cp's OEIS Frontend

A176481 Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 2, where b(n) = A001333(n).

%I A176481 #19 Sep 08 2022 08:45:52
%S A176481 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 A176481 1,141,197,217,217,197,141,1,1,339,477,531,545,531,477,339,1,1,817,
%U A176481 1153,1289,1337,1337,1289,1153,817,1,1,1971,2785,3119,3249,3283,3249,3119,2785,1971,1
%N A176481 Triangle, read by rows, defined by T(n, k) = b(n) - b(k) - b(n-k) + 2, where b(n) = A001333(n).
%C A176481 Row sums are: {1, 2, 5, 12, 37, 118, 369, 1112, 3241, 9194, 25533, ...}.
%H A176481 G. C. Greubel, <a href="/A176481/b176481.txt">Rows n=0..100 of triangle, flattened</a>
%H A176481 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 A176481 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 A176481 Triangle begins as:
%e A176481   1;
%e A176481   1,    1;
%e A176481   1,    3,    1;
%e A176481   1,    5,    5,    1;
%e A176481   1,   11,   13,   11,    1;
%e A176481   1,   25,   33,   33,   25,    1;
%e A176481   1,   59,   81,   87,   81,   59,    1;
%e A176481   1,  141,  197,  217,  217,  197,  141,    1;
%e A176481   1,  339,  477,  531,  545,  531,  477,  339,    1;
%e A176481   1,  817, 1153, 1289, 1337, 1337, 1289, 1153,  817,    1;
%e A176481   1, 1971, 2785, 3119, 3249, 3283, 3249, 3119, 2785, 1971, 1;
%t A176481 b[n_]:= LucasL[n, 2]/2; T[n_, k_]:= b[n] -b[k] -b[n-k] +2;
%t A176481 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 06 2019 *)
%o A176481 (PARI)
%o A176481 {b(n) = round(((1+sqrt(2))^n + (1-sqrt(2))^n)/2)};
%o A176481 {T(n,k) = b(n) -b(k) -b(n-k) +2};
%o A176481 for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 06 2019
%o A176481 (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 A176481 (Sage)
%o A176481 def b(m): return lucas_number2(m,2,-1)/2
%o A176481 def T(n, k): return b(n) - b(k) - b(n-k) + 2
%o A176481 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 06 2019
%Y A176481 Cf. A001333.
%K A176481 nonn,tabl
%O A176481 0,5
%A A176481 _Roger L. Bagula_, Apr 18 2010
%E A176481 Edited by _G. C. Greubel_, May 06 2019