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Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 2, where b(n) = A001333(n).
1

%I #19 Sep 08 2022 08:45:52

%S 1,1,1,1,3,1,1,5,5,1,1,11,13,11,1,1,25,33,33,25,1,1,59,81,87,81,59,1,

%T 1,141,197,217,217,197,141,1,1,339,477,531,545,531,477,339,1,1,817,

%U 1153,1289,1337,1337,1289,1153,817,1,1,1971,2785,3119,3249,3283,3249,3119,2785,1971,1

%N Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 2, where b(n) = A001333(n).

%C Row sums are: {1, 2, 5, 12, 37, 118, 369, 1112, 3241, 9194, 25533, ...}.

%H G. C. Greubel, <a href="/A176481/b176481.txt">Rows n=0..100 of triangle, flattened</a>

%H B. Adamczewski, Ch. Frougny, A. Siegel and W. Steiner, <a href="https://arxiv.org/abs/0907.0206">Rational numbers with purely periodic beta-expansion</a>, arXiv:0907.0206 [math.NT], 2009-2010; Bull. Lond. Math. Soc. 42:3 (2010), pp. 538-552.

%F Let b(n) = ((1+sqrt(2))^n + (1-sqrt(2))^n)/2 = A001333(n), then T(n, k) = b(n) - b(k) - b(n-k) + 2.

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 5, 5, 1;

%e 1, 11, 13, 11, 1;

%e 1, 25, 33, 33, 25, 1;

%e 1, 59, 81, 87, 81, 59, 1;

%e 1, 141, 197, 217, 217, 197, 141, 1;

%e 1, 339, 477, 531, 545, 531, 477, 339, 1;

%e 1, 817, 1153, 1289, 1337, 1337, 1289, 1153, 817, 1;

%e 1, 1971, 2785, 3119, 3249, 3283, 3249, 3119, 2785, 1971, 1;

%t b[n_]:= LucasL[n, 2]/2; T[n_, k_]:= b[n] -b[k] -b[n-k] +2;

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 06 2019 *)

%o (PARI)

%o {b(n) = round(((1+sqrt(2))^n + (1-sqrt(2))^n)/2)};

%o {T(n,k) = b(n) -b(k) -b(n-k) +2};

%o for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 06 2019

%o (Magma) b:= func< n| Round(((1+Sqrt(2))^n + (1-Sqrt(2))^n)/2) >; [[b(n)-b(k)-b(n-k)+2: k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, May 06 2019

%o (Sage)

%o def b(m): return lucas_number2(m,2,-1)/2

%o def T(n, k): return b(n) - b(k) - b(n-k) + 2

%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 06 2019

%Y Cf. A001333.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Apr 18 2010

%E Edited by _G. C. Greubel_, May 06 2019