A123245 Triangle A079487 with reversed rows.
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 3, 3, 3, 1, 1, 3, 4, 5, 4, 3, 1, 1, 3, 5, 7, 7, 6, 4, 1, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 4, 8, 13, 16, 17, 14, 10, 5, 1, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1
Offset: 0
Examples
{1},
{1, 1},
{1, 1, 1},
{1, 1, 2, 1},
{1, 2, 2, 2, 1},
{1, 2, 3, 3, 3, 1},
{1, 3, 4, 5, 4, 3, 1},
{1, 3, 5, 7, 7, 6, 4, 1},
{1, 4, 7, 10, 11, 10, 7, 4, 1},
{1, 4, 8, 13, 16, 17, 14, 10, 5, 1},
{1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1}
Links
- Sébastien Labbé and Mélodie Lapointe, The q-analog of the Markoff injectivity conjecture over the language of a balanced sequence, Comb. Theor. (2022) Vol. 2, No. 1, #9. See also arXiv:2106.15886 [math.CO], 2021.
- Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, arXiv:1812.00170 [math.CO], 2018-2020.
- Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365 [math.QA], 2019.
- Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and q-continued fractions, (2019) [math].
- Sophie Morier-Genoud and Valentin Ovsienko, Quantum real numbers and q-deformed Conway-Coxeter friezes, arXiv:2011.10809 [math.QA], 2020.
- Sophie Morier-Genoud and Valentin Ovsienko, q-deformed rationals and irrationals, arXiv:2503.23834 [math.CO], 2025. See p. 7.
- Valentin Ovsienko, Modular invariant q-deformed numbers: first steps, 2023.
Programs
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Mathematica
p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = If[Mod[k, 2] == 0, x*p[k - 1, x] + p[k - 2, x], p[k - 1, x] + x^2*p[k - 2, x]]; Table[CoefficientList[p[n, x], x], {n, 0, 10}] // Flatten
Formula
p(k, x) = x*p(k - 1, x) + p(k - 2, x) for k even, otherwise p(k, x) = p(k - 1, x) + x^2*p(k - 2, x).
Extensions
Edited by Joerg Arndt, May 26 2015
Offset corrected by Andrey Zabolotskiy, Sep 22 2017
Comments