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.

A060924 Bisection of Lucas triangle A060922: odd-indexed members of column sequences of A060922 (not counting leading zeros).

Original entry on oeis.org

3, 7, 6, 18, 38, 9, 47, 158, 120, 12, 123, 566, 753, 280, 15, 322, 1880, 3612, 2568, 545, 18, 843, 5964, 15040, 16220, 7043, 942, 21, 2207, 18342, 57366, 83780, 57560, 16536, 1498, 24, 5778, 55162, 206115
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Row sums give A060927. Column sequences (without leading zeros) are, for m=0..5: A005248(n+1), 2*A061171, A061172, 4*A061173, A061174, 2*A061175.
Companion triangle A060923 (even part).

Examples

			{3}; {7,6}; {18,38,9}; {47,158,120,12}; ...; pLo(2,x)= 2*(3+x-2*x^2).

Crossrefs

Cf. A005248.

Formula

a(n, m) = A060922(2*n+1-m, m).
a(n, m) = ((2*n-m+1)*A060923(n, m-1) + 2*(2*(2*n+1)-3*m)*a(n-1, m-1) + 4*(2*n-m)*A060923(n-1, m-1))/(5*m), m >= n >= 1; a(n, 0) = A005248(n); otherwise 0.
G.f. for column m >= 0: x^m*pLo(m+1, x)/(1-3*x+x^2)^(m+1), where pLo(n, x) := Sum_{m=0..n+floor((n-1)/2)} A061187(n-1, m)*x^m are the row polynomials of the (signed) staircase A061187.
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) = 3, T(1,0) = 7, T(1,1) = 6, T(2,0) = 18, T(2,1) = 38, T(2,2) = 9, 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.

A060926 Row sums of triangle A060923 (even part of bisection of Lucas triangle).

Original entry on oeis.org

1, 5, 29, 149, 765, 3941, 20301, 104565, 538589, 2774149, 14289005, 73599381, 379093501, 1952623525, 10057515149, 51803949749, 266830242845, 1374381274821, 7079122173101, 36462931836885
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Links

Crossrefs

Cf. A060927 (Row sums of A060924).

Programs

  • Magma
    I:=[1,5,29]; [n le 3 select I[n] else 5*Self(n-1) + 4*Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 09 2021
  • Mathematica
    LinearRecurrence[{5,0,4}, {1,5,29}, 31] (* G. C. Greubel, Apr 09 2021 *)
  • Sage
    def A060926_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1+4*x^2)/(1-5*x-4*x^3) ).list()
A060926_list(30) # G. C. Greubel, Apr 09 2021

Formula

a(n) = Sum_{j=0..n} A060923(n, j).
a(n) = A060928(n) + 4*A060928(n-2), n >= 2, otherwise A060928(n).
G.f.: (1+4*x^2)/(1-5*x-4*x^3).
Showing 1-3 of 3 results.

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