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A038197
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4-wave sequence.
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12
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1, 1, 1, 1, 2, 3, 4, 7, 9, 10, 19, 26, 30, 56, 75, 85, 160, 216, 246, 462, 622, 707, 1329, 1791, 2037, 3828, 5157, 5864, 11021, 14849, 16886, 31735, 42756, 48620, 91376, 123111, 139997, 263108, 354484, 403104, 757588, 1020696, 1160693, 2181389
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OFFSET
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0,5
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COMMENTS
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This sequence is related to the nonagon or 9-gon.
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LINKS
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Eric Weisstein's World of Mathematics, Nonagon.
Index entries for linear recurrences with constant coefficients, signature (1,-1,3,-3,3,0,0,0,-1,1,-1).
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FORMULA
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a(n) = a(n-1)+a(n-2) if n=3*m+1, a(n) = a(n-1)+a(n-4) if n=3*m+2, a(n) = a(n-1)+a(n-6) if n=3*m. Also: a(n) = 2*a(n-3)+3*a(n-6)-a(n-9)-a(n-12).
G.f.: -(-1-x-x^2+x^3-x^5+x^6)/(1-2*x^3-3*x^6+x^9+x^12)
a(n-1) = sequence(sequence(T(n,k), k=2..4), n>=2) with a(0)=1; T(n,k) = sum(T(n-1,k1), k1 = 5-k..4) with T(1,1) = T(1,2) = T(1,3) = 0 and T(1,4) = 1; n>=1 and 1 <= k <= 4. [Steinbach]
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EXAMPLE
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The first few rows of the T(n,k) array are, n>=1, 1 <= k <=4:
0, 0, 0, 1
1, 1, 1, 1
1, 2, 3, 4
4, 7, 9, 10
10, 19, 26, 30
30, 56, 75, 85
85, 160, 216, 246
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MAPLE
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m:=4: nmax:=15: for k from 1 to m-1 do T(1, k):=0 od: T(1, m):=1: for n from 2 to nmax do for k from 1 to m do T(n, k):= add(T(n-1, k1), k1=m-k+1..m) od: od: for n from 1 to nmax/2 do seq(T(n, k), k=1..m) od; a(0):=1: Tx:=1: for n from 2 to nmax do for k from 2 to m do a(Tx):= T(n, k): Tx:=Tx+1: od: od: seq(a(n), n=0..Tx-1); # Johannes W. Meijer, Aug 03 2011
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MATHEMATICA
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LinearRecurrence[{1, -1, 3, -3, 3, 0, 0, 0, -1, 1, -1}, {1, 1, 1, 1, 2, 3, 4, 7, 9, 10, 19}, 50] (* Harvey P. Dale, Oct 02 2015 *)
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CROSSREFS
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Cf. A120747 (m = 5: hendecagon or 11-gon)
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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