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A309536
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Total number of triangular numbers in all compositions of n.
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2
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0, 1, 2, 6, 14, 33, 77, 174, 389, 860, 1885, 4098, 8853, 19020, 40668, 86593, 183698, 388421, 818892, 1721884, 3611968, 7560337, 15793474, 32932549, 68556300, 142495004, 295754816, 613039248, 1269137729, 2624393922, 5421024773, 11186523404, 23061994524
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} x^(k*(k+1)/2)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = 0.1604081401637884665734606925563573585565153844... - Vaclav Kotesovec, Aug 18 2019
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EXAMPLE
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a(4) = 14: (1)(1)(1)(1), 2(1)(1), (1)2(1), (1)(1)2, 22, (3)(1), (1)(3), 4.
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MAPLE
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a:= proc(n) option remember; add(a(n-j)+
`if`(issqr(8*j+1), ceil(2^(n-j-1)), 0), j=1..n)
end:
seq(a(n), n=0..33);
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MATHEMATICA
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CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)) - 1)*((1 - x)^2/(1 - 2*x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Aug 18 2019 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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