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A249018
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Decimal expansion of the Flajolet-Prodinger constant 'K', a constant related to asymptotically enumerating level number sequences for binary trees.
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0
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2, 5, 4, 5, 0, 5, 5, 2, 3, 5, 6, 5, 3, 1, 9, 5, 1, 3, 3, 7, 0, 8, 8, 1, 7, 7, 0, 0, 3, 1, 5, 4, 6, 1, 5, 6, 0, 4, 6, 4, 9, 3, 7, 4, 1, 7, 2, 5, 0, 6, 1, 9, 4, 4, 4, 9, 8, 4, 5, 5, 0, 0, 0, 6, 3, 8, 6, 3, 8, 9, 2, 3, 9, 0, 0, 8, 8, 3, 1, 6, 8, 6, 0, 0, 2, 5, 8, 1, 2, 2, 6, 3, 5, 5, 8, 6, 1, 8, 7, 7
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OFFSET
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0,1
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5 Kalmar's composition constant, p. 294.
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LINKS
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FORMULA
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H(n) ~ K*nu^n, where H(n) is number of level number sequences associated to binary trees (Cf. A002572) and 'nu' is the constant A102375.
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EXAMPLE
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0.254505523565319513370881770031546156046493741725...
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MATHEMATICA
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digits = 105; m0 = 5; dm = 2; Clear[f, g, v, K]; v[c_, d_] := v[c, d] = If[d<0 || c<0, 0, If[d == c, 1, Sum[v[i, d-c], {i, 1, 2*c}]]]; H[n_] := v[1, n]; H[1] = 1; f[x_, m_] := Sum[((-1)^(j+1)*x^(2^(j+1)-2-j))/Product[1-x^(2^k-1), {k, 1, j}], {j, 1, m}] // N[#, digits]&; g[m_] := g[m] = (1/x /. FindRoot[f[x, m] == 1, {x, 5/9, 4/9, 6/9}, WorkingPrecision -> digits]); g[m0]; g[m = m0+dm]; While[RealDigits[g[m], 10, digits+5] != RealDigits[g[m-dm], 10, digits+5], m = m+dm]; nu = g[m]; K[m_] := K[m] = H[m]/nu^m; dm=100; K[m = 100]; K[m = m+dm]; While[Print[m]; RealDigits[K[m], 10, digits+5] != RealDigits[K[m-dm], 10, digits+5], m = m+dm]; RealDigits[K[m], 10, digits-5] // First
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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