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%I #51 Mar 15 2024 21:27:20
%S 1,2,7,16,41,98,239,576,1393,3362,8119,19600,47321,114242,275807,
%T 665856,1607521,3880898,9369319,22619536,54608393,131836322,318281039,
%U 768398400,1855077841,4478554082,10812186007,26102926096,63018038201
%N a(2n) = A002315(n), a(2n+1) = A082639(n+1).
%C From _Paul D. Hanna_, Oct 22 2005: (Start)
%C 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.
%C Equals the self-convolution of integer sequence A113281. (End)
%C 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
%C 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
%H C. Dement, <a href="http://fumba.eu/sitelayout/Floretion.php">Floretion Online Multiplier</a>. [broken link]
%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).
%F G.f.: (1+x^2)/((x-1)*(x+1)*(x^2+2*x-1).
%F a(n+2) - a(n+1) - a(n) = A100828(n+1).
%F 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
%F 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
%F a(n) = A001333(n+1) - A000035(n). - _R. J. Mathar_, Apr 12 2011
%F a(n) = floor((1+sqrt(2))^(n+1)/2). - _Bruno Berselli_, Feb 06 2013
%F From _Peter Bala_, Mar 19 2015: (Start)
%F a(n) = (1/2) * A129744(n+1).
%F exp( Sum_{n >= 1} 2*a(n-1)*x^n/n ) = 1 + 2*Sum_{n >= 1} Pell(n) *x^n. (End)
%F a(n) = A105635(n-1) + A105635(n+1). - _R. J. Mathar_, Mar 23 2023
%t 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_ *)
%t CoefficientList[Series[(1 + x^2) / ((x^2 - 1) (x^2 + 2 x - 1)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Mar 20 2015 *)
%t LinearRecurrence[{2,2,-2,-1},{1,2,7,16},30] (* _Harvey P. Dale_, Oct 10 2017 *)
%o (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_
%o (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 */
%o (Magma) [Floor((1+Sqrt(2))^(n+1)/2): n in [0..30]]; // _Vincenzo Librandi_, Mar 20 2015
%Y Cf. A113225, A002315, A082639, A100828, A113281, A113282, A113283, A113284, A129744.
%K nonn,easy
%O 0,2
%A _Creighton Dement_, Oct 18 2005
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