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A173568
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Triangle T(n, k) = f(k, n-k+1) + f(n-k+1, k), where f(n, k) = round( ((1+sqrt(k))^(2*n+1) - (1-sqrt(k))^(2*n+1))/(2*sqrt(k))) - 1, read by rows.
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1
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6, 19, 19, 68, 56, 68, 261, 211, 211, 261, 1030, 1044, 654, 1044, 1030, 4103, 5819, 2993, 2993, 5819, 4103, 16392, 33560, 19102, 9840, 19102, 33560, 16392, 65545, 195147, 137571, 52989, 52989, 137571, 195147, 65545, 262154, 1136836, 1019606, 412700, 182270, 412700, 1019606, 1136836, 262154
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OFFSET
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1,1
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LINKS
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FORMULA
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T(n, k) = f(k, n-k+1) + f(n-k+1, k), where f(n, k) = round( ((1+sqrt(k))^(2*n+1) - (1-sqrt(k))^(2*n+1))/(2*sqrt(k)) ) - 1.
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EXAMPLE
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Triangle begins as:
6;
19, 19;
68, 56, 68;
261, 211, 211, 261;
1030, 1044, 654, 1044, 1030;
4103, 5819, 2993, 2993, 5819, 4103;
16392, 33560, 19102, 9840, 19102, 33560, 16392;
65545, 195147, 137571, 52989, 52989, 137571, 195147, 65545;
262154, 1136836, 1019606, 412700, 182270, 412700, 1019606, 1136836, 262154;
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MATHEMATICA
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f[n_, k_]:= Round[((1+Sqrt[k])^(2*n+1) - (1-Sqrt[k])^(2*n+1))/(2*Sqrt[k])] - 1;
T[n_, k_]:= f[k, n-k+1] + f[n-k+1, k];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 26 2021 *)
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PROG
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(Sage)
@CachedFunction
def f(n, k): return round(((1+sqrt(k))^(2*n+1) -(1-sqrt(k))^(2*n+1))/(2*sqrt(k))) -1
def T(n, k): return f(k, n-k+1) + f(n-k+1, k)
flatten([[T(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 26 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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