A077951 -id:A077951 - cp's OEIS frontend

A078015 Expansion of (1-x)/(1-x+x^2-2*x^3).

1, 0, -1, 1, 2, -1, -1, 4, 3, -3, 2, 11, 3, -4, 15, 25, 2, 7, 55, 52, 11, 69, 162, 115, 91, 300, 439, 321, 482, 1039, 1199, 1124, 2003, 3277, 3522, 4251, 7283, 10076, 11295, 15785, 24642, 31447, 38375, 56212, 80731, 101269, 132962, 193155, 262731, 335500, 459079, 649041

Offset: 0

N. J. A. Sloane, Nov 17 2002

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-1,2).

Cf. A077951.

a:=[1,0,-1];; for n in [4..60] do a[n]:=a[n-1]-a[n-2]+2*a[n-3]; od; a; # G. C. Greubel, Jun 29 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1-x)/(1-x+x^2-2*x^3) )); // G. C. Greubel, Jun 29 2019

LinearRecurrence[{1,-1,2}, {1,0,-1}, 60] (* or *) CoefficientList[Series[ (1-x)/(1-x+x^2-2*x^3), {x,0,60}], x] (* G. C. Greubel, Jun 29 2019 *)

my(x='x+O('x^60)); Vec((1-x)/(1-x+x^2-2*x^3)) \\ G. C. Greubel, Jun 29 2019

((1-x)/(1-x+x^2-2*x^3)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Jun 29 2019

G.f.: (1-x)/(1-x+x^2-2*x^3).

a(n) = A077951(n) - A077951(n-1). - G. C. Greubel, Jun 29 2019

A331394 Number of ways of 4-coloring the Fibonacci square spiral tiling of n squares with colors introduced in order.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 7, 11, 16, 19, 25, 38, 51, 63, 88, 127, 165, 214, 303, 419, 544, 731, 1025, 1382, 1819, 2487, 3432, 4583, 6125, 8406, 11447, 15291, 20656, 28259, 38185, 51238, 69571, 94703, 127608, 172047, 233845

Offset: 1

Michael C. Case, Jan 15 2020

nonn

The Fibonacci square spiral tiling is the pattern formed by tiling the plane using squares with side-lengths of successive Fibonacci numbers (so the k-th square is of size F(k)), in a spiral pattern.
The Fibonacci square spiral tiling for 6 squares:
   ___ ___ _____________
  |     |   |               |
  |     | |               |
  |___|_|_|               |
  |         |               |
  |         |               |
  |         |               |
  |         |               |
  |_______|_______________|
In a 4-coloring of the Fibonacci square spiral tiling, the square k cannot be the same color as squares k-4, k-3, or k-1. When k-1 is the same color as k-3, k can be colored in 2 different ways.
The first 3 squares must be colored ABC but for k>3 square k can be the same color as square k-2.

			There are 3 ways to 4-color a Fibonacci square spiral tiling of 5 squares:
   _____ ___   _____ ___   _____ ___
  |     | C | |     | C | |     | C |
  |  B  |_ _| |  B  |_ _| |  D  |_ _|
  |_____|A|B| |_____|A|B| |_____|A|B|
  |         | |         | |         |
  |         | |         | |         |
  |    C    | |    D    | |    C    |
  |         | |         | |         |
  |_________| |_________| |_________| so a(5)=3.
There are 7 ways to 4-color a Fibonacci square spiral tiling of 8 squares (ABCBCADA, ABCBCDAD, ABCBDADA, ABCBDADC, ABCDCABA, ABCDCDAB, ABCDCDBA), so a(7) = 8.

Michael C. Case, Table of n, a(n) for n = 1..200 [a(192) corrected by _Georg Fischer_, Apr 15 2020]
Wikipedia, Fibonacci number
Index entries for linear recurrences with constant coefficients, signature (1, -1, 2).

Cf. A000045, A216790, A077951.

Rest@ CoefficientList[Series[x (1 + x) (1 - x + 2 x^2 - 2 x^3 + 2 x^4)/(1 - x + x^2 - 2 x^3), {x, 0, 42}], x] (* Michael De Vlieger, Jan 31 2020 *)

a(n) = a(n-1) - a(n-2) + 2*a(n-3) for n >= 7.
G.f.: x*(1 + x)*(1 - x + 2*x^2 - 2*x^3 + 2*x^4)/(1 - x + x^2 - 2*x^3).
a(n)/a(n-1) approaches the only real solution of x^3 - x^2 + x - 2 = 0, x = (1 - 2*(2/(47 + 3*sqrt(249)))^(1/3) + ((47 + 3*sqrt(249))/2)^(1/3))/3 = 1.35320996419932... .

cp's OEIS Frontend

A078015 Expansion of (1-x)/(1-x+x^2-2*x^3).

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