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Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3*n-2) = T(n,3*n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).
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%I #8 Sep 14 2020 12:18:00

%S 1,1,2,2,2,2,2,3,3,4,4,4,4,3,3,4,4,6,7,8,8,8,7,6,4,4,5,5,8,10,13,15,

%T 16,16,15,13,10,8,5,5,6,6,10,13,18,23,28,31,32,31,28,23,18,13,10,6,6,

%U 7,7,12,16,23,31,41,51,59,63,63,59,51,41,31,23,16,12,7,7

%N Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3*n-2) = T(n,3*n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

%C The number of terms in row n is 3*n-1 = A016789(n-1).

%C The sum of row n is equal to 2*A094002(n-1) = 2*A188589(n).

%C Fibonacci(n) = T(n+k,n) - T(n+k-1,n) for n >= 1, k = 1,2,3,...

%C The elements b(k) of the main diagonal, superdiagonal 1 and all subdiagonals have the recursive formula: b(k) = 2*b(k-1) + b(k-2) - 2*b(k-3) - b(k-4) for k > 4.

%F T(n,k) = T(n,3*k-n) for 1 <= k <= 3*n-1.

%F T(n,k) = Sum_{u=2*(n-k)+3..2*n-k+1} ceiling(u/2)*A065941(k-2,u-2*(n-k)-3) for n >= 3, 3 <= k <= n.

%F T(n,k) = Sum_{m1=1..k-n} A208354(m1)*binomial(n-m1-1, k-n-m1) + Sum_{m2=1..2*n-k} A208354(m2)*binomial(n-m2-1, 2*n-k-m2) for n >= 2, n+1 <= k <= 2*n-1.

%F T(n,k) = Sum_{u=2*(k-2*n)+3..k-n+1} ceiling(u/2)*A065941(3*n-k-2,u-2*(k-2*n)-3) for n>= 3, 2*n <= k <= 3*(n-1).

%F T(n,k) = A208354(k) + (n-k)*Fibonacci(k) for n >= 3, 3 <= k <= n.

%F T(n,k) = A029907(k-1) + (n-k+1)*Fibonacci(k) for n >= 2, 3 <= k <= n+1.

%e Triangle begins:

%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20...

%e 1 1 1

%e 2 2 2 2 2 2

%e 3 3 3 4 4 4 4 3 3

%e 4 4 4 6 7 8 8 8 7 6 4 4

%e 5 5 5 8 10 13 15 16 16 15 13 10 8 5 5

%e 6 6 6 10 13 18 23 28 31 32 31 28 23 18 13 10 6 6

%e 7 7 7 12 16 23 31 41 51 59 63 63 59 51 41 31 23 16 12 7 7

%e ...

%Y Superdiagonal 1 is A029907 for n >= 1.

%Y The main diagonal is A208354 for n >= 1.

%Y Subdiagonal 1 is A102702(n-1) for n >= 1.

%Y Subdiagonal 2 is A206268(n+2) for n >= 1 (conjectured).

%Y Subdiagonal 3 is A191830(n+3) for n >= 1.

%Y Cf. A007318, A051597, A228196.

%K nonn,tabf

%O 1,3

%A _Lechoslaw Ratajczak_, Jul 04 2020