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A077419
Largest Whitney number of Fibonacci lattices J(Z_n).
6
1, 1, 1, 2, 2, 3, 5, 7, 11, 17, 26, 40, 63, 97, 153, 238, 376, 587, 931, 1458, 2317, 3640, 5794, 9124, 14545, 22951, 36631, 57904, 92512, 146461, 234205, 371281, 594169, 943045, 1510192, 2399460, 3844787, 6114555, 9802895, 15603339, 25027296
OFFSET
0,4
COMMENTS
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
LINKS
Emanuele Munarini, Mar 05 2007, Table of n, a(n) for n = 0..100
Brian Kent, Sarah Racz, and Sanjit Shashi, Scrambling in quantum cellular automata, arXiv:2301.07722 [quant-ph], 2023.
E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
FORMULA
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
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
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
MAPLE
with(FormalPowerSeries): with(LREtools): # requires Maple 2022
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));
re:= FindRE(gf, x, a(n));
inits:= {seq(a(i-1)=[1, 1, 1, 2, 2, 3, 5, 7, 11, 17, 26, 40, 63, 97, 153][i], i=1..14)};
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;
minre:= MinimalRecurrence(re, a(n), inits); minrm:= MinimalRecurrence(rm, a(n), inits); # shows that Mathar's recurrence is equivalent
f:= REtoproc(re, a(n), inits); seq(f(n), n=0..40); # Georg Fischer, Oct 22 2022
MATHEMATICA
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]);
Table[SeriesCoefficient[gf[x], {x, 0, n}], {n, 0, 40}] (* Hugo Pfoertner, Oct 22 2022 *)
CROSSREFS
Sequence in context: A240487 A066889 A214040 * A125189 A226498 A196375
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 19 2003
EXTENSIONS
More terms from Emanuele Munarini, Mar 05 2007
STATUS
approved