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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
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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
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[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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