A078808 -id:A078808 - cp's OEIS frontend

A077419 Largest Whitney number of Fibonacci lattices J(Z_n).

1, 1, 1, 2, 2, 3, 5, 7, 11, 17, 26, 40, 63, 97, 153, 238, 376, 587, 931, 1458, 2317, 3640, 5794, 9124, 14545, 22951, 36631, 57904, 92512, 146461, 234205, 371281, 594169, 943045, 1510192, 2399460, 3844787, 6114555, 9802895, 15603339, 25027296

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

A051286 and A051291, interleaved. a(n) is the maximal element in the n-th row of A079487 or A123245 and in the (n+2)-th row of A078807 or A078808. - Andrey Zabolotskiy, Sep 21 2017

Emanuele Munarini, Mar 05 2007, Table of n, a(n) for n = 0..100
Brian Kent, Sarah Racz, and Sanjit Shashi, Scrambling in quantum cellular automata, arXiv:2301.07722 [quant-ph], 2023.
E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

with(FormalPowerSeries): with(LREtools): # requires Maple 2022
gf:= (1 + 2*x + 2*x^4 - x^6 - (1-x^2)*sqrt(1 - 2*x^2 - x^4 - 2*x^6 + x^8))/(2*x*sqrt(1 - 2*x^2 - x^4 - 2*x^6 + x^8));
re:= FindRE(gf,x,a(n));
inits:= {seq(a(i-1)=[1,1,1,2,2,3,5,7,11,17,26,40,63,97, 153][i],i=1..14)};
rm:=  (n+1)*a(n) +(n-2)*a(n-1) +2*(-n+1)*a(n-2) +2*(-n+1)*a(n-3) +(-n-3)*a(n-4) +(-n+8)*a(n-5) +2*(-n+6)*a(n-6) +2*(-n+7)*a(n-7) +(n-9)*a(n-8) +(n-10)*a(n-9)=0;
minre:= MinimalRecurrence(re, a(n), inits); minrm:= MinimalRecurrence(rm, a(n), inits); # shows that Mathar's recurrence is equivalent
f:= REtoproc(re,a(n),inits); seq(f(n),n=0..40); # Georg Fischer, Oct 22 2022

gf[x_] = (1 + 2 x + 2 x^4 - x^6 - (1 - x^2) Sqrt[1 - 2 x^2 - x^4 - 2 x^6 + x^8])/(2 x Sqrt[1 - 2 x^2 - x^4 - 2 x^6 + x^8]);
Table[SeriesCoefficient[gf[x], {x, 0, n}], {n, 0, 40}] (* Hugo Pfoertner, Oct 22 2022 *)

G.f.: (1 + 2 x + 2 x^4 - x^6 - (1-x^2) sqrt(1 - 2 x^2 - x^4 - 2 x^6 + x^8) )/(2x sqrt(1 - 2 x^2 - x^4 - 2 x^6 + x^8)). - Emanuele Munarini, Mar 05 2007
a(n) ~ phi^(n+2) / (5^(1/4) * sqrt(2*Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 22 2017
D-finite with recurrence: (n+1)*a(n) +(n-2)*a(n-1) +2*(-n+1)*a(n-2) +2*(-n+1)*a(n-3) +(-n-3)*a(n-4) +(-n+8)*a(n-5) +2*(-n+6)*a(n-6) +2*(-n+7)*a(n-7) +(n-9)*a(n-8) +(n-10)*a(n-9)=0. - R. J. Mathar, Nov 19 2019

More terms from Emanuele Munarini, Mar 05 2007

A078807 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, all runlengths odd and first letter 0.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 3, 3, 3, 2, 1, 0, 0, 1, 3, 4, 5, 4, 3, 1, 1, 4, 6, 7, 7, 5, 3, 1, 0, 0, 1, 4, 7, 10, 11, 10, 7, 4, 1, 1, 5, 10, 14, 17, 16, 13, 8, 4, 1, 0, 0, 1, 5, 11, 18, 24, 26, 24, 18, 11, 5, 1, 1, 6, 15, 25, 35, 40, 39, 32, 22, 12, 5, 1, 0, 0

Offset: 1

Clark Kimberling, Dec 07 2002

Row sums: 1,1,2,3,5,8,13,..., the Fibonacci numbers (A000045).

			T(6,2) counts the words 010001 and 000101. Top of triangle:
1 = T(1,0)
0 1 = T(2,0) T(2,1)
1 1 0
0 1 1 1
1 2 1 1 0

Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.

Cf. A078808, A078821, A079487, A123245, A077419.

T(n, k)=T(n-1, n-k-1)+T(n-3, n-k-3)+...+T(n-2m-1, n-k-2m-1), where m=[(n-1)/2] and (by definition) T(i, j)=0 if i<0 or j<0 or i=j.

Row 0 removed to stick to the triangle format by Andrey Zabolotskiy, Sep 22 2017

A123245 Triangle A079487 with reversed rows.

Original entry on oeis.org

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

Roger L. Bagula, Oct 07 2006

Row sums give Fibonacci numbers (A000045).

			{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}

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.

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

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

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).

Edited by Joerg Arndt, May 26 2015

Offset corrected by Andrey Zabolotskiy, Sep 22 2017

A078821 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's and all runlengths odd.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0, 2, 2, 2, 0, 1, 2, 2, 2, 2, 1, 0, 2, 4, 4, 4, 2, 0, 1, 3, 4, 5, 5, 4, 3, 1, 0, 2, 6, 8, 10, 8, 6, 2, 0, 1, 4, 7, 10, 12, 12, 10, 7, 4, 1, 0, 2, 8, 14, 20, 22, 20, 14, 8, 2, 0, 1, 5, 11, 18, 25, 29, 29, 25, 18, 11, 5, 1, 0, 2, 10, 22, 36, 48, 52, 48, 36, 22, 10, 2, 0

Offset: 0

Clark Kimberling, Dec 07 2002

Rows are symmetric. Row sums (0,2,2,4,6,10,16,26,...) are given by 2*F(n), where F(n) is the n-th Fibonacci number, A000045(n).

			T(6,2) counts the words 010001, 000101, 101000 and 100010.
Top of triangle:
  0
  1 1
  0 2 0
  1 1 1 1
  0 2 2 2 0
  1 2 2 2 2 1

Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol.10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.

Zachary Greenberg, Dani Kaufman, and Anna Wienhard, SL_2-like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers, arXiv:2402.19300 [math.RA], 2024. See p. 38.

Cf. A000045, A078807, A078808.

T(n, k) = A078807(n, k) + A078808(n, k).

cp's OEIS Frontend

A077419 Largest Whitney number of Fibonacci lattices J(Z_n).

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A078807 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's, all runlengths odd and first letter 0.

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A123245 Triangle A079487 with reversed rows.

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A078821 Triangular array T given by T(n,k) = number of 01-words of length n having exactly k 1's and all runlengths odd.

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