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URL: https://oeis.org/A123245

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A123245
Triangle A079487 with reversed rows.
4
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
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.
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
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