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A247309 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1), (1,-2). 4
1, 0, 0, 1, 1, 1, 2, 3, 3, 5, 8, 8, 13, 21, 21, 34, 55, 55, 89, 144, 144, 233, 377, 377, 610, 987, 987, 1597, 2584, 2584, 4181, 6765, 6765, 10946, 17711, 17711, 28657, 46368, 46368, 75025, 121393, 121393, 196418, 317811, 317811, 514229, 832040, 832040 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Every member of T is a Fibonacci number, and the sum of the numbers in column n is A000045(2n+2).

LINKS

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

FORMULA

Let F = A000045 (Fibonacci numbers); then

(row 0, the bottom row) = (F(2n)), n >= 0;

(row 1, the middle row) = (F(2n)), n >= 0;

(row 2, the top row) = (F(2n-1)), n >= 0.

(n-th column sum) = (F(2n+2)), n >= 0.

EXAMPLE

First 10 columns:

0 .. 1 .. 3 .. 8 .. 21 .. 55 .. 144 .. 377 .. 987 ... 2584

0 .. 1 .. 3 .. 8 .. 21 .. 55 .. 144 .. 377 .. 987 ... 2584

1 .. 1 .. 2 .. 5 .. 13 .. 34 .. 89 ... 233 .. 610 ... 1597

T(2,2) counts these 3 paths, given as vector sums applied to (0,0):

(1,2) + (1,0); (1,1) + (1,1); (1,0) + (1,2).

MATHEMATICA

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 1;

t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];

t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];

t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2]

TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)

Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (*  A247309 *)

CROSSREFS

Cf.  A247049, A247310, A000045.

Sequence in context: A295606 A154690 A046937 * A069831 A017820 A129577

Adjacent sequences:  A247306 A247307 A247308 * A247310 A247311 A247312

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Sep 12 2014

STATUS

approved

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Last modified July 11 20:03 EDT 2020. Contains 335652 sequences. (Running on oeis4.)