Abstract
A relationship between the MacWilliams transform matrices and the classical integer Fibonacci, Lucas, and Padovan sequences is established. Namely, it is proved that the summation over some naturally chosen planes in the pyramid composed of these matrices yields a new integer sequence, which is the convolution of the Fibonacci numbers and the (alternating) Padovan numbers. In turn, this convolution is linearly represented in terms of the Lucas numbers and the Padovan numbers.
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MacWilliams, F.J. and Sloane, N.J.A., The Theory of Error-Correcting Codes, Amsterdam: North-Holland, 1977. Translated under the title Teoriya kodov, ispravlyayushchikh oshibki, Moscow: Radio i Svyaz’, 1979.
Online Encyclopedia of Integer Sequences, http://www.research.att.com/:_njas/sequences/.
Hall, M., Jr., Combinatorics Theory, Waltham, Mass.: Blaisdell, 1967. Translated under the title Kombinatorika, Moscow: Mir, 1970.
Vorob’ev, N.N., Chisla Fibonachchi (Fibonacci Numbers), Moscow: Nauka, 1978.
Graham, R.L., Knuth, D.E., and Patashnik, O., Concrete Mathematics, Addison-Wesley, 1994.
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Original Russian Text © N.D. Gogin, A.A. Myllari, 2007, published in Programmirovanie, 2007, Vol. 33, No. 2.
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Gogin, N.D., Myllari, A.A. The Fibonacci-Padovan sequence and MacWilliams transform matrices. Program Comput Soft 33, 74–79 (2007). https://doi.org/10.1134/S0361768807020041
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DOI: https://doi.org/10.1134/S0361768807020041