A079487 Triangle read by rows giving Whitney numbers T(n,k) of Fibonacci lattices.
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 3, 2, 1, 1, 3, 4, 5, 4, 3, 1, 1, 4, 6, 7, 7, 5, 3, 1, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 5, 10, 14, 17, 16, 13, 8, 4, 1, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1, 1, 6, 15, 25, 35, 40, 39, 32, 22, 12, 5, 1, 1
Offset: 0
Examples
Triangle begins:
{1},
{1, 1},
{1, 1, 1},
{1, 2, 1, 1},
{1, 2, 2, 2, 1},
{1, 3, 3, 3, 2, 1},
{1, 3, 4, 5, 4, 3, 1},
{1, 4, 6, 7, 7, 5, 3, 1},
{1, 4, 7, 10, 11, 10, 7, 4, 1},
{1, 5, 10, 14, 17, 16, 13, 8, 4, 1},
{1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1}
Links
- Giovanni Resta, Rows n=0..139 of triangle, flattened
- Robert G. Donnelly, Molly W. Dunkum, Sasha V. Malone, and Alexandra Nance, Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras, arXiv:2012.14991 [math.CO], 2020.
- A. Khrabrov and K. Kokhas, Points on a line, shoelace and dominoes, arXiv:1505.06309 [math.CO], (23-May-2015).
- 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.
- Emanuele Munarini and Norma Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
- Valentin Ovsienko, Modular invariant q-deformed numbers: first steps, 2023.
Crossrefs
Largest element in each row gives A077419.
Programs
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Mathematica
p[0, x] = 1; p[1, x] = x + 1; p[k_, x_] := p[k, x] = Expand@ If[Mod[k, 2] == 1, x*p[k - 1, x] + p[k - 2, x], p[k - 1, x] + x^2*p[k - 2, x]]; Flatten[ Table[CoefficientList[p[n, x], x], {n, 0, 10}]] (* Roger L. Bagula, Oct 07 2006 *) T[ n_, k_] := (T[n, k] = Which[k<0 || k>n, 0, k==0, 1, True, T[n-1, k-Mod[n, 2]] + T[n-2, k-Mod[n+1, 2]*2]]); (* Michael Somos, Dec 12 2023 *) -
PARI
{T(n, k) = if(k<0 || k>n, 0, k==0, 1, T(n-1, k-(n%2)) + T(n-2, k-(n+1)%2*2))}; /* Michael Somos, Dec 12 2023 */
Formula
Define polynomials by: if k is odd then p(k, x) = x*p(k - 1, x) + p(k - 2, x); if k is even then: p(k, x) = p(k - 1, x) + x^2*p(k - 2, x). Triangle gives array of coefficients. - Roger L. Bagula, Oct 07 2006
Extensions
Mma program editing and a(67)-a(79) from Giovanni Resta, May 26 2015
Comments