zeta_recurrence.tex

Paper II: Refinement Towers, Weighted Transfer, and Integral Separation of Operator Packets

zeta_recurrence.tex

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round]
2 \draw[fill=black!8,draw=black!45,line width=.35pt,] (0.00000,1.75000) rectangle ++(0.87000,0.65000);
3 \node[] at (0.43500,2.07500) {$0$};
4 \draw[fill=black!20,draw=black!45,line width=.35pt,] (0.00000,1.00000) rectangle ++(0.87000,0.65000);
5 \node[] at (0.43500,1.32500) {$0$};
6 \draw[fill=black!8,draw=black!45,line width=.35pt,] (0.87000,1.75000) rectangle ++(0.87000,0.65000);
7 \node[] at (1.30500,2.07500) {$1$};
8 \draw[fill=black!20,draw=black!45,line width=.35pt,] (0.87000,1.00000) rectangle ++(0.87000,0.65000);
9 \node[] at (1.30500,1.32500) {$0$};
10 \draw[fill=black!8,draw=black!45,line width=.35pt,] (1.74000,1.75000) rectangle ++(0.87000,0.65000);
11 \node[] at (2.17500,2.07500) {$2$};
12 \draw[fill=black!20,draw=black!45,line width=.35pt,] (1.74000,1.00000) rectangle ++(0.87000,0.65000);
13 \node[] at (2.17500,1.32500) {$2$};
14 \draw[fill=black!8,draw=black!45,line width=.35pt,] (2.61000,1.75000) rectangle ++(0.87000,0.65000);
15 \node[] at (3.04500,2.07500) {$3$};
16 \draw[fill=black!20,draw=black!45,line width=.35pt,] (2.61000,1.00000) rectangle ++(0.87000,0.65000);
17 \node[] at (3.04500,1.32500) {$2$};
18 \draw[fill=black!8,draw=black!45,line width=.35pt,] (3.48000,1.75000) rectangle ++(0.87000,0.65000);
19 \node[] at (3.91500,2.07500) {$4$};
20 \draw[fill=black!20,draw=black!45,line width=.35pt,] (3.48000,1.00000) rectangle ++(0.87000,0.65000);
21 \node[] at (3.91500,1.32500) {$2$};
22 \draw[fill=black!8,draw=black!45,line width=.35pt,] (4.35000,1.75000) rectangle ++(0.87000,0.65000);
23 \node[] at (4.78500,2.07500) {$5$};
24 \draw[fill=black!20,draw=black!45,line width=.35pt,] (4.35000,1.00000) rectangle ++(0.87000,0.65000);
25 \node[] at (4.78500,1.32500) {$6$};
26 \draw[fill=black!8,draw=black!45,line width=.35pt,] (5.22000,1.75000) rectangle ++(0.87000,0.65000);
27 \node[] at (5.65500,2.07500) {$6$};
28 \draw[fill=black!20,draw=black!45,line width=.35pt,] (5.22000,1.00000) rectangle ++(0.87000,0.65000);
29 \node[] at (5.65500,1.32500) {$10$};
30 \draw[fill=black!8,draw=black!45,line width=.35pt,] (6.09000,1.75000) rectangle ++(0.87000,0.65000);
31 \node[] at (6.52500,2.07500) {$7$};
32 \draw[fill=black!20,draw=black!45,line width=.35pt,] (6.09000,1.00000) rectangle ++(0.87000,0.65000);
33 \node[] at (6.52500,1.32500) {$14$};
34 \draw[fill=black!8,draw=black!45,line width=.35pt,] (6.96000,1.75000) rectangle ++(0.87000,0.65000);
35 \node[] at (7.39500,2.07500) {$8$};
36 \draw[fill=black!20,draw=black!45,line width=.35pt,] (6.96000,1.00000) rectangle ++(0.87000,0.65000);
37 \node[] at (7.39500,1.32500) {$26$};
38 \draw[fill=black!8,draw=black!45,line width=.35pt,] (7.83000,1.75000) rectangle ++(0.87000,0.65000);
39 \node[] at (8.26500,2.07500) {$9$};
40 \draw[fill=black!20,draw=black!45,line width=.35pt,] (7.83000,1.00000) rectangle ++(0.87000,0.65000);
41 \node[] at (8.26500,1.32500) {$46$};
42 \draw[fill=black!8,draw=black!45,line width=.35pt,] (8.70000,1.75000) rectangle ++(0.87000,0.65000);
43 \node[] at (9.13500,2.07500) {$10$};
44 \draw[fill=black!20,draw=black!45,line width=.35pt,] (8.70000,1.00000) rectangle ++(0.87000,0.65000);
45 \node[] at (9.13500,1.32500) {$74$};
46 \draw[fill=black!8,draw=black!45,line width=.35pt,] (9.57000,1.75000) rectangle ++(0.87000,0.65000);
47 \node[] at (10.00500,2.07500) {$11$};
48 \draw[fill=black!20,draw=black!45,line width=.35pt,] (9.57000,1.00000) rectangle ++(0.87000,0.65000);
49 \node[] at (10.00500,1.32500) {$126$};
50 \draw[fill=black!8,draw=black!45,line width=.35pt,] (10.44000,1.75000) rectangle ++(0.87000,0.65000);
51 \node[] at (10.87500,2.07500) {$12$};
52 \draw[fill=black!20,draw=black!45,line width=.35pt,] (10.44000,1.00000) rectangle ++(0.87000,0.65000);
53 \node[] at (10.87500,1.32500) {$218$};
54 \node[anchor=east] at (-0.15000,2.07500) {$n$};
55 \node[anchor=east] at (-0.15000,1.32500) {$a_n$};
56 \node[] at (5.65000,3.10000) {$Z_{0,1}(t)=\dfrac{2t^2}{1-t-2t^3}$};
57 \node[] at (5.65000,0.00000) {$a_8=a_7+2a_5=14+2\cdot6=26$};
58 \end{tikzpicture}

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