A131022 Triangular array T read by rows: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.

1, 1, 2, 1, 2, 4, 0, 1, 3, 7, 0, 0, 1, 4, 11, 0, 0, 0, 1, 5, 16, 1, 1, 1, 1, 2, 7, 23, 1, 2, 3, 4, 5, 7, 14, 37, 1, 2, 4, 7, 11, 16, 23, 37, 74, 0, 1, 3, 7, 14, 25, 41, 64, 101, 175, 0, 0, 1, 4, 11, 25, 50, 91, 155, 256, 431, 0, 0, 0, 1, 5, 16, 41, 91, 182, 337, 593, 1024

Offset

1,3

Comments

All columns are periodic with period length 6. The (3+6*i)-th row equals the first (3+6*i) terms of main diagonal (i >= 0).

Links

Michel Marcus, Rows n = 1..100 of triangle, flattened

Formula

From Werner Schulte, Jul 22 2017: (Start)

T(n,k) = 2^(k-2) + 2*sqrt(3)^(k-3) * sin(Pi/6*(2*n-k)) for 1 < k <= n, and

T(n,1) = 1 - floor((n-1)/3) mod 2 for n >= 1. (End)

Example

First seven rows of T are
[ 1 ]
[ 1, 2 ]
[ 1, 2, 4 ]
[ 0, 1, 3, 7 ]
[ 0, 0, 1, 4, 11 ]
[ 0, 0, 0, 1, 5, 16 ]
[ 1, 1, 1, 1, 2, 7, 23 ].

Mathematica

T[j_, 1] := If[Mod[j-1, 6]<3, 1, 0]; T[j_, k_] := T[j, k] = T[j-1, k-1]+T[j, k-1]; Table[T[j, k], {j, 1, 13}, {k, 1, j}] // Flatten (* Jean-François Alcover, Mar 06 2014 *)

Prog

(PARI) {m=13; M=matrix(m, m); for(j=1, m, M[j, 1]=if((j-1)%6<3, 1, 0)); for(k=2, m, for(j=k, m, M[j, k]=M[j-1, k-1]+M[j, k-1])); for(j=1, m, for(k=1, j, print1(M[j, k], ", ")))}
(Magma) m:=13; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; &cat[ [ M[j, k]: k in [1..j] ]: j in [1..m] ];

Crossrefs

Cf. A129339 (main diagonal of T), A131023 (first subdiagonal of T), A131024 (row sums of T), A131025 (antidiagonal sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.

Sequence in context: A206425 A227428 A265255 * A137408 A007461 A181302

Adjacent sequences: A131019 A131020 A131021 * A131023 A131024 A131025

Keyword

nonn,tabl

Author

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

Status

approved