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 A173568 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. 1

%I

%S 6,19,19,68,56,68,261,211,211,261,1030,1044,654,1044,1030,4103,5819,

%T 2993,2993,5819,4103,16392,33560,19102,9840,19102,33560,16392,65545,

%U 195147,137571,52989,52989,137571,195147,65545,262154,1136836,1019606,412700,182270,412700,1019606,1136836,262154

%N 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.

%H G. C. Greubel, <a href="/A173568/b173568.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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.

%e Triangle begins as:

%e 6;

%e 19, 19;

%e 68, 56, 68;

%e 261, 211, 211, 261;

%e 1030, 1044, 654, 1044, 1030;

%e 4103, 5819, 2993, 2993, 5819, 4103;

%e 16392, 33560, 19102, 9840, 19102, 33560, 16392;

%e 65545, 195147, 137571, 52989, 52989, 137571, 195147, 65545;

%e 262154, 1136836, 1019606, 412700, 182270, 412700, 1019606, 1136836, 262154;

%t f[n_, k_]:= Round[((1+Sqrt[k])^(2*n+1) - (1-Sqrt[k])^(2*n+1))/(2*Sqrt[k])] - 1;

%t T[n_, k_]:= f[k, n-k+1] + f[n-k+1, k];

%t Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by _G. C. Greubel_, Apr 26 2021 *)

%o (Sage)

%o @CachedFunction

%o def f(n,k): return round(((1+sqrt(k))^(2*n+1) -(1-sqrt(k))^(2*n+1))/(2*sqrt(k))) -1

%o def T(n,k): return f(k,n-k+1) + f(n-k+1,k)

%o flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 26 2021

%K nonn,tabl,less

%O 1,1

%A _Roger L. Bagula_, Feb 22 2010

%E Edited by _G. C. Greubel_, Apr 26 2021

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Last modified December 7 08:24 EST 2022. Contains 358649 sequences. (Running on oeis4.)