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A038201
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5-wave sequence.
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9
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1, 1, 1, 1, 1, 2, 3, 4, 5, 9, 12, 14, 15, 29, 41, 50, 55, 105, 146, 175, 190, 365, 511, 616, 671, 1287, 1798, 2163, 2353, 4516, 6314, 7601, 8272, 15873, 22187, 26703, 29056, 55759, 77946, 93819, 102091, 195910, 273856, 329615, 358671, 688286, 962142
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OFFSET
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0,6
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COMMENTS
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This sequence is related to the hendecagon or 11-gon, see A120747.
Sequence of perfect distributions for a cascade merge with six tapes according to Knuth. - Michael Somos, Feb 07 2004
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REFERENCES
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D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.4.3, Eq. (1).
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LINKS
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Eric Weisstein's World of Mathematics, Hendecagon.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,3,0,0,0,-4,0,0,0,-1,0,0,0,1).
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FORMULA
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a(n) = a(n-1)+a(n-2) if n=4*m+1, a(n) = a(n-1)+a(n-4) if n=4*m+2, a(n) = a(n-1)+a(n-6) if n=4*m+3 and a(n) = a(n-1)+a(n-8) if n=4*m.
G.f.: -(1+x+x^2+x^3-2*x^4-x^5+x^7-x^8-x^11+x^12)/(-1+3*x^4+3*x^8-4*x^12-x^16+x^20).
a(n) = 3*a(n-4)+3*a(n-8)-4*a(n-12)-a(n-16)+a(n-20).
a(n-1) = sequence(sequence(T(n,k), k=2..5), n>=2) with a(0)=1; T(n,k) = sum(T(n-1,k1), k1 = 6-k..5) with T(1,1) = T(1,2) = T(1,3) = T(1,4) = 0 and T(1,5) = 1; n>=1 and 1 <= k <= 5. [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 <=5:
0, 0, 0, 0, 1
1, 1, 1, 1, 1
1, 2, 3, 4, 5
5, 9, 12, 14, 15
15, 29, 41, 50, 55
55, 105, 146, 175, 190
190, 365, 511, 616, 671
G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 9*x^8 + 12*x^9 + ...
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MAPLE
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m:=5: nmax:=12: 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[{0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, 1}, {1, 1, 1, 1, 1, 2, 3, 4, 5, 9, 12, 14, 15, 29, 41, 50, 55, 105, 146, 175}, 50] (* Harvey P. Dale, Dec 13 2012 *)
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PROG
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(PARI) {a(n) = local(m); if( n<=0, n==0, m = (n-1)\4 * 4; sum(k=2*m - n, m, a(k)))};
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CROSSREFS
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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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