A123245 - OEIS

%I #42 Jan 17 2024 09:35:52

%S 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,

%T 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,

%U 18,11,5,1

%N Triangle A079487 with reversed rows.

%C Row sums give Fibonacci numbers (A000045).

%H Sébastien Labbé and Mélodie Lapointe, <a href="https://doi.org/10.5070/C62156881">The q-analog of the Markoff injectivity conjecture over the language of a balanced sequence</a>, Comb. Theor. (2022) Vol. 2, No. 1, #9. See also <a href="https://arxiv.org/abs/2106.15886">arXiv:2106.15886</a> [math.CO], 2021.

%H Sophie Morier-Genoud and Valentin Ovsienko, <a href="https://arxiv.org/abs/1812.00170">q-deformed rationals and q-continued fractions</a>, arXiv:1812.00170 [math.CO], 2018-2020.

%H Sophie Morier-Genoud and Valentin Ovsienko, <a href="https://arxiv.org/abs/1908.04365">On q-deformed real numbers</a>, arXiv:1908.04365 [math.QA], 2019.

%H Sophie Morier-Genoud and Valentin Ovsienko, <a href="https://hal.archives-ouvertes.fr/hal-02270545/">q-deformed rationals and q-continued fractions</a>, (2019) [math].

%H Sophie Morier-Genoud and Valentin Ovsienko, <a href="https://arxiv.org/abs/2011.10809">Quantum real numbers and q-deformed Conway-Coxeter friezes</a>, arXiv:2011.10809 [math.QA], 2020.

%H Valentin Ovsienko, <a href="https://amathr.org/wp-content/uploads/2023/03/QOMBINUMReview-1.pdf">Modular invariant q-deformed numbers: first steps</a>, 2023.

%F 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).

%e {1},

%e {1, 1},

%e {1, 1, 1},

%e {1, 1, 2, 1},

%e {1, 2, 2, 2, 1},

%e {1, 2, 3, 3, 3, 1},

%e {1, 3, 4, 5, 4, 3, 1},

%e {1, 3, 5, 7, 7, 6, 4, 1},

%e {1, 4, 7, 10, 11, 10, 7, 4, 1},

%e {1, 4, 8, 13, 16, 17, 14, 10, 5, 1},

%e {1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1}

%t p[0, x] = 1; p[1, x] = x + 1;

%t 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]];

%t Table[CoefficientList[p[n, x], x], {n, 0, 10}] // Flatten

%Y Cf. A000045, A079487, A078807, A078808, A077419.

%K nonn,tabl

%O 0,9

%A _Roger L. Bagula_, Oct 07 2006

%E Edited by _Joerg Arndt_, May 26 2015

%E Offset corrected by _Andrey Zabolotskiy_, Sep 22 2017