The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A022026 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A022026

Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(2,15).
(history; published version)

#42 by Michael De Vlieger at Sat Aug 06 07:20:34 EDT 2022

STATUS

proposed

approved

#41 by Michel Marcus at Sat Aug 06 03:49:50 EDT 2022

STATUS

editing

proposed

#40 by Michel Marcus at Sat Aug 06 03:49:36 EDT 2022

LINKS

M. Desjarlais and R. Molina, <a href="https://wwwweb.yumpuarchive.comorg/enweb/document20060904233419/viewhttp:/52170177/counting-spanning-trees-in-grid-graphs-othello.alma-college.edu/~molina/papers/pdf%20papers/spantree.pdf">Counting Spanning Trees in Grid Graphs</a>

STATUS

proposed

editing

Discussion

Sat Aug 06

03:49

Michel Marcus: change of mind

#39 by Michel Marcus at Sat Aug 06 01:44:06 EDT 2022

STATUS

editing

proposed

#38 by Michel Marcus at Sat Aug 06 01:43:51 EDT 2022

COMMENTS

1.2 1-2 1.2 1.2 1.2 1-2 1-2 1-2 1.2 1.2 1.2 1.2 1-2 1-2 1-2

. . . . . | . . | . . | . . | . . | | | | . | | | . | | . |

4.3 4.3 4.3 4-3 4.3 4.3 4-3 4.3 4-3 4.3 4-3 4-3 4-3 4.3 4-3

.o -o .o -o .o -o .o -o

.. .. .. .. .| .| .| .|

.o .o -o -o .o .o -o -o

.o-o -o-o .o-o -o-o

.| | .| | .| | .| |

.o-o .o-o -o-o -o-o (End)

LINKS

M. Desjarlais and R. Molina, <a href="httphttps://othellowww.yumpu.com/en/document/view/52170177/counting-spanning-trees-in-grid-graphs-alma.edu/~molina/papers/pdf%20papers/spantree.pdf-college">Counting Spanning Trees in Grid Graphs</a>

<a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8, -4).

FORMULA

From Peter Bala, May 03 2014: (Start)

a(n) = sum of the entries in the 2 X 2 matrix A^n where A is the 2 X 2 matrix [4, 4; 3, 4]. a(n) = (1 + 7*sqrt(3)/12)*(4 + 2*sqrt(3))^n + (1 - 7*sqrt(3)/12)*(4 - 2*sqrt(3))^n. See Desjarlais and Molina. - _Peter Bala_, May 03 2014

a(n) = (1 + 7*sqrt(3)/12)*(4 + 2*sqrt(3))^n + (1 - 7*sqrt(3)/12)*(4 - 2*sqrt(3))^n. See Desjarlais and Molina. (End)

STATUS

proposed

editing

#37 by Jon E. Schoenfield at Sat Aug 06 01:17:30 EDT 2022

STATUS

editing

proposed

#36 by Jon E. Schoenfield at Sat Aug 06 01:17:27 EDT 2022

COMMENTS

a(n) is also the number of forests in the 2x2 X (n+1) grid.

a(0)=2, because there are 2 forests in the 2x1 2 X 1 grid: 1.2 and 1-2.

a(1)=15, because there are 15 forests in the 2x2 2 X 2 grid:

FORMULA

a(n) = sum of the entries in the 2X2 2 X 2 matrix A^n where A is the 2X2 2 X 2 matrix [4, 4; 3, 4]. a(n) = (1 + 7*sqrt(3)/12)*(4 + 2*sqrt(3))^n + (1 - 7*sqrt(3)/12)*(4 - 2*sqrt(3))^n. See Desjarlais and Molina. - Peter Bala, May 03 2014

STATUS

approved

editing

#35 by OEIS Server at Wed Feb 24 17:02:48 EST 2016

LINKS

Alois P. Heinz, <a href="/A022026/b022026_1.txt">Table of n, a(n) for n = 0..1000</a>

#34 by Alois P. Heinz at Wed Feb 24 17:02:48 EST 2016

STATUS

editing

approved

Discussion

Wed Feb 24

17:02

OEIS Server: Installed new b-file as b022026.txt.  Old b-file is now b022026_1.txt.

#33 by Alois P. Heinz at Wed Feb 24 17:02:44 EST 2016

LINKS

Alois P. Heinz, <a href="/A022026/b022026_1.txt">Table of n, a(n) for n = 0..4001000</a>

KEYWORD

nonn,easy,changed

STATUS

approved

editing