OFFSET
1,6
LINKS
G. C. Greubel, Rows n = 1..100 of the triangle, flattened
FORMULA
By columns, Fibonacci(k) interspersed with alternate zeros in every column, k=1,2,3,...
Sum_{k=1..n} T(n, k) = A052952(n-1) (row sums).
From G. C. Greubel, Mar 17 2024: (Start)
T(n, k) = (1/2)*(1 + (-1)^(n+k))*Fibonacci(k).
T(n, n) = A000045(n).
T(2*n-1, n) = (1/2)*(1-(-1)^n)*A000045(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A052952(n-1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1 - (-1)^n)*(Fibonacci((n+ 5)/2) - 1).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n) * A355020(floor((n-1)/2)). (End)
EXAMPLE
First few rows of the triangle are:
1;
0, 1;
1, 0, 2;
0, 1, 0, 3;
1, 0, 2, 0, 5;
0, 1, 0, 3, 0, 8;
1, 0, 2, 0, 5, 0, 13;
0, 1, 0, 3, 0, 8, 0, 21;
1, 0, 2, 0, 5, 0, 13, 0, 34;
0, 1, 0, 3, 0, 8, 0, 21, 0, 55;
1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89;
...
MATHEMATICA
Table[Fibonacci[k]*Mod[n-k+1, 2], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Mar 17 2024 *)
PROG
(Magma) [((n+k+1) mod 2)*Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024
(SageMath) flatten([[((n-k+1)%2)*fibonacci(k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 17 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Mar 14 2007
EXTENSIONS
a(6) corrected and more terms from Georg Fischer, May 30 2023
STATUS
approved