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

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

Author

Creighton Dement, Oct 18 2005

Keywords

• nonn
• easy

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

• Creighton Dement, Floretion Online Multiplier. [broken link]
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

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