On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers
Abstract
:1. Introduction
- (a)
- For , it holds that
- (i)
- If , then either or
- (ii)
- If , then
- (b)
- For , it holds that
2. Auxiliary Results
3. The Proofs
3.1. The Proof of Theorem 1
3.1.1. The Case
3.1.2. The Case
3.2. The Proof of Theorem 2
t[n_, r_] := t[n, r] = Which[n == 0, 0, 0 < n < r, 1, n >= r, Sum[t[n - i, r], {i, 1, r}]];
Catch[Do[{ n, m, r,l}; If[t[m,r] == Sum[Factorial[n+i], {i,0,l}], Print[{n,m,r,l}]], {l,1,5}, {n, 4, 109},{r, 2, 1638,2}, {m, r+1, 3276}]]returns as the only solution.
Catch[Do[{ n, m, r,l}; If[t[m,r] == Sum[Factorial[n+i], {i,0,l}], Print[{n,m,r,l}]], {l,1,5}, {n, 1, 3}, {r, 2, 1638,2}, {m, 2r+1, 3276}]]returns and as solutions. This finishes the proof.
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
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Trojovská , E.; Trojovský, P. On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers. Mathematics 2021, 9, 962. https://doi.org/10.3390/math9090962
Trojovská E, Trojovský P. On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers. Mathematics. 2021; 9(9):962. https://doi.org/10.3390/math9090962
Chicago/Turabian StyleTrojovská , Eva, and Pavel Trojovský. 2021. "On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers" Mathematics 9, no. 9: 962. https://doi.org/10.3390/math9090962