A129339 Main diagonal of triangular array T: 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, 2, 4, 7, 11, 16, 23, 37, 74, 175, 431, 1024, 2291, 4825, 9650, 18571, 34955, 65536, 124511, 242461, 484922, 989527, 2038103, 4194304, 8565755, 17308657, 34617314, 68703187, 135812051, 268435456, 532087943, 1059392917, 2118785834

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Paul Curtz, Comments on this sequence

Index entries for linear recurrences with constant coefficients, signature (5,-9,6).

Formula

G.f.: x*(1-x)^3/((1-2*x)*(1-3*x+3*x^2)). [multiplied by x to match the offset by R. J. Mathar, Jul 22 2009]

a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 7; for n > 4, a(n) = 5*a(n-1) - 9*a(n-2) + 6*a(n-3).

Binomial transform of A088911. - Klaus Brockhaus, Jun 17 2007

a(n+1) = A057083(n)/3+2^(n-1), n > 1. - R. J. Mathar, Jul 22 2009

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

a[n_] := 2^(n-2) + 2*3^((n-3)/2)*Sin[n*Pi/6]; a[1]=1; Table[a[n], {n, 1, 33}] (* Jean-François Alcover, Aug 13 2012 *)
CoefficientList[Series[(1 - x)^3 / ((1 - 2 x) (1 - 3 x + 3 x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 13 2018 *)

Prog

(PARI) {m=33; v=concat([1, 2, 4, 7], vector(m-4)); for(n=5, m, v[n]=5*v[n-1]-9*v[n-2]+6*v[n-3]); v} \\ Klaus Brockhaus, Jun 10 2007
(Magma) m:=33; 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; [ M[n, n]: n in [1..m] ]; // Klaus Brockhaus, Jun 10 2007
(Magma) m:=33; S:=[ [1, 1, 1, 0, 0, 0][(n-1) mod 6 + 1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; // Klaus Brockhaus, Jun 17 2007
(Magma) I:=[1, 2, 4, 7]; [n le 4 select I[n] else 5*Self(n-1)-9*Self(n-2)+6*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 13 2018

Crossrefs

Cf. A038504, A131022 (T read by rows), 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: A005253 A212364 A320591 * A196719 A011912 A063676

Adjacent sequences: A129336 A129337 A129338 * A129340 A129341 A129342

Keyword

nonn,easy

Author

Paul Curtz, May 28 2007

Extensions

Edited and extended by Klaus Brockhaus, Jun 10 2007

Status

approved