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%I #7 Sep 08 2022 08:45:52
%S 1,1,1,1,0,1,1,0,0,1,1,-1,-1,-1,1,1,-2,-3,-3,-2,1,1,-4,-6,-7,-6,-4,1,
%T 1,-7,-11,-13,-13,-11,-7,1,1,-12,-19,-23,-24,-23,-19,-12,1,1,-20,-32,
%U -39,-42,-42,-39,-32,-20,1,1,-33,-53,-65,-71,-73,-71,-65,-53,-33,1
%N Triangle, T(n, k) = Fibonacci(k+1) + Fibonacci(n-k+1) - Fibonacci(n+1), read by rows.
%C Row sums are: {1, 2, 2, 2, -1, -8, -25, -60, -130, -264, -515,...}.
%C This sequence is a particular case of a symmetrical triangular sequence dependent upon a recurrence. The triangle is given by T(n, k, q) = b(k+1, q) + b(n-k+1, q) - b(n+1, q) where b(n, q) satisfies the recurrence b(n, q) = b(n-1, q) + q*b(n-2, 1). This sequence is for q=1. - _G. C. Greubel_, Nov 23 2019
%H G. C. Greubel, <a href="/A176259/b176259.txt">Rows n = 0..100 of triangle, flattened</a>
%F From _G. C. Greubel_, Nov 23 2019: (Start)
%F T(n, k) = Fibonacci(k+1) + Fibonacci(n-k+1) - Fibonacci(n+1).
%F Sum_{k=0..n} T(n,k) = Fibonacci(n+4) - n*Fibonacci(n+1) - 2 (row sums). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 0, 1;
%e 1, 0, 0, 1;
%e 1, -1, -1, -1, 1;
%e 1, -2, -3, -3, -2, 1;
%e 1, -4, -6, -7, -6, -4, 1;
%e 1, -7, -11, -13, -13, -11, -7, 1;
%e 1, -12, -19, -23, -24, -23, -19, -12, 1;
%e 1, -20, -32, -39, -42, -42, -39, -32, -20, 1;
%e 1, -33, -53, -65, -71, -73, -71, -65, -53, -33, 1;
%p with(combinat); f:=fibonacci; seq(seq( f(k+1) + f(n-k+1) - f(n+1), k=0..n), n = 0..12); # _G. C. Greubel_, Nov 23 2019
%t (* q = 0..10 *)
%t b[n_, q_]:= b[n, q]= If[n<2, n, b[n-1, q] + q*b[n-2, q]];
%t T[n_, k_, q_]:= T[n, k, q]= b[k+1, q] + b[n-k+1, q] - b[n+1, q];
%t Table[Flatten[Table[T[n, k, q], {n, 0, 12}, {k, 0, n}]], {q, 0, 10}] (* modified by _G. C. Greubel_, Nov 23 2019 *)
%t (* Second program *)T[n_, k_]= Fibonacci[k+1] +Fibonacci[n-k+1] -Fibonacci[n+1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 23 2019 *)
%o (PARI) T(n,k) = my(f=fibonacci); f(k+1) + f(n-k+1) - f(n+1); \\ _G. C. Greubel_, Nov 23 2019
%o (Magma) F:=Fibonacci; [F(k+1) +F(n-k+1) -F(n+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 23 2019
%o (Sage) f=fibonacci; [[f(k+1) + f(n-k+1) - f(n+1) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 23 2019
%o (GAP) F:=Fibonacci;; Flat(List([0..12], n-> List([0..n], k-> F(k+1) + F(n-k+1) - F(n+1) ))); # _G. C. Greubel_, Nov 23 2019
%Y Cf. A000045.
%K sign,tabl
%O 0,17
%A _Roger L. Bagula_, Apr 13 2010
%E Edited by _G. C. Greubel_, Nov 23 2019
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