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A137742
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a(n) = (n-1)*(n+4)*(n+6)/6 for n > 1, a(1)=1.
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9
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1, 8, 21, 40, 66, 100, 143, 196, 260, 336, 425, 528, 646, 780, 931, 1100, 1288, 1496, 1725, 1976, 2250, 2548, 2871, 3220, 3596, 4000, 4433, 4896, 5390, 5916, 6475, 7068, 7696, 8360, 9061, 9800, 10578, 11396, 12255, 13156, 14100, 15088, 16121, 17200, 18326, 19500
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OFFSET
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1,2
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COMMENTS
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Also the number of different strings of length n+3 obtained from "123...n" by iteratively duplicating any substring (see A137743 for comments and examples). This is the principal (although not simplest) definition of this sequence and explains why a(1)=1 and not 0.
For n >= 3, sequence appears (not yet proved by induction) to give the number of multiplications between two nonzero matrix elements in calculating the product of two n X n Hessenberg matrices (square matrices which have 0's below the subdiagonal, other elements being in general nonzero). - John M. Coffey, Jun 21 2016
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LINKS
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FORMULA
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G.f.: x*(1+4*x-5*x^2+x^4)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
E.g.f.: 4 + x + (-24 + 24*x + 12*x^2 + x^3)*exp(x)/6.
Sum_{n>=1} 1/a(n) = 1542/1225. (End)
a(n) = binomial(n+4,3) - 2*(n+4) for n > 1. - Michael Chu, Dec 09 2021
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EXAMPLE
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MATHEMATICA
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Join[{1}, Table[Binomial[n, 3]-2*n, {n, 6, 60}]] (*or*) Join[{1}, Table[(n-1)(n+4)(n+6)/6, {n, 2, 56}]] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)
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PROG
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(PARI) A137742(n)=if(n<2, 1, (n - 1)*(n + 4)*(n + 6)/6)
(Magma) [1] cat [(n^3+9*n^2+14*n-24)/6: n in [2..46]]; // Bruno Berselli, Aug 23 2011
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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