login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A247049 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,0) to (n,k), where 0 >= k <= 2, consisting of segments given by the vectors (1,1), (1,2), (1,-1). 14
1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 8, 8, 8, 13, 13, 13, 21, 21, 21, 34, 34, 34, 55, 55, 55, 89, 89, 89, 144, 144, 144, 233, 233, 233, 377, 377, 377, 610, 610, 610, 987, 987, 987, 1597, 1597, 1597, 2584, 2584, 2584, 4181, 4181, 4181 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

COMMENTS

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, and if i > 0, then s(i) is in {0,1,2} and s(i) - s(i-1) is in {1,2,-1}.  Every row of T is a Fibonacci sequence (A000045), as is the sequence of column sums.

This is a 3-rowed array read upwards by columns. - N. J. A. Sloane, Sep 14 2014

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

FORMULA

Let F = A000045, the Fibonacci numbers.  Then (row 0, the bottom row) = F(n-1) for n >= 0; (row 1, the middle row) = F(n) for n >=0; (row 2, the top row) = (row 1).

EXAMPLE

First 10 columns:

0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 .. 13 .. 21 .. 34

0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 .. 13 .. 21 .. 34

1 .. 0 .. 1 .. 1 .. 2 .. 3 .. 5 .. 8 ... 13 .. 21

T(4,1) counts these 3 paths, given as vector sums applied to (0,0):

(1,2) + (1,-1) + (1,1) + (1,-1);

(1,1) + (1,-1) + (1,2) + (1,-1);

(1,2) + (1,-1) + (1,-1) + (1,1).

Partial sums of second components in each vector sum give the 3 integer strings described in Comments:  (0,2,1,2,1), (0,1,0,2,1), (0,2,1,0,1).

MATHEMATICA

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1]; t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2]; t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1]; TableForm[ Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (* array *)

u = Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* sequence *)

CROSSREFS

Cf.  A000045, A247050, A247051, A247309, A247310, A247311.

Sequence in context: A179269 A108711 A261736 * A029059 A035449 A161555

Adjacent sequences:  A247046 A247047 A247048 * A247050 A247051 A247052

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Sep 11 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 15:10 EDT 2017. Contains 284273 sequences.