5*x - 4*x^3).

Original entry on oeis.org

1, 5, 25, 129, 665, 3425, 17641, 90865, 468025, 2410689, 12416905, 63956625, 329425881, 1696797025, 8739811625, 45016761649, 231870996345, 1194314228225, 6151638187721, 31685674923985, 163205631532825
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Links

Crossrefs

Cf. A060926, A060927.

Programs

  • Magma
    I:=[1, 5, 25, 129]; [n le 3 select I[n] else 5*Self(n-1) + 4*Self(n-3): n in [1..31]]; // G. C. Greubel, Apr 07 2021
  • Maple
    m:= 40; S:= series( 1/(1-5*x-4*x^3), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Apr 07 2021
  • Mathematica
    CoefficientList[Series[1/(1-5x-4x^3),{x,0,30}],x] (* or *) LinearRecurrence[ {5,0,4}, {1,5,25}, 30] (* Harvey P. Dale, Apr 09 2018 *)
  • PARI
    { for (n=0, 30, if (n>2, a=5*a1 + 4*a3; a3=a2; a2=a1; a1=a, if (n==0, a=a3=1, if (n==1, a=a2=5, a=a1=25))); print1(a, ", "); ) } \\ Harry J. Smith, Jul 14 2009
  • Sage
    def A060928_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1/(1-5*x-4*x^3) ).list()
A060928_list(30) # G. C. Greubel, Apr 07 2021

Formula

G.f.: 1/(1 - 5*x - 4*x^3).
a(n) = 5*a(n-1) + 4*a(n-3), n >= 3, a(n) = 5^n, n = 0, 1, 2.

A060927 Row sums of triangle A060924 (odd part of bisection of Lucas triangle).

Original entry on oeis.org

3, 13, 65, 337, 1737, 8945, 46073, 237313, 1222345, 6296017, 32429337, 167036065, 860364393, 4431539313, 22825840825, 117570661697, 605579465737, 3119200691985, 16066286106713, 82753748396513
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Links

Crossrefs

Cf. A060926 (row sums of A060923 companion triangle).
Cf. A060924, A060928.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30);
Coefficients(R!( (3-2*x)/(1-5*x-4*x^3) )); // G. C. Greubel, Apr 07 2021
  • Mathematica
    CoefficientList[Series[(3-2*x)/(1-5*x-4*x^3), {x, 0, 30}], x] (* G. C. Greubel, Apr 07 2021 *)
  • Sage
    def A060927_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (3-2*x)/(1-5*x-4*x^3) ).list()
A060927_list(30) # G. C. Greubel, Apr 07 2021

Formula

a(n) = Sum_{j=0..n} A060924(n, j).
G.f.: (3-2*x)/(1-5*x-4*x^3).
a(n) = 3*A060928(n) - 2*A060928(n-1), n >= 1; a(0)=3.
Showing 1-3 of 3 results.

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