Iterated Partial Sums of the k-Fibonacci Sequences
Round 1
Reviewer 1 Report
In remark 1, {αn}n>=0 should be {αk,n}n>=0 (twice)
Comments for author File: 
Comments.pdf
Author Response
In remark 1, {αn}n>=0 has been changed to {αk,n}n>=0 (twice) as suggested. Thank you
Reviewer 2 Report
See the attachment for the report.
Comments for author File: Comments.pdf
Author Response
Reviewer: 1. The statement "or k = 1 the respective sequences are" in line 49 should before the Table 3.
Answer: The statement appears now before the Table 2, as suggested. Thank you.
Reviewer: 2. The example in line 52 should be written as the form of the definition of the iterated partial sums of the k-Fibonacci numbers, that is $S_{2.5}(4) = S_{2,1}(3)+S_{2,2}(3)+S_{2,3}(3)+S_{2,4}(3)+S_{2,5}(3)" should be $S^{4)}_{2,5} = S^{3)}_{2,1} + S^{3)}_{2,2} + S^{3)}_{2,3} + S^{3)}_{2,4} + S^{3)}_{2,5}$";
Answer: The example in line 52 has been changed as suggested. Thank you.
Reviewer: 3. It is recommended to modify the theorem labelling. There are two Theorem 1 in the paper;
Answer: 3. Theorem labelling has been changed. Thank you.
Reviewer: 4. The sum "$\sum^{n+i}_{i=0}$" should be "$\sum^{n+a}_{i=0}$" in line 69;
Answer: 4. The sum has been changed as suggested. Thank you.
Reviewer: 5. Theorem 1 has given the relationship between the iterated partial sums $S^{r)}_{k,n}$ and the k-Fibonacci numbers $F_{k,j}, (0 \leq j \leq n)$. Theorems 3, 4, and 5 are equivalent to the combination of Theorem 1 and formula (3) to give the special cases of Theorem 1 in r = 2, 3, 4, respectively. Therefore, I don't consider Theorem 3, 4 and 5 as significant, and these can be presented as the corollaries of Theorem 1.
Answer: 5. Theorems 3, 4 and 5 are presented no as Corollaries as suggested. Thank you.
Reviewer 3 Report
Dear Authors,
Only I have two comments in the file in attachment.
Best wishes
Comments for author File: Comments.pdf
Author Response
Reviewer: 1. The sentence in line 49 should be written after Table 2
Answer: 1. That sentence appears now after Table 2 as suggested. Thank you.
Reviewer: 2. Theorem 2 should begin with "For $r=2$".
Answer: 2. "For $r=2$" has been included as suggested. Thank you.
Reviewer 4 Report
Comments for author File: Comments.pdf
Author Response
Reviewer: Observation 0. AngelPlaza – pease insert space between the names.
Answer: Space between the names has been inserted. Thank you.
Reviewer: Observation 1. Please give supplementary information about 4TLE partition of triangles.
Answer: We have included in the introduction section a definition of the 4TLE partition of triangles and two new references on the topic. Thank you.
