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User: Rick Mabry

Rick Mabry has authored 2 sequences.

A343273 a(n) is the number of geometrically distinct edge-unfoldings of the regular n-gonal cupola.

308, 3030, 29757, 294327, 2911142, 28814940, 285214743, 2823311133, 27947663768, 276653115090, 2738581182417, 27109156615827, 268352962161482, 2656420444277880, 26295851254778283, 260302091898387033, 2576725065493516028, 25506948561006315150

Offset: 3

Rick Mabry, Apr 10 2021

The term "regular" applies only to the regular n-gon and 2n-gon (the "top and bottom" of the cupola), the other faces (the "sides") being n isosceles triangles and n sufficiently long rectangles. For n=3,4,5, regular triangles and squares can be used for the sides. That applies to n=6 if a two-sided (flat) polyhedron is allowed.

The first 25 terms of the auxiliary sequence c(n) in the Formula and Mathematica program match the 25 terms listed for sequence A085376.

Zsolt Lengvárszky and Rick Mabry, Enumerating nets of prism-like polyhedra, Acta Sci. Math. (Szeged) 83 (2017), no. 3-4, 377-392.
Wikipedia, Cupola
Index entries for linear recurrences with constant coefficients, signature (11,-1,-109,109,1,-11,1).

Cf. A085376; see the sequence c(n) in the Formula and Mathematica program, but note that A085376 has only been conjectured to be the same as c(n).

a[n_]:=Sum[c[k],{k,1,2n-1}]+(1/2)c[2n]+If[OddQ[n],(1/2)c[n],c[n]];
c[1] = 1; c[2] = 3; c[3] = 11; c[4] = 30;
c[m_] := c[m] = 10 c[m - 2] - c[m - 4];

Recursively define the sequence c(m) as follows: Let c(1) = 1, c(2) = 3, c(3) = 11, c(4) = 30, and for m > 4, let c(m) = 10*c(m-2) - c(m-4). Then for all n >= 3, the sequence a(n) can be given by a(n) = (c(2*n+1) + 5*c(2*n) - c(2*n-1) - c(2*n-2) - 5)/8 + (3 + (-1)^n)*c(n)/4.
a(n) = (c(2*n+1) + 5*c(2*n) - c(2*n-1) - c(2*n-2) - 5)/8 + (3 + (-1)^n)*c(n)/4 for n >= 3 where c(m) = 10*c(m-2) - c(m-4) for m > 4 and c(1) = 1, c(2) = 3, c(3) = 11, c(4) = 30.
G.f.: x^3*(308 - 358*x - 3265*x^2 + 3602*x^3 - 360*x^5 + 33 x^6)/(1 - 11*x + x^2 + 109*x^3 - 109*x^4 - x^5 + 11*x^6 - x^7). - Stefano Spezia, Apr 10 2021

A094907 Number of different nontrivial two-digit cancellations of the form (xy)/(zx) = y/z in base n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 2, 2, 4, 0, 4, 0, 2, 6, 7, 0, 4, 0, 4, 10, 6, 0, 6, 6, 4, 6, 10, 0, 6, 0, 4, 8, 6, 6, 21, 0, 2, 6, 18, 0, 6, 0, 4, 18, 10, 0, 8, 10, 10, 12, 12, 0, 6, 16, 22, 14, 6, 0, 10, 0, 2, 12, 21, 12, 20, 0, 4, 10, 22, 0, 10, 0, 2, 12, 20, 14, 24, 0, 8, 24, 8, 0, 10, 28, 6, 6, 18, 0, 10

Offset: 2

Rick Mabry (rmabry(AT)pilot.lsus.edu), Jun 16 2004

Trivial cancellations are of the form xx/xx=x/x, e.g. 44/44 = 4/4.

			a(10) = 4 because we have the four nontrivial base-10 cancellations 64/16 = 4/1, 65/26 = 5/2, 95/19 = 5/1, 98/49 = 8/4.

Boas, R. P. "Anomalous Cancellation," Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113-129, 1979.

Ludovic Schwob, Table of n, a(n) for n = 2..1000
R. P. Boas, Anomalous Cancellation, The Two Year College Mathematics Journal, Vol. 3, No. 2 (Autumn 1972), 21-24.
Eric Weisstein's World of Mathematics, Anomalous Cancellation.

a[n_]:= Length[(DeleteCases[ #1, {u_, u_, u_}] & )[ Position[Table[(n*x + y)/(n*z + x) == y/z, {x, 1, n - 1}, {y, 1, x - 1}, {z, 1, y - 1}], True]]]

From Ludovic Schwob, Nov 10 2020: (Start)
a(n)=0 if and only if n is prime.
a(n) is odd if and only if n is an even square. (End)

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User: Rick Mabry

A343273 a(n) is the number of geometrically distinct edge-unfoldings of the regular n-gonal cupola.

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