On Generalized Reverse 3-primes Numbers

Main Article Content

Yuksel Soykan


In this paper, we introduce and investigate the generalized reverse 3-primes sequences and we deal with, in detail, three special cases which we call them reverse 3-primes, reverse Lucas 3-primes and reverse modified 3-primes sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

Reverse 3-primes numbers, reverse Lucas 3-primes numbers, 3-primes numbers, Lucas 3-primes numbers, Tribonacci numbers.

Article Details

How to Cite
Soykan, Y. (2020). On Generalized Reverse 3-primes Numbers. Journal of Scientific Research and Reports, 26(6), 1-20. https://doi.org/10.9734/jsrr/2020/v26i630267
Original Research Article


Fibonacci Quarterly. 1984;22(3):244-246.
Catalani M. Identities for Tribonacci-related sequences; 2012. arXiv:math/0209179 Choi E. Modular Tribonacci numbers by matrix method. Journal of the Korean Society of Mathematical Education Series B: Pure and Applied. Mathematics. ;20(3):207-221.

Elia M. Derived sequences, the tribonacci recurrence and cubic forms. Fibonacci Quarterly. 2001;39(2):107-115.

Er MC. Sums of Fibonacci numbers by matrix methods. Fibonacci Quarterly. ;22(3):204-207.

Lin PY. De moivre-type identities for the tribonacci numbers. Fibonacci Quarterly. ;26:131-134.

Pethe S. Some Identities for Tribonacci sequences. Fibonacci Quarterly. ;26(2):144-151.

Scott A, Delaney T, Hoggatt Jr. V. The Tribonacci sequence. Fibonacci Quarterly.

Shannon A. Tribonacci numbers and Pascal’s pyramid. Fibonacci Quarterly. ;15(3):268&275.

Soykan Y. Tribonacci and Tribonacci-lucas sedenions. Mathematics. 2019;7(1):74.

Spickerman W. Binet’s formula for the Tribonacci sequence. Fibonacci Quarterly.

Yalavigi CC. Properties of Tribonacci numbers. Fibonacci Quarterly.;10(3):231-246.

Yilmaz N, Taskara N. Tribonacci and Tribonacci-lucas numbers via the determinants of special Matrices. Applied
Mathematical Sciences. 2014;8(39):1947-

Howard FT, Saidak F. Zhou’s theory of constructing identities. Congress Numer.

Soykan Y. On generalized grahaml numbers. Journal of Advances in Mathematics and Computer Science.
DOI: 10.9734/JAMCS/2020/v35i230248

Sloane NJA. The on-line encyclopedia of integer sequences. Available:http://oeis.org/

Kilic¸ E, Stanica P. A Matrix approach for general higher order linear recurrences.

Bulletin of the Malaysian Mathematical Sciences Society. 2011;34(1):51- Soykan Y. Simson identity of generalized
m-step fibonacci numbers. Int. J. Adv. Appl.
Math. and Mech. 2019;7(2):45-56.

Soykan Y. Summing formulas for generalized tribonacci numbers. Universal Journal of Mathematics and Applications.
DOI: https://doi.org/10.32323/ujma.637876
Soykan Y. Generalized tribonacci numbers:

Summing formulas. Int. J. Adv. Appl. Mathand Mech. 2020;7(3):57-76. (ISSN: 2347-

Soykan Y. On the sums of squares of generalized tribonacci numbers: Closed formulas of
Σn k=0 xkW2 k . Archives of Current Research International.;20(4):22-47.
DOI: 10.9734/ACRI/2020/v20i430187

Soykan Y. A closed formula for the sums of squares of generalized tribonacci numbers. Journal of Progressive Research in Mathematics. 2020;16(2):2932-2941.

Kalman D. Generalized fibonacci numbers by matrix methods. Fibonacci Quarterly.

Koshy T. Fibonacci and lucas numbers with applications. Wiley-Interscience. New York; Marohni´c L, Strmeˇcki T. Plastic number: Construction and applications.

Advanced Research in Scientific Areas.

Padovan R. Dom hans van der laan and the plastic number. In: Williams K, Ostwald M. (Eds) Architecture and Mathematics from Antiquity to the Future. Birkh¨auser, Cham ;407-419.
-00143-2 27
(First published as: Richard Padovan

“Dom Hans van Der Laan and the Plastic Number”, pp. 181–193 in Nexus IV: Architecture and Mathematics, Kim
Williams and Jose Francisco Rodrigues, eds. Fucecchio (Florence): Kim Williams Books; 2002. Available:http://www.nexusjournal.com/conferences/

Padovan R. Dom hans van der laan: Modern primitive. Architectura and Natura
Press; 1994.

Shtayat J, Al-Kateeb A. An encodingdecoding algorithm based on Padovan numbers; 2019. arXiv:1907.02007
Basu M, Das M. Tribonacci matrices and a new coding theory. Discrete

Mathematics, Algorithms and Applications. ;6(1):1450008.

Tas¸cıD, Acar H. Gaussian Padovan and Gaussian Pell-Padovan numbers.

Commun. Fac. Sci. Ank. Ser. A1 Math. Stat. ;67(2):82-88.

TascıD. Padovan and Pell-Padovan quaternions. Journal of Science and Arts. 2018;1(42):125-132.

Deveci, O¨ , Shannon AG. Pell–Padovan- Circulant sequences and their applications. Notes on Number Theory and Discrete Mathematics. 2017;23(3):100-114.

Cerda-Morales G. Identities for third order Jacobsthal quaternions. Advances in Applied Clifford Algebras. 2017;27(2):1043- Cerda-Morales G. On a generalization of Tribonacci quaternions. Mediterranean Journal of Mathematics. 2017;14(239):1-12.