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

0,2

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

Table of n, a(n) for n=0..28.

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).

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

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 *)

Prog

(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
(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 */
(Magma) [Floor((1+Sqrt(2))^(n+1)/2): n in [0..30]]; // Vincenzo Librandi, Mar 20 2015

Crossrefs

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

Sequence in context: A131727 A320236 A073371 * A178945 A309561 A026571

Adjacent sequences: A113221 A113222 A113223 * A113225 A113226 A113227

Keyword

nonn,easy

Author

Creighton Dement, Oct 18 2005

Status

approved