A188935 -id:A188935 - cp's OEIS frontend

A182281 a(n) = floor(a(n-1)/3)+a(n-2) with a(0)=2, a(1)=3.

2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 15, 17, 20, 23, 27, 32, 37, 44, 51, 61, 71, 84, 99, 117, 138, 163, 192, 227, 267, 316, 372, 440, 518, 612, 722, 852, 1006, 1187, 1401, 1654, 1952, 2304, 2720, 3210, 3790, 4473, 5281, 6233, 7358, 8685, 10253, 12102

Offset: 0

Bruno Berselli, Apr 21 2012

nonn

a(n)/a(n-1) tends to (1+sqrt(37))/6 = 1.180460421716369948...

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

Cf. A064650, A182229, A182280; A188935.

a182281 n = a182281_list !! n
a182281_list = 2 : 3 : zipWith (+)
                       a182281_list (map (flip div 3) $ tail a182281_list)
-- Reinhard Zumkeller, Apr 30 2015

[n le 2 select n+1 else Floor(Self(n-1)/3)+Self(n-2): n in [1..55]];

RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == Floor[a[n - 1]/3] + a[n - 2]}, a, {n, 54}]
Transpose[NestList[{#[[2]],Floor[#[[2]]/3]+#[[1]]}&,{2,3},60]][[1]] (* Harvey P. Dale, Nov 26 2015 *)

A188934 Decimal expansion of (1+sqrt(17))/4.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a (1/2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A (1/2)-extension rectangle matches the continued fraction [1,3,1,1,3,1,1,3,1,1,3,...] for the shape L/W=(1+sqrt(17))/4.  This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...].  Specifically, for the (1/2)-extension rectangle, 1 square is removed first, then 3 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(17))/4 is partitioned into an infinite collection of squares.
Conjecture: This number is an eigenvalue to infinitely many n*n submatrices of A191898, starting in the upper left corner, divided by the row index. For the first few characteristic polynomials see A260237 and A260238. - Mats Granvik, May 12 2016.

			1.2807764064044151374553524639935192562...

Cf. A188640, A188935.

Essentially the same as A188485.

r = 1/2; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
(* for the continued fraction *) ContinuedFraction[t, 120]
RealDigits[(1 + Sqrt@ 17)/4, 10, 111][[1]] (* Or *)
RealDigits[Exp@ ArcSinh[1/4], 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)

(sqrt(17)+1)/4 \\ Charles R Greathouse IV, May 12 2016

cp's OEIS Frontend

A182281 a(n) = floor(a(n-1)/3)+a(n-2) with a(0)=2, a(1)=3.

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