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

0,9

Comments

Row sums give Fibonacci numbers (A000045).

Links

Table of n, a(n) for n=0..65.

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.

Valentin Ovsienko, Modular invariant q-deformed numbers: first steps, 2023.

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).

Example

{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}

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

Crossrefs

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

Sequence in context: A343856 A053874 A170906 * A262611 A110535 A033941

Adjacent sequences: A123242 A123243 A123244 * A123246 A123247 A123248

Keyword

nonn,tabl

Author

Roger L. Bagula, Oct 07 2006

Extensions

Edited by Joerg Arndt, May 26 2015

Offset corrected by Andrey Zabolotskiy, Sep 22 2017

Status

approved