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.

A113224 a(2n) = A002315(n), a(2n+1) = A082639(n+1).

Original entry on oeis.org

1, 2, 7, 16, 41, 98, 239, 576, 1393, 3362, 8119, 19600, 47321, 114242, 275807, 665856, 1607521, 3880898, 9369319, 22619536, 54608393, 131836322, 318281039, 768398400, 1855077841, 4478554082, 10812186007, 26102926096, 63018038201
Offset: 0

Views

Author

Creighton Dement, Oct 18 2005

Keywords

Comments

From Paul D. Hanna, Oct 22 2005: (Start)
The logarithmic derivative of this sequence is twice the g.f. of A113282, where a(2*n) = A113282(2*n), a(4*n+1) = A113282(4*n+1) - 3, a(4*n+3) = A113282(4*n+3) - 1.
Equals the self-convolution of integer sequence A113281. (End)
With an offset of 1, this sequence is the case P1 = 2, P2 = 0, Q = -1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 19 2015
Floretion Algebra Multiplication Program, FAMP Code: -2ibaseiseq[B*C], B = - .5'i + .5'j - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'; C = + .5'i + .5i' + .5'ii' + .5e

Links

Crossrefs

Cf. A113225, A002315, A082639, A100828, A113281, A113282, A113283, A113284, A129744.

Programs

  • Magma
    [Floor((1+Sqrt(2))^(n+1)/2): n in [0..30]]; // Vincenzo Librandi, Mar 20 2015
  • Mathematica
    a[n_] := n*Sum[ Sum[ Binomial[i, n-k-i]*Binomial[k+i-1, k-1], {i, Ceiling[(n-k)/2], n-k}]*(1-(-1)^k)/(2*k), {k, 1, n}]; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
CoefficientList[Series[(1 + x^2) / ((x^2 - 1) (x^2 + 2 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{2,2,-2,-1},{1,2,7,16},30] (* Harvey P. Dale, Oct 10 2017 *)
  • Maxima
    a(n):=n*sum(sum(binomial(i,n-k-i)*binomial(k+i-1,k-1),i,ceiling((n-k)/2),n-k)*(1-(-1)^k)/(2*k),k,1,n); /* Vladimir Kruchinin, Apr 11 2011 */
  • PARI
    {a(n)=local(x=X+X*O(X^n));polcoeff((1+x^2)/(1-x^2)/(1-2*x-x^2),n,X)} \\ Paul D. Hanna

Formula

G.f.: (1+x^2)/((x-1)*(x+1)*(x^2+2*x-1)).
a(n+2) - a(n+1) - a(n) = A100828(n+1).
a(n) = -(u^(n+1)-1)*(v^(n+1)-1)/2 with u = 1+sqrt(2), v = 1-sqrt(2). - Vladeta Jovovic, May 30 2007
a(n) = n * Sum_{k=1..n} Sum_{i=ceiling((n-k)/2)..n-k} binomial(i,n-k-i)*binomial(k+i-1,k-1)*(1-(-1)^k)/(2*k). - Vladimir Kruchinin, Apr 11 2011
a(n) = A001333(n+1) - A000035(n). - R. J. Mathar, Apr 12 2011
a(n) = floor((1+sqrt(2))^(n+1)/2). - Bruno Berselli, Feb 06 2013
From Peter Bala, Mar 19 2015: (Start)
a(n) = (1/2) * A129744(n+1).
exp( Sum_{n >= 1} 2*a(n-1)*x^n/n ) = 1 + 2*Sum_{n >= 1} Pell(n) *x^n. (End)
a(n) = A105635(n-1) + A105635(n+1). - R. J. Mathar, Mar 23 2023

A113281 Self-convolution equals A113224.

Original entry on oeis.org

1, 1, 3, 5, 11, 23, 51, 113, 255, 579, 1325, 3047, 7039, 16319, 37951, 88489, 206799, 484255, 1135969, 2668951, 6279509, 14793169, 34889553, 82372723, 194664283, 460436551, 1089943229, 2582033519, 6120967411, 14519686915, 34463104999
Offset: 0

Views

Author

Paul D. Hanna, Oct 22 2005

Keywords

Comments

Convolution of bisections A113283 and A113284 yields A082639, which is a bisection of the self-convolution of this sequence. Terms of the logarithmic derivative A113282 are related to the self-convolution A113224 by: A113282(2*n) = A113224(2*n), A113282(4*n+1) = 3 + A113224(4*n+1), A113282(4*n+3) = 1 + A113224(4*n+3).

Crossrefs

Cf. A113282, A113283, A113284.

Formula

G.f.: A(x) = ( (1+x^2)/(1-x^2)/(1-2*x-x^2) )^(1/2).
a(n) ~ (1 + sqrt(2))^(n + 1/2) / sqrt(2*Pi*n). - Vaclav Kotesovec, Oct 18 2020

A113283 Even bisection of A113281: a(n) = A113281(2*n).

Original entry on oeis.org

1, 3, 11, 51, 255, 1325, 7039, 37951, 206799, 1135969, 6279509, 34889553, 194664283, 1089943229, 6120967411, 34463104999, 194474062663, 1099571123853, 6227893795649, 35329149864161, 200691916063033, 1141489886332555
Offset: 0

Views

Author

Paul D. Hanna, Oct 22 2005

Keywords

Crossrefs

Cf. A113281, A113282, A113284.

Programs

  • PARI
    {a(n)=local(x=X+X*O(X^n));polcoeff( ((1+x)/(1-x)/(1-6*x+x^2)*(1-x+(1-6*x+x^2)^(1/2))/2)^(1/2),n,X)}

Formula

G.f.: ( (1+x)/(1-x)/(1-6*x+x^2)*(1-x+(1-6*x+x^2)^(1/2))/2 )^(1/2).
a(n) ~ (1 + sqrt(2))^(2*n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 18 2020

A113284 Odd bisection of A113281: a(n) = A113281(2*n+1).

Original entry on oeis.org

1, 5, 23, 113, 579, 3047, 16319, 88489, 484255, 2668951, 14793169, 82372723, 460436551, 2582033519, 14519686915, 81845419777, 462319557311, 2616334071987, 14830559353869, 84189874112659, 478559722392493
Offset: 0

Views

Author

Paul D. Hanna, Oct 22 2005

Keywords

Crossrefs

Cf. A113281, A113282, A113283.

Programs

  • PARI
    {a(n)=local(x=X+X*O(X^n));polcoeff( (2*(1+x)/(1-x)/(1-6*x+x^2)/(1-x+(1-6*x+x^2)^(1/2)) )^(1/2),n,X)}

Formula

G.f.: ( (1+x)/(1-x)/(1-6*x+x^2)*(1-x-(1-6*x+x^2)^(1/2))/2/x )^(1/2).
a(n) ~ (1 + sqrt(2))^(2*n + 3/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 18 2020
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.