cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A307083 Expansion of 1/(1 - x/(1 - 3*x/(1 - 4*x/(1 - 7*x/(1 - 11*x/(1 - 18*x/(1 - ... - Lucas(k)*x/(1 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 4, 28, 292, 4408, 97432, 3231256, 164789104, 13170099856, 1670220282544, 338692348412320, 110327835695333920, 57892877044109184160, 49019180876700301391680, 67044425508546158335526080, 148216012413625321252632612160, 529829556146109541834263919553920
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2019

Keywords

Links

Crossrefs

Cf. A000204, A001622, A206742, A307082.

Programs

  • Maple
    L:= proc(n) option remember; (<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1] end:
b:= proc(x, y) option remember; `if`(x=0 and y=0, 1,
     `if`(x>0, b(x-1, y)*L(y-x+1), 0)+`if`(y>x, b(x, y-1), 0))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, Nov 12 2023
  • Mathematica
    nmax = 17; CoefficientList[Series[1/(1 + ContinuedFractionK[-LucasL[k] x, 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

a(n) ~ c * phi^(n*(n+1)/2), where phi = A001622 is the golden ratio and c = 5.62026823201787715079864730026619553810473701484813959397175006212578036... - Vaclav Kotesovec, Sep 18 2021

A367252 a(n) is the number of ways to tile an n X n square as explained in comments.

Original entry on oeis.org

1, 0, 1, 4, 88, 3939, 534560, 185986304, 175655853776, 437789918351688, 2898697572048432368, 50698981110982431863735, 2342038257118692026082013568, 285250169294740386915765591840768, 91531011920509198679773321121428857296, 77312253225939431362091700178995800855209496
Offset: 0

Views

Author

Anna Tscharre, Nov 11 2023

Keywords

Comments

Draw a Dyck path from (0,0) to (n,n) so the path always stays above the diagonal. Now section the square into horizontal rows of height one to the left of the path and tile these rows using 1 X 2 and 1 X 1 tiles. Similarly, section the part to the right of the path into columns with width one and tile these using 2 X 1 and 1 X 1 tiles. Furthermore, no 1 X 1 tiles are allowed in the bottom row.

Links

Crossrefs

Special case of A003150.
Cf. A055010, A307082.

Programs

  • Maple
    b:= proc(x, y) option remember; (F->
     `if`(x=0 and y=0, 1, `if`(x>0, b(x-1, y)*F(y-1), 0)+
     `if`(y>x, b(x, y-1)*F(x+1), 0)))(combinat[fibonacci])
    end:
a:= n-> b(n$2):
seq(a(n), n=0..15);  # Alois P. Heinz, Nov 11 2023
  • Mathematica
    b[x_, y_] := b[x, y] = With[{F = Fibonacci},
     If[x == 0 && y == 0, 1,
     If[x > 0, b[x - 1, y]*F[y - 1], 0] +
     If[y > x, b[x, y - 1]*F[x + 1], 0]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Nov 14 2023, after Alois P. Heinz *)

Formula

a(n) == 1 (mod 2) <=> n in { A055010 }. - Alois P. Heinz, Nov 11 2023
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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