A177707 -id:A177707 - cp's OEIS frontend

A177706 Period 5: repeat [1, 1, 1, 1, 2].

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2

Offset: 0

Klaus Brockhaus, May 11 2010

Continued fraction expansion of (5+sqrt(65))/8.

Decimal expansion of 3704/33333.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A130782 (repeat 1, 1, 2, 1, 1), A177707 (decimal expansion of (5+sqrt(65))/8).

Magma
```
&cat[ [1, 1, 1, 1, 2]: k in [1..21] ];
```

A177706:=n->floor(6*(n+1)/5)-floor(6*n/5): seq(A177706(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2014

Table[Floor[6 (n + 1)/5] - Floor[6 n/5], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 24 2014 *)

a(n) = a(n-5) for n > 4; a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 2.
G.f.: (1+x+x^2+x^3+2*x^4)/(1-x^5).
a(n) = A130782(n+3).
a(n+4) = A198517(n+2) + A198517(n+1) + A198517(n). - Bruno Berselli, Nov 02 2011
a(n) = floor((n+1)*6/5) - floor((n)*6/5). - Hailey R. Olafson, Jul 23 2014
a(n) = (2/5)*(3 + cos(4*(n-4)*Pi/5) + cos(2*(n+1)*Pi/5)). - Wesley Ivan Hurt, Oct 05 2018
a(n) = 2 - ((n+1)^4 mod 5). - Aaron J Grech, Aug 30 2024

A188656 Decimal expansion of (1+sqrt(65))/8.

Original entry on oeis.org

1, 1, 3, 2, 7, 8, 2, 2, 1, 8, 5, 3, 7, 3, 1, 8, 7, 0, 6, 5, 4, 5, 8, 2, 6, 6, 5, 3, 7, 8, 7, 9, 7, 1, 3, 9, 1, 3, 9, 1, 7, 9, 9, 5, 3, 8, 2, 0, 1, 0, 7, 1, 6, 7, 3, 4, 9, 2, 0, 7, 4, 0, 4, 8, 6, 5, 7, 9, 8, 4, 3, 6, 8, 8, 7, 8, 2, 1, 1, 0, 2, 5, 3, 7, 0, 0, 1, 9, 2, 8, 3, 3, 3, 9, 6, 5, 3, 8, 3, 0, 4, 5, 4, 6, 8, 0, 3, 0, 8, 2, 6, 7, 4, 9, 3, 2, 3, 9, 0, 2, 6, 7, 1, 8, 5, 8, 1, 5, 1, 5

Offset: 1

Clark Kimberling, Apr 09 2011

Apart from the second digit the same as A177707.
Decimal expansion of the shape of a (1/4)-extension rectangle.
See A188640 for definitions of shape and r-extension rectangle.
A (1/4)-extension rectangle matches the continued fraction [1,7,1,1,7,1,1,7,1,1,7,1,1,7,...] for the shape L/W= (1+sqrt(65))/8.  This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,...].  Specifically, for the (4/3)-extension rectangle, 1 square is removed first, then 7 squares, then 1 square, then 1 square, then 7 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

			length/width = 1.13278221853731870654582665....

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

Cf. A188640, A105395.

RealDigits[(1 + Sqrt[65])/8, 10, 111][[1]] (* Robert G. Wilson v, Aug 19 2011 *)

cp's OEIS Frontend

A177706 Period 5: repeat [1, 1, 1, 1, 2].

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