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Integer sequences Somos-4. (English. Russian original) Zbl 1464.11020
Dokl. Math. 98, No. 1, 357-359 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 481, No. 5, 471-473 (2018).
From the text: A new three-parameter family of integer sequences Somos-4 is constructed. The main result is as follows:
Theorem. Let $$x, y$$, and $$z$$ be independent variables. Let $\alpha = yz-x(1+y),\quad \beta = x(1+y) - y^2z + xy(1+y),$ $A(-1) = y, \quad A(0) = A(1) =1,\quad A(2) = 1+y.$
Then, for any $$n$$, $$A(n)$$ is a polynomial in $$x, y$$, and $$z$$ with integer coefficients.
Remark. For $$x = y = 1$$ and $$z = 3$$, we obtain the sequence from [M. Somos, “Problem 1470”, Crux. Math. 15, 208 (1989)].

##### MSC:
 11B37 Recurrences
Full Text:
##### References:
 [1] Somos, M., No article title, Crux Mathematicorum, 15, 208, (1989) [2] Gale, D., No article title, Math. Intel., 13, 40-42, (1991) [3] Somos Polynomials. http://somos.crg4.com/somospol.html. [4] Fomin, S.; Zelevinsky, A., No article title, Adv. Appl. Math., 28, 119-144, (2002) · Zbl 1012.05012 [5] Hone, A. N. W.; Swart, C. S., No article title, Math. Proc. Cambridge Philos. Soc., 145, 65-85, (2008) · Zbl 1165.11018 [6] Hone, A. N. W., No article title, Bull. London Math. Soc., 37, 161-171, (2005) · Zbl 1166.11333 [7] Hone, A. N. W., No article title, Trans. Am. Math. Soc., 359, 5019-5034, (2007) · Zbl 1162.11011 [8] Ma, X., No article title, Discrete Math., 310, 1-5, (2010) · Zbl 1217.11016 [9] Bykovskii, V. A., No article title, Funct. Anal. Appl., 50, 193-203, (2016) · Zbl 1360.30023
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