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Decimal expansion of Sum_{k>=0} (-1)^k * k / ((k+1) (2k)!).

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allocated for Clark KimberlingDecimal expansion of Sum_{k>=0} (-1)^k k / ((k+1) (2k)!).

2, 2, 3, 2, 4, 4, 2, 7, 5, 4, 8, 3, 9, 3, 2, 7, 3, 0, 7, 0, 5, 9, 4, 1, 2, 5, 0, 7, 0, 3, 5, 7, 4, 6, 0, 2, 9, 7, 7, 4, 3, 6, 5, 4, 2, 2, 1, 4, 6, 6, 4, 3, 5, 9, 0, 1, 5, 6, 0, 0, 6, 7, 5, 3, 6, 4, 9, 2, 1, 2, 0, 3, 5, 5, 6, 9, 5, 1, 1, 0, 2, 4, 1, 5, 2, 3

0,1

Equals 2 - cos(1) - 2 sin(1).

-0.2232442754839327307059412507035746029774...

s = Sum[(-1)^k k/((k + 1) (2 k)!), {k, 0, Infinity}]

d = N[s, 100]

First[RealDigits[d]]

Cf. A372339, A372340.

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Clark Kimberling, May 01 2024

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Maximum number of vertices of an irreducible multipole with n semiedges.

0, 1, 2, 5

2,3

Multipoles are the pieces we obtain by cutting some edges of a cubic graph at one or more points. Every 3-edge-coloring of a multipole induces a coloring or color-state of its semiedges. Multipoles have been extensively used in the study of snarks, that is, cubic graphs which are not 3-edge-colorable.

Given two multipoles M1 and M2 with the same number of semiedges, we say that M1 is reducible to M2 if the color-state set of M2 is a nonempty subset of the color state set of M1, and M2 has fewer vertices than M1.

Finding the maximum number a(n) of vertices of an irreducible multipole with n semiedges could be seen as the non-planar version of the study of irreducible configurations in the Four-Color Theorem.

The only known values are a(2) = 0, a(3) = 1 (both are trivial results), a(4) = 2 (Goldberg, 1981), and a(5) = 5 (Cameron, Chetwynd, and Watkins, 1987). For n = 6, Karabáš, Máčajová, and Nedela (2013) proved that a(6) >= 12.

It is known that a(n) is, at least, partially monotone in the sense that a(n) >= max{n, a(n-1)-1, a(n-2)}.

A conjecture of Jaeger and Swart (1980) states that every snark contains a cycle separating edge-cut of size at most six. If this were true, then a(6) would be the most interesting unknown value of a(n).

M. A. Fiol, A Boolean algebra approach to the construction of snarks, in Graph Theory, Combinatorics and Applications, vol. 1 (Eds. Y. Alavi, G. Chartrand, O. R. Oellermann, and A. J. Schwenk), John Wiley & Sons, New York (1991), 493-524.

M. A. Fiol, G. Mazzuoccolo, and E. Steffen, <a href="https://doi.org 10.37236/6848">Measures of edge-uncolorability of cubic graphs</a>, Electron. J. Combin. 25 (2018), no. 4, P4.54.

M. A. Fiol and J. Vilaltella, <a href="https://doi.org/10.37236/3629">Some results on the structure of multipoles in the study of snarks</a>, Electron J. Combin. 22(1) (2015), #P1.45.

M. Gardner, <a href="https://doi.org 10.1038%2Fscientificamerican0476-126"> Mathematical Games: Snarks, Boojums and other conjectures related to the four-color-map theorem</a>, Sci. Am. 234 (1976), no. 4, 126-130.

R. Isaacs, <a href="https://doi.org/10.2307/2319844">Infinite families of nontrivial trivalent graphs which are not Tait colorable</a>, Amer. Math. Monthly 82 (1975), no. 3, 221-239.

J. Karabáš, E. Máčajová, and R. Nedela, <a href="https://doi.org/10.1016/j.ejc.2012.07.019">6-decomposition of snarks</a>, European J. Combin. 34 (2013), no. 1, 111-122.

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Miquel A. Fiol, Apr 05 2024

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Given two multipoles M1 and M2 with the same number of semiedges, we say that M1 is reducible to M2 if the color-state set of M2 is a non-empty nonempty subset of the color state set of M1, and M2 has fewer vertices than M1.