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.

A372239 Expansion of (1 + 2*x) / ((1 - 2*x)*sqrt(1 - 4*x)).

Original entry on oeis.org

1, 6, 22, 76, 262, 916, 3260, 11800, 43334, 161028, 604052, 2283048, 8681116, 33171144, 127260088, 489870896, 1891057222, 7317881444, 28378110628, 110251755656, 429040567732, 1672032067544, 6524678847688, 25490986350416, 99696437839132, 390298689482216
Offset: 0

Views

Author

Mélika Tebni, Apr 23 2024

Keywords

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 6 == 0 (mod p).
Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 2 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028283, A029759, A082590, A126966, A188622.

Programs

  • Maple
    a := n -> binomial(2*n,n) + 4*add(2^(n-k-1)*binomial(2*k,k), k = 0 .. n-1):
seq(a(n), n = 0 .. 25);
# Second program:
a:= proc(n) option remember; `if`(n=0,1,2*a(n-1)+2*binomial(2*n-2, n-1)*(3*n-1)/n) end: seq(a(n), n = 0 .. 25);
# Recurrence:
a := proc(n) option remember; if n < 2 then return [1, 6][n + 1] fi;
((-18*(n - 2)^2 - 42*n + 66)*a(n - 1) + 4*(3*n - 1)*(2*n - 3)*a(n - 2)) / (n*(4 - 3*n)) end: seq(a(n), n = 0..25);  # Peter Luschny, Apr 23 2024

Formula

a(n) = 5*A000984(n) - 4* A029759(n) = binomial(2*n,n) + 4*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
a(n) = 2*a(n-1) + A028283(n) = 2*a(n-1) + 2*binomial(2n-2, n-1)*(3*n-1)/n for n >= 1.
a(n) = 2*A082590(n-1) + A082590(n) for n >= 1.
a(n) = 2*A188622(n) - A126966(n).
D-finite with recurrence n*a(n) +2*(-2*n-1)*a(n-1) +4*(-n+6)*a(n-2) +8*(2*n-5)*a(n-3)=0. - R. J. Mathar, Apr 24 2024
E.g.f.: exp(2*x)*(BesselI(0, 2*x)*(1 + 4*x + 2*Pi*x*StruveL(1, 2*x)) - 2*Pi*x*BesselI(1, 2*x)*StruveL(0, 2*x)). - Stefano Spezia, Aug 29 2025

A372420 Expansion of (1 + x) / ((1 - 2*x)*sqrt(1 - 4*x)).

Original entry on oeis.org

1, 5, 18, 62, 214, 750, 2676, 9708, 35718, 132926, 499228, 1888644, 7186876, 27478508, 105474216, 406182552, 1568563014, 6071812638, 23552366796, 91525132692, 356242058004, 1388588519268, 5419533876696, 21176597444712, 82834229300124, 324326668721100
Offset: 0

Views

Author

Mélika Tebni, Apr 30 2024

Keywords

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 5 == 0 (mod p).
Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028270, A029759, A082590, A126966, A188622.

Programs

  • Maple
    a := n -> -2^(n-1)*3*I + binomial(2*n, n)*(1-3/2*hypergeom([1, n+1/2], [n+1], 2)):
seq(simplify(a(n)), n = 0 .. 25);
  • PARI
    my(x='x+O('x^40)); Vec((1 + x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ Michel Marcus, Apr 30 2024

Formula

a(n) = 4*A000984(n) - 3* A029759(n) = binomial(2*n,n) + 3*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
a(n) = 2*a(n-1) + A028270(n) = 2*a(n-1) + binomial(2*n, n) + binomial(2*n-2, n-1) for n >= 1.
a(n) = - 2^(n-1)*3*i + binomial(2*n,n)*(1-3/2*hypergeom([1,n+1/2],[n + 1],2)).
a(n) = A082590(n-1) + A082590(n) for n >= 1.
a(n) = (5*A188622(n) - 2*A126966(n)) / 3.
D-finite with recurrence n*a(n) -5*n*a(n-1) +2*(n+5)*a(n-2) +4*(2*n-5)*a(n-3)=0. - R. J. Mathar, May 01 2024

A372611 Expansion of (1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x)).

Original entry on oeis.org

1, 7, 26, 90, 310, 1082, 3844, 13892, 50950, 189130, 708876, 2677452, 10175356, 38863780, 149045960, 573559240, 2213551430, 8563950250, 33203854460, 128978378620, 501839077460, 1955475615820, 7629823818680, 29805375256120, 116558646378140, 456270710243332
Offset: 0

Views

Author

Mélika Tebni, May 07 2024

Keywords

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 7 == 0 (mod p).
Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 3 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028322, A029759, A082590, A126966, A188622, A372239, A372420.

Programs

  • Maple
    a := n -> -2^(n-1)*5*I + binomial(2*n, n)*(1-5/2*hypergeom([1, n+1/2], [n+1], 2)): seq(simplify(a(n)), n = 0 .. 25);
  • PARI
    my(x='x+O('x^30)); Vec((1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ Michel Marcus, May 07 2024

Formula

a(n) = 6*A000984(n) - 5* A029759(n) = binomial(2*n,n) + 5*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
a(n) = 2*a(n-1) + A028322(n) = 2*a(n-1) + binomial(2*n, n) + 3*binomial(2*n-2, n-1) for n >= 1.
a(n) = - 2^(n-1)*5*i + binomial(2*n,n)*(1-5/2*hypergeom([1, n + 1/2], [n + 1], 2)).
a(n) = 3*A082590(n-1) + A082590(n) for n >= 1.
a(n) = (7*A188622(n) - 4*A126966(n))/3.
a(n) = 2*A372239(n) - A372420(n).
Showing 1-3 of 3 results.

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