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A261745
Decimal expansion of -sm(-1), where sm(t) is the Dixonian elliptic function sm(t).
0
1, 2, 0, 5, 4, 1, 5, 1, 5, 1, 4, 0, 2, 9, 8, 3, 1, 5, 4, 8, 3, 1, 4, 1, 1, 3, 7, 5, 7, 8, 4, 4, 8, 8, 0, 1, 2, 0, 7, 2, 7, 0, 4, 1, 9, 1, 8, 8, 2, 2, 4, 9, 5, 8, 1, 0, 9, 3, 2, 7, 1, 8, 2, 3, 5, 4, 4, 7, 6, 4, 8, 8, 1, 0, 6, 5, 5, 1, 1, 2, 5, 5, 6, 3, 2, 1, 7, 0, 3, 6, 5, 2, 2, 9, 3, 7, 9, 8, 9, 0, 8, 0, 6, 7, 9
OFFSET
1,2
COMMENTS
In the context of particle physics and in the case of a Yule branching process with two types of particles, this constant appears in the asymptotic expression of the probability that all particles be of the second type at time t as exp(-t)*smh(1).
LINKS
Eric van Fossen Conrad and Philippe Flajolet, The Fermat cubic, elliptic functions, continued fractions and a combinatorial excursion, Séminaire Lotharingien de Combinatoire 54 (2006), Article B54g, page 20.
Eric Weisstein's MathWorld, Weierstrass Elliptic Function
EXAMPLE
1.205415151402983154831411375784488012072704191882249581...
MATHEMATICA
sm[z_] := 6*WeierstrassP[z, {0, 1/27}]/(1 - 3*WeierstrassPPrime[z, {0, 1/27}]); N[-sm[-1], 105] // RealDigits // First
(* or, without using the Weierstrass P function: *) nint[y_?NumericQ] := NIntegrate[1/(1 + w^3)^(2/3), {w, 0, y}, WorkingPrecision -> 105]; smh[t_] := y /. FindRoot[nint[y] == t, {y, t}, WorkingPrecision -> 105]; N[smh[1], 105] // RealDigits // First
CROSSREFS
Sequence in context: A086280 A349950 A164976 * A083714 A215481 A262933
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved