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 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. 15
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Paul Curtz, Comments on this sequence Index to sequences with 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)). 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 *) 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 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: A065095 A005253 A212364 * 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 G.f multiplied by x to match the offset - R. J. Mathar, Jul 22 2009 STATUS approved

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