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A137712 Triangle read by rows: T(n,k) = T(n-1, k-1) - T(n-k, k-1); with left border = the Fibonacci sequence.

%I A137712 #15 Apr 02 2022 10:11:56
%S A137712 1,1,1,2,0,1,3,1,0,1,5,1,0,0,1,8,2,1,0,0,1,13,3,1,0,0,0,1,21,5,2,1,0,
%T A137712 0,0,1,34,8,3,2,0,0,0,0,1,55,13,5,2,2,0,0,0,0,1,89,21,8,4,2,1,0,0,0,0,
%U A137712 1,144,34,13,6,4,2,1,0,0,0,0,1,233,55,21,10,5,4,1,1,0,0,0,0,1
%N A137712 Triangle read by rows: T(n,k) = T(n-1, k-1) - T(n-k, k-1); with left border = the Fibonacci sequence.
%C A137712 Row sums = A137713: (1, 2, 3, 5, 7, 12, 18, 30, 48, 78, 126, ...).
%C A137712 A137710 is the analogous triangle with left border = (1, 2, 4, 8, 16, 32, ...).
%H A137712 Robert Israel, <a href="/A137712/b137712.txt">Table of n, a(n) for n = 1..10011</a>(rows 1 to 141, flattened)
%F A137712 T(n,k) = T(n-1, k-1) - T(n-k, k-1), given left border = (1, 1, 2, 3, 5, 8, 13, ...).
%F A137712 Here T(n,k) = T(n-1,k-1) if n-k < k-1. - _Robert Israel_, Aug 20 2018
%e A137712 First few rows of the triangle:
%e A137712     1;
%e A137712     1,  1;
%e A137712     2,  0,  1;
%e A137712     3,  1,  0,  1;
%e A137712     5,  1,  0,  0, 1;
%e A137712     8,  2,  1,  0, 0, 1;
%e A137712    13,  3,  1,  0, 0, 0, 1;
%e A137712    21,  5,  2,  1, 0, 0, 0, 1;
%e A137712    34,  8,  3,  2, 0, 0, 0, 0, 1;
%e A137712    55, 13,  5,  2, 2, 0, 0, 0, 0, 1;
%e A137712    89, 21,  8,  4, 2, 1, 0, 0, 0, 0, 1;
%e A137712   144, 34, 13,  6, 4, 2, 1, 0, 0, 0, 0, 1;
%e A137712   233, 55, 21, 10, 5, 4, 1, 1, 0, 0, 0, 0, 1;
%e A137712   377, 89, 34, 16, 8, 5, 4, 1, 1, 0, 0, 0, 0, 1;
%e A137712   ...
%p A137712 for n from 1 to 20 do
%p A137712   T[n,1]:= combinat:-fibonacci(n);
%p A137712   for k from 2 to n do
%p A137712     if n >= 2*k-1 then T[n,k]:= T[n-1,k-1] - T[n-k,k-1]
%p A137712     else T[n,k]:= T[n-1,k-1]
%p A137712     fi
%p A137712   od:
%p A137712 od:
%p A137712 seq(seq(T[n,k],k=1..n),n=1..20); # _Robert Israel_, Aug 20 2018
%Y A137712 Cf. A137713, A137710.
%K A137712 nonn,tabl
%O A137712 1,4
%A A137712 _Gary W. Adamson_, Feb 08 2008
%E A137712 Corrected by _Robert Israel_, Aug 20 2018