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A247352 Rectangular array read by upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1). 4
0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 2, 1, 2, 2, 3, 4, 2, 5, 8, 5, 5, 10, 12, 13, 10, 17, 28, 22, 17, 38, 49, 45, 38, 66, 100, 87, 66, 138, 191, 166, 138, 257, 370, 329, 257, 508, 724, 627, 508, 981, 1392, 1232, 981, 1900, 2721, 2373, 1900, 3702, 5254, 4621, 3702 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,10

COMMENTS

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 1, s(n) = k, and for i > 0,  s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n.

LINKS

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

FORMULA

The four rows and column sums all empirically satisfy the linear recurrence r(n) = 3*r(n-2) + 2*r(n-3) - r(n-4), with g.f. of the form p(x)/q(x), where q(x) = 1 - 3 x^2 - 2 x^3 + x^4.  Initial terms and p(x) follow:

(row 0, the bottom row):  0,1,0,2; x - x^3

(row 1):  1,0,0,2; 1 - x^2

(row 2):  0,1,2,3; x +2*x^2

(row 3):  0,1,1,4; x + x^2 + x^3

(n-th column sum) = 1,3,5,11; 1 + 3*x + 2*x^2.

EXAMPLE

First 10 columns:

0 .. 1 .. 1 .. 4 .. 5 .. 13 .. 22 .. 45 .. 87 ... 166

0 .. 1 .. 2 .. 3 .. 8 .. 12 .. 28 .. 49 .. 100 .. 191

1 .. 0 .. 2 .. 2 .. 5 .. 10 .. 17 .. 38 .. 66 ... 138

0 .. 1 .. 0 .. 2 .. 2 .. 5 ... 10 .. 17 .. 38 ... 66

T(5,0) counts these 5 paths, given as vector sums applied to (0,1):

(1,1) + (1,1) + (1,-1) + (1,-1) + (1 -1)

(1,1) + (1,-1) + (1,1) + (1,-1) + (1,-1)

(1,-1) + (1,1) + (1,1) + (1,-1) + (1,-1)

(1,1) + (1,-1) + (1,-1) + (1,1) + (1,-1)

(1,-1) + (1,1) + (1,-1) + (1,1) + (1,-1)

Partial sums of second components in each vector sum give the 3 integer strings described in comments:

(1,2,3,2,1,0),

(1,2,1,2,1,0),

(1,0,1,2,1,0),

(1,2,1,0,1,0),

(1,0,1,0,1,0).

MATHEMATICA

z = 50; t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0; t[0, 3] = 0;

t[1, 3] = 1; 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] + t[n - 1, 3]

t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2]

u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, 3}]] (* A247352 *)

TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 3}]]]] (* array  *)

v = Map[Total, u1]  (* A247353 *)

Table[t[n, 0], {n, 0, z}]   (* row 0, A247354*)

Table[t[n, 1], {n, 0, z}]   (* row 1, cf. row 0 *)

Table[t[n, 2], {n, 0, z}]   (* row 2, A247355 *)

Table[t[n, 3], {n, 0, z}]   (* row 3, A247325 *)

CROSSREFS

Cf.  A247049, A247321, A247353, A247354, A247355.

Sequence in context: A289497 A029328 A109831 * A097266 A226983 A112175

Adjacent sequences:  A247349 A247350 A247351 * A247353 A247354 A247355

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Sep 15 2014

STATUS

approved

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Last modified August 16 13:26 EDT 2017. Contains 290623 sequences.