A077419 - OEIS

%I #36 Jan 31 2023 15:11:35

%S 1,1,1,2,2,3,5,7,11,17,26,40,63,97,153,238,376,587,931,1458,2317,3640,

%T 5794,9124,14545,22951,36631,57904,92512,146461,234205,371281,594169,

%U 943045,1510192,2399460,3844787,6114555,9802895,15603339,25027296

%N Largest Whitney number of Fibonacci lattices J(Z_n).

%C A051286 and A051291, interleaved. a(n) is the maximal element in the n-th row of A079487 or A123245 and in the (n+2)-th row of A078807 or A078808. - _Andrey Zabolotskiy_, Sep 21 2017

%H Emanuele Munarini, Mar 05 2007, <a href="/A077419/b077419.txt">Table of n, a(n) for n = 0..100</a>

%H Brian Kent, Sarah Racz, and Sanjit Shashi, <a href="https://arxiv.org/abs/2301.07722">Scrambling in quantum cellular automata</a>, arXiv:2301.07722 [quant-ph], 2023.

%H E. Munarini and N. Zagaglia Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00378-3">On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns</a>, Discrete Mathematics 259 (2002), 163-177.

%F G.f.: (1 + 2 x + 2 x^4 - x^6 - (1-x^2) sqrt(1 - 2 x^2 - x^4 - 2 x^6 + x^8) )/(2x sqrt(1 - 2 x^2 - x^4 - 2 x^6 + x^8)). - _Emanuele Munarini_, Mar 05 2007

%F a(n) ~ phi^(n+2) / (5^(1/4) * sqrt(2*Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Sep 22 2017

%F D-finite with recurrence: (n+1)*a(n) +(n-2)*a(n-1) +2*(-n+1)*a(n-2) +2*(-n+1)*a(n-3) +(-n-3)*a(n-4) +(-n+8)*a(n-5) +2*(-n+6)*a(n-6) +2*(-n+7)*a(n-7) +(n-9)*a(n-8) +(n-10)*a(n-9)=0. - _R. J. Mathar_, Nov 19 2019

%p with(FormalPowerSeries): with(LREtools): # requires Maple 2022

%p gf:= (1 + 2*x + 2*x^4 - x^6 - (1-x^2)*sqrt(1 - 2*x^2 - x^4 - 2*x^6 + x^8))/(2*x*sqrt(1 - 2*x^2 - x^4 - 2*x^6 + x^8));

%p re:= FindRE(gf,x,a(n));

%p inits:= {seq(a(i-1)=[1,1,1,2,2,3,5,7,11,17,26,40,63,97, 153][i],i=1..14)};

%p rm:=  (n+1)*a(n) +(n-2)*a(n-1) +2*(-n+1)*a(n-2) +2*(-n+1)*a(n-3) +(-n-3)*a(n-4) +(-n+8)*a(n-5) +2*(-n+6)*a(n-6) +2*(-n+7)*a(n-7) +(n-9)*a(n-8) +(n-10)*a(n-9)=0;

%p minre:= MinimalRecurrence(re, a(n), inits); minrm:= MinimalRecurrence(rm, a(n), inits); # shows that Mathar's recurrence is equivalent

%p f:= REtoproc(re,a(n),inits); seq(f(n),n=0..40); # _Georg Fischer_, Oct 22 2022

%t gf[x_] = (1 + 2 x + 2 x^4 - x^6 - (1 - x^2) Sqrt[1 - 2 x^2 - x^4 - 2 x^6 + x^8])/(2 x Sqrt[1 - 2 x^2 - x^4 - 2 x^6 + x^8]);

%t Table[SeriesCoefficient[gf[x], {x, 0, n}], {n, 0, 40}] (* _Hugo Pfoertner_, Oct 22 2022 *)

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Jan 19 2003

%E More terms from _Emanuele Munarini_, Mar 05 2007