<?xml version='1.0' encoding='UTF-8'?>
<doi_batch version="5.4.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.crossref.org/schema/5.4.0" xsi:schemaLocation="http://www.crossref.org/schema/5.4.0 https://www.crossref.org/schemas/crossref5.4.0.xsd" xmlns:jats="http://www.ncbi.nlm.nih.gov/JATS1" xmlns:fr="http://www.crossref.org/fundref.xsd" xmlns:ai="http://www.crossref.org/AccessIndicators.xsd" xmlns:rel="http://www.crossref.org/relations.xsd" xmlns:mml="http://www.w3.org/1998/Math/MathML">
  <head>
    <doi_batch_id>NONE</doi_batch_id>
    <timestamp>20260624103630184</timestamp>
    <depositor>
      <depositor_name>wseas/wseas</depositor_name>
      <email_address>content-registration-form@crossref.org</email_address>
    </depositor>
    <registrant>content-registration-form</registrant>
  </head>
  <body>
    <journal>
      <journal_metadata>
        <full_title>PROOF</full_title>
        <issn media_type="print">2944-9162</issn>
        <issn media_type="electronic">2732-9941</issn>
      </journal_metadata>
      <journal_article>
        <titles>
          <title>Relations Among Hn, Tc and Fibonacci Sequences</title>
        </titles>
        <contributors>
          <person_name sequence="first" contributor_role="author">
            <given_name>Hasan</given_name>
            <surname>Keleş</surname>
            <affiliations>
              <institution>
                <institution_name>Department of Mathematics Karadeniz Technical University Campus of Kanuni, Ortahisar, 61080 Trabzon TÜRKÍYE</institution_name>
              </institution>
            </affiliations>
          </person_name>
        </contributors>
        <publication_date media_type="print">
          <month>06</month>
          <day>24</day>
          <year>2026</year>
        </publication_date>
        <publication_date media_type="online">
          <month>06</month>
          <day>24</day>
          <year>2026</year>
        </publication_date>
        <pages>
          <first_page>8</first_page>
        </pages>
        <publisher_item>
          <item_number item_number_type="article_number">2</item_number>
        </publisher_item>
        <ai:program name="AccessIndicators">
          <ai:license_ref>https://creativecommons.org/licenses/by/4.0/deed.en_US</ai:license_ref>
        </ai:program>
        <doi_data>
          <doi>10.37394/232020.2026.6.2</doi>
          <resource>https://wseas.com/journals/proof/2026/a04proof-002(2026).pdf</resource>
        </doi_data>
        <citation_list>
          <citation key="ref0">
            <unstructured_citation>Shen, N., Jiang, Z., Li, J. On Explicit Determinants of the RFMLR and RLMFL Circulant Matrices Involving Certain Famous Numbers, WSEAS Transactions on Mathematics, 12, 2013. h t t p s : //wseas.com/journals/article s.php?id=5850 (Accessed: 2026-01-27).</unstructured_citation>
          </citation>
          <citation key="ref1">
            <unstructured_citation>Jiang, Z., Li, J., Shen, N. On the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers, WSEAS Transactions on Mathematics, 12, 2013. https://wseas. com/journals/articles.php?id=5 822 (Accessed: 2026-01-27).</unstructured_citation>
          </citation>
          <citation key="ref2">
            <unstructured_citation>Deger, A. H., Be¸senk, M., Güler, B. O. On ˘ suborbital graphs and related continued fractions, Applied Mathematics and Computation, 218, 2011. h t t p s : / / d o i.org/10.1016/j.amc.2011.03.065</unstructured_citation>
          </citation>
          <citation key="ref3">
            <unstructured_citation>Güler, B. O., Be¸senk, M., Deger, A. H., Kader, ˘ S. Elliptic elements and circuits in suborbital graphs, Hacet. J. Math Stat., 40(2), 2011, 203-210. https://dergipark.org.tr /tr/download/article-file/86607 (Accessed: 2026-01-27).</unstructured_citation>
          </citation>
          <citation key="ref4">
            <unstructured_citation>Güler, B. O., Kör, T., ¸Sanlı, Z. Solution to some congruence equations via suborbital graphs, SpringerPlus, 5, 2016, 1327. https://doi. org/10.1186/s40064-016-3016-5</unstructured_citation>
          </citation>
          <citation key="ref5">
            <unstructured_citation>Jones, G. A., Singerman, D., Wicks, K. The Modular Group and Generalized Farey Graphs, London Math. Soc. Lecture Note Series, CUP, Cambridge, 160, 1991, 316-338. https:// scispace.com/pdf/solutions-to-s ome-congruence-equations-via-s uborbital-graphs-ypll4hxyzc.pdf (Accessed: 2026-01-27).</unstructured_citation>
          </citation>
          <citation key="ref6">
            <unstructured_citation>Lee, G.Y., Kim, J.S., Cho, S.H. Some combinatorial identities via Fibonacci numbers, Discrete Applied Mathematics, 130(3), 2003, 527-534. https://doi.org/10.1016/ S0166-218X(03)00331-7</unstructured_citation>
          </citation>
          <citation key="ref7">
            <unstructured_citation>Koshy, T. Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, 2001.http s://doi.org/10.1002/9781118033 067</unstructured_citation>
          </citation>
          <citation key="ref8">
            <unstructured_citation>Falcón, S., Plaza, A. The k-Fibonacci sequence and Pascal 2-triangle, Chaos, Solitons and Fractals, 33(1), 2007, 38-49. https://doi. org/10.1016/j.chaos.2006.10.022</unstructured_citation>
          </citation>
          <citation key="ref9">
            <unstructured_citation>Kader, S., Güler, B. O. On suborbital graphs for extended modular group ! ˆ , Graphs and Combinatorics, 29, 2013, 1813-1825. https: //doi.org/10.1007/s00373-012-1 226-3</unstructured_citation>
          </citation>
          <citation key="ref10">
            <unstructured_citation>Köroglu, T., Güler, B. O., ¸Sanl, Z. Suborbital ˘ graphs for the Atkin Lehner group, Turk. J. of Math., 41, 2017, 235-243. https://doi.or g/10.3906/mat-1602-10</unstructured_citation>
          </citation>
          <citation key="ref11">
            <unstructured_citation>Kele¸s, H. On the Digits of Numbers in the System Logic B3, J. Appl. &amp; Pure Math., 6(1-2), 2024, 97-103. https://doi.org/10.230 91/japm.2024.097</unstructured_citation>
          </citation>
          <citation key="ref12">
            <unstructured_citation>Sanlı, Z., Köroglu, T. Some Group Actions and ˘ Fibonacci Numbers, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 71(1), 2022, 273-284. https://doi.org/10.31801/cfsua smas.939096</unstructured_citation>
          </citation>
          <citation key="ref13">
            <unstructured_citation>Negri, M. An algebraic completeness proof for Kleene’s 3-valued logic, Bollettino dell’Unione Matematica Italiana, 5-B(2), 2002, 447-467. http://www.bdim.eu/item?id=BU MI_2002_8_5B_2_447_0&amp;fmt=pdf (Accessed: 2026-01-27).</unstructured_citation>
          </citation>
          <citation key="ref14">
            <unstructured_citation>Naish, L. A three-valued semantics for logic programmers, Theory and Practice of Logic Programming, 6(5), 2006, 509-538. https: //doi.org/10.1017/S14710684060 02742</unstructured_citation>
          </citation>
          <citation key="ref15">
            <unstructured_citation>McCall, S., Meyer, R. K. Pure three-valued ukasiewiczian implication, Journal of Symbolic Logic, 31(3), 1966, 399-405. https://doi. org/10.2307/2270455</unstructured_citation>
          </citation>
          <citation key="ref16">
            <unstructured_citation>Garbacz, P. Philosophical Motivations of Jan ukasiewicz’s Three-valued Logic, Roczniki Filozoficzne, 45(1), 1997/2020. https://bi bliotekanauki.pl/articles/1917 762 (Accessed: 2026-01-27).</unstructured_citation>
          </citation>
          <citation key="ref17">
            <unstructured_citation>Grant, J., Hunter, A. Semantic inconsistency measures using 3-valued logics, International Journal of Approximate Reasoning, 156, 2023, 38-60. https://doi.org/10.1016/j. ijar.2023.02.008</unstructured_citation>
          </citation>
          <citation key="ref18">
            <unstructured_citation>Grigolia, R. Three-Valued Gödel Logic With Constants and Involution for Application to R-Functions, ISSBG 2022 (Athena Publishing), 2023. https://doi.org/10.55060/s .atmps.231115.005</unstructured_citation>
          </citation>
          <citation key="ref19">
            <unstructured_citation>Rosenmann, A. A Multiple-Valued Logic Approach to the Design and Verification of Hardware Circuits, arXiv preprint, 2015. http s://doi.org/10.48550/arXiv.150 2.05748</unstructured_citation>
          </citation>
          <citation key="ref20">
            <unstructured_citation>Kele¸s, H., ¸Sahin, H. . On Some Logical Properties in B3, Journal of Applied Mathematics and Informatics, 43(2), 2025, 267-275. https://doi.org/10.14317 /JAMI.2025.267</unstructured_citation>
          </citation>
          <citation key="ref21">
            <unstructured_citation>Kele¸s, H., ¸Sahin, H. . On Conditional Propositions and Equivalence in Logical B3, International Journal of Applied Mathematics, Computational Science and Systems Engineering, 7(2), 2025, 19-26. https://doi.org/10.37394/23202 6.2025.7.2</unstructured_citation>
          </citation>
          <citation key="ref22">
            <unstructured_citation>Kele¸s, H. Spin Half-Adder in B3, Journal of Applied and Pure Mathematics, 5(3), 2023, 187-196. https://doi.org/10.23091 /japm.2023.187</unstructured_citation>
          </citation>
          <citation key="ref23">
            <unstructured_citation>Kele¸s, H. On The Basic Structure of Logic B3, in: slamolu E. (ed.), Advanced Research in Mathematics, Technology and Social Sciences, BIDGE Publications, Ankara, 2024, 4-20. ht tps://www.bidgeyayinlari.com.t r/yayinlar-2/ (Accessed: 2026-01-27).</unstructured_citation>
          </citation>
          <citation key="ref24">
            <unstructured_citation>Degirmen, N., Sa ˘ gır, B. A New ˘ D-Normed Banach Bicomplex BC-Module Derived Using Matrix of Hyperbolic Fibonacci Numbers, WSEAS Transactions on Mathematics, 24, 2025, 203-208. https://doi.org/10.3 7394/23206.2025.24.20</unstructured_citation>
          </citation>
          <citation key="ref25">
            <unstructured_citation>Özkan, E., Akku¸s, H. On Generalizations of Dickson k-Fibonacci Polynomials, WSEAS Transactions on Mathematics, 23, 2024, 856-862. https://doi.org/10.37394 /23206.2024.23.88</unstructured_citation>
          </citation>
          <citation key="ref26">
            <unstructured_citation>Nilsrakoo, W., Nilsrakoo, A. On the Integral Representations of the k-Fibonacci and k-Lucas Numbers, WSEAS Transactions on Mathematics, 23, 2024, 791-801. https://doi.org/10.37394/232 06.2024.23.82</unstructured_citation>
          </citation>
          <citation key="ref27">
            <unstructured_citation>Akku¸s, H., Özkan, E. Hyperbolic (s,t)-Fibonacci and (s,t)-Lucas Quaternions, Proof, 4, 2024, 97-105. https://doi.or g/10.37394/232020.2024.4.9</unstructured_citation>
          </citation>
          <citation key="ref28">
            <unstructured_citation>Potek, R. On One Series of the Reciprocals of the Product of two Fibonacci Numbers Whose Indices Differ by an Even Number, Equations, 4, 2024, 24-31. https://doi.org/10.3 7394/232021.2024.4.4</unstructured_citation>
          </citation>
          <citation key="ref29">
            <unstructured_citation>Kumari, M., Prasad, K., Kuloglu, B., Özkan, ˘ E. The k-Fibonacci Group and Periods of the k-step Fibonacci Sequences, WSEAS Transactions on Mathematics, 21, 2022, 838-843. https://doi.org/10.37394 /23206.2022.21.95</unstructured_citation>
          </citation>
          <citation key="ref30">
            <unstructured_citation>Sikhwal, O., Rastogi, A. Domination, Independence and Fibonacci Numbers in Graphs Containing Disjoint Cycles, International Journal of Computational and Applied Mathematics &amp; Computer Science, 2, 2022, 65-68. https://doi.org/10.3 7394/232028.2022.2.12</unstructured_citation>
          </citation>
          <citation key="ref31">
            <unstructured_citation>Beletsky, A. Generalized Galois and Fibonacci Matrices in Cryptographic Applications, WSEAS Transactions on Circuits and Systems, 21, 2022, 1-19. https://doi.org/10.3 7394/23201.2022.21.1</unstructured_citation>
          </citation>
          <citation key="ref32">
            <unstructured_citation>Yu, Y., Jiang, Z. On the Norms and Spreads of Fermat, Mersenne and Gaussian Fibonacci RFMLR Circulant Matrices, WSEAS Transactions on Mathematics, 15, 2016, 34-43. https://wseas.com/journals/mat hematics/2016/a085806-844.pdf (Accessed: 2026-01-27).</unstructured_citation>
          </citation>
          <citation key="ref33">
            <unstructured_citation>Ryoo, C. S. Phenomena and Properties of Roots of Bernoulli-Fibonacci Polynomials, J. Appl. &amp; Pure Math., 6(1-2), 2024, 47-54. https:// doi.org/10.23091/japm.2024.047</unstructured_citation>
          </citation>
          <citation key="ref34">
            <unstructured_citation>Kim, Y. R., Choi, J. E., Ryoo, C. S. Zeros of the Euler-Fibonacci Polynomials, Journal of Applied and Pure Mathematics, 6(3-4), 2024, 119-126. https://doi.org/10.23091 /japm.2024.119</unstructured_citation>
          </citation>
          <citation key="ref35">
            <unstructured_citation>Kim, A. Zeros of the Degenerate Bernoulli-Fibonacci Polynomials, J. Appl. &amp; Pure Math., 6(5-6), 2024, 231-240. https:// doi.org/10.23091/japm.2024.231</unstructured_citation>
          </citation>
          <citation key="ref36">
            <unstructured_citation>Verma, K. L. A Comprehensive Generalization of Classical Fibonacci Sequences, Binet Formula and Identities, J. Appl. &amp; Pure Math., 6(5-6), 2024, 283-299. https://doi.org/ 10.23091/japm.2024.283</unstructured_citation>
          </citation>
          <citation key="ref37">
            <unstructured_citation>Kim, A., Kim, Y. R. A Study the Sine Tangent-Fibonacci Polynomials and Cosine Tangent-Fibonacci Polynomials, J. Appl. &amp; Pure Math., 6(5-6), 2024, 337-351. https:// doi.org/10.23091/japm.2024.337</unstructured_citation>
          </citation>
          <citation key="ref38">
            <unstructured_citation>Ryoo, C. S. Distribution Zeros of the q-Bernoulli-Fibonacci Polynomials, J. Appl. &amp; Pure Math., 7(1-2), 2025, 1-10. https://do i.org/10.23091/japm.2025.001</unstructured_citation>
          </citation>
          <citation key="ref39">
            <unstructured_citation>Park, S. Y., Ryoo, C. S., Lee, H. Y. On the Apostol Tangent Fibonacci Polynomials and Their Analytical Properties, J. Appl. &amp; Pure Math., 7(5-6), 2025, 383-392. https://do i.org/10.23091/japm.2025.383</unstructured_citation>
          </citation>
          <citation key="ref40">
            <unstructured_citation>Choi, J. E., Ryoo, C. S. Numerical Investigation of the Distribution of Zeros of the (p,q)-Bernoulli-Fibonacci Polynomials, J. Appl. &amp; Pure Math., 7(3-4), 2025, 137-147. https://doi.org/10.23091/japm. 2025.137</unstructured_citation>
          </citation>
        </citation_list>
      </journal_article>
    </journal>
  </body>
</doi_batch>
