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A176259
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Triangle, T(n, k) = Fibonacci(k+1) + Fibonacci(n-k+1) - Fibonacci(n+1), read by rows.
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1
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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, 1, -7, -11, -13, -13, -11, -7, 1, 1, -12, -19, -23, -24, -23, -19, -12, 1, 1, -20, -32, -39, -42, -42, -39, -32, -20, 1, 1, -33, -53, -65, -71, -73, -71, -65, -53, -33, 1
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OFFSET
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0,17
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COMMENTS
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Row sums are: {1, 2, 2, 2, -1, -8, -25, -60, -130, -264, -515,...}.
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
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LINKS
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FORMULA
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T(n, k) = Fibonacci(k+1) + Fibonacci(n-k+1) - Fibonacci(n+1).
Sum_{k=0..n} T(n,k) = Fibonacci(n+4) - n*Fibonacci(n+1) - 2 (row sums). (End)
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EXAMPLE
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Triangle begins as:
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;
1, -7, -11, -13, -13, -11, -7, 1;
1, -12, -19, -23, -24, -23, -19, -12, 1;
1, -20, -32, -39, -42, -42, -39, -32, -20, 1;
1, -33, -53, -65, -71, -73, -71, -65, -53, -33, 1;
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MAPLE
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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
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MATHEMATICA
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(* q = 0..10 *)
b[n_, q_]:= b[n, q]= If[n<2, n, b[n-1, q] + q*b[n-2, q]];
T[n_, k_, q_]:= T[n, k, q]= b[k+1, q] + b[n-k+1, q] - b[n+1, q];
Table[Flatten[Table[T[n, k, q], {n, 0, 12}, {k, 0, n}]], {q, 0, 10}] (* modified by G. C. Greubel, Nov 23 2019 *)
(* 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 *)
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PROG
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(PARI) T(n, k) = my(f=fibonacci); f(k+1) + f(n-k+1) - f(n+1); \\ G. C. Greubel, Nov 23 2019
(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
(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
(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
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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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