A188656 -id:A188656 - cp's OEIS frontend

A182280 a(n) = floor(a(n-1)/4)+a(n-2) with a(0)=3, a(1)=4.

3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 11, 12, 14, 15, 17, 19, 21, 24, 27, 30, 34, 38, 43, 48, 55, 61, 70, 78, 89, 100, 114, 128, 146, 164, 187, 210, 239, 269, 306, 345, 392, 443, 502, 568, 644, 729, 826, 935, 1059, 1199, 1358, 1538, 1742, 1973, 2235, 2531, 2867

Offset: 0

Bruno Berselli, Apr 24 2012

nonn

a(n)/a(n-1) tends to (1+sqrt(65))/8 = 1.132782218537318706...

Bruno Berselli, Table of n, a(n) for n = 0..1000

Cf. A064650, A182230, A182281; A188656.

a182280 n = a182280_list !! n
a182280_list = 3 : 4 : zipWith (+)
                       a182280_list (map (flip div 4) $ tail a182280_list)
-- Reinhard Zumkeller, Apr 30 2015

[n le 2 select n+2 else Floor(Self(n-1)/4)+Self(n-2): n in [1..59]];

RecurrenceTable[{a[0] == 3, a[1] == 4, a[n] == Floor(a[n - 1]/4) + a[n - 2]}, a, {n, 58}]

A188657 Decimal expansion of (3+sqrt(73))/8.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 09 2011

Decimal expansion of the length/width ratio of a (3/4)-extension rectangle.
See A188640 for definitions of shape and r-extension rectangle for ratio r.
A (3/4)-extension rectangle matches the continued fraction [1,2,3,1,7,1,3,2,1,1,2,3,1,7,1,3,2,...] for the shape L/W= (3+sqrt(73))/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 (3/4)-extension rectangle, 1 square is removed first, then 2 squares, then 3 squares, then 1 square, then 7 squares,..., so that the original rectangle is partitioned into an infinite collection of squares.

			1.4430004681646...

Clark Kimberling, A Visual Euclidean Algorithm, The Mathematics Teacher 76 (1983) 108-109.

Cf. A188640, A188656.

evalf(3+sqrt(73))/8 ; # R. J. Mathar, Apr 11 2011

RealDigits[(3 + Sqrt[73])/8, 10, 111][[1]] (* Robert G. Wilson v, Aug 19 2011 *)

(sqrt(73)+3)/8 \\ Charles R Greathouse IV, Apr 25 2016

cp's OEIS Frontend

A182280 a(n) = floor(a(n-1)/4)+a(n-2) with a(0)=3, a(1)=4.

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