%I #34 Mar 15 2024 21:27:56

%S 1,1,3,6,15,35,85,204,493,1189,2871,6930,16731,40391,97513,235416,

%T 568345,1372105,3312555,7997214,19306983,46611179,112529341,271669860,

%U 655869061,1583407981,3822685023,9228778026,22280241075,53789260175

%N a(2n) = A011900(n), a(2n+1) = A001109(n+1).

%C a(n+1) - a(n) = A097075(n+1), a(n) + a(n+1) = A024537(n+1), a(n+2) - a(n+1) - a(n) = A105635(n+1).

%C For n >= 1, a(n) is also the edge cover number and edge cut count of the n-Pell graph. - _Eric W. Weisstein_, Aug 01 2023

%C Also the independence number, Lovasz number, and Shannon capacity of the n-Pell graph. - _Eric W. Weisstein_, Aug 01 2023

%C Floretion Algebra Multiplication Program, FAMP Code: -2jbasejseq[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

%D C. Dement, Floretion Integer Sequences (work in progress).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCoverNumber.html">Edge Cover Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCut.html">Edge Cut</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependenceNumber.html">Independence Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LovaszNumber">Lovasz Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellGraph.html">Pell Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ShannonCapacity.html">Shannon Capacity</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).

%F G.f.: y/(y^2-1) where y=x/(x^2+x-1) if offset=1. - _Michael Somos_, Sep 09 2006

%F G.f.: (-1+x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)).

%F Diagonal sums of A119468. - _Paul Barry_, May 21 2006

%F a(n) = (1 + (-1)^n + 2 A000129(n+1))/4. - _Eric W. Weisstein_, Aug 01 2023

%F a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4). - _Eric W. Weisstein_, Aug 01 2023

%p seq(iquo(fibonacci(n,2),1)-iquo(fibonacci(n,2),2),n=1..30); # _Zerinvary Lajos_, Apr 20 2008

%p with(combinat):seq(ceil(fibonacci(n,2)/2), n=1..30); # _Zerinvary Lajos_, Jan 12 2009

%t Ceiling[Fibonacci[Range[20], 2]/2]

%t Table[(1 + (-1)^n + 2 Fibonacci[n + 1, 2])/4, {n, 0, 20}] // Expand

%t CoefficientList[Series[-(-1 + x + x^2)/(1 - 2 x - 2 x^2 + 2 x^3 + x^4), {x, 0, 20}], x]

%t LinearRecurrence[{2, 2, -2, -1}, {1, 1, 3, 6}, 20]

%o (PARI) {a(n)=local(y); if(n<0, 0, n++; y=x/(x^2+x-1)+x*O(x^n); polcoeff( y/(y^2-1), n))} /* _Michael Somos_, Sep 09 2006 */

%Y Cf. A113224, A002315, A082639, A100828.

%K easy,nonn

%O 0,3

%A _Creighton Dement_, Oct 18 2005