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A309535
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Total number of square parts in all compositions of n.
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2
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0, 1, 2, 5, 13, 30, 69, 156, 348, 769, 1682, 3653, 7884, 16924, 36160, 76944, 163137, 344770, 726533, 1527052, 3202076, 6700096, 13992080, 29167936, 60703424, 126141953, 261754114, 542448645, 1122778124, 2321317916, 4794159168, 9891365008, 20388823360
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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^2)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = (EllipticTheta[3, 0, 1/2] - 1)/8 = 0.1411171034014846448336823185681189155765645674... - Vaclav Kotesovec, Aug 18 2019, updated Mar 17 2024
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EXAMPLE
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a(4) = 13: (1)(1)(1)(1), (1)(1)2, (1)2(1), 2(1)(1), 22, (1)3, 3(1), (4).
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MAPLE
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a:= proc(n) option remember; add(a(n-j)+
`if`(issqr(j), 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[3, 0, x]-1)*(1-x)^2/(2*(1-2*x)^2), {x, 0, 30}], x] (* Vaclav Kotesovec, Aug 18 2019 *)
Table[Sum[If[k == n, 1, (2^(n - k - 2)*(3 + n - k))] * If[IntegerQ[Sqrt[k]], 1, 0], {k, 1, n}], {n, 0, 30}] (* 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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