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A129339
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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.
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15
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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
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OFFSET
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1,2
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LINKS
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Table of n, a(n) for n=1..33.
Paul Curtz, Comments on this sequence
Index to sequences with linear recurrences with constant coefficients, signature (5,-9,6).
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FORMULA
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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
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EXAMPLE
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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 ].
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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Paul Curtz, May 28 2007
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EXTENSIONS
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Edited and extended by Klaus Brockhaus, Jun 10 2007
G.f multiplied by x to match the offset - R. J. Mathar, Jul 22 2009
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STATUS
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approved
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