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Polynomials s_k(x) which form the Sheffer sequence for f(t)=-(2t)/(1-t^2) (1) and have exponential generating function ...

A set of numbers obeying a pattern like the following: 91·37 = 3367 (1) 9901·3367 = 33336667 (2) 999001·333667 = 333333666667 (3) 99990001·33336667 = 3333333366666667 (4) 4^2 ...

An odd power is a number of the form m^n for m>0 an integer and n a positive odd integer. The first few odd powers are 1, 8, 27, 32, 64, 125, 128, 216, 243, 343, 512, ... ...

An out-shuffle, also known as a perfect shuffle (Golomb 1961), is a riffle shuffle in which the top half of the deck is placed in the right hand, and cards are then ...

Specifying three sides uniquely determines a triangle whose area is given by Heron's formula, K=sqrt(s(s-a)(s-b)(s-c)), (1) where s=1/2(a+b+c) (2) is the semiperimeter of the ...

A surface given by the parametric equations x(u,v) = u (1) y(u,v) = v (2) z(u,v) = 1/3u^3-1/2v^2. (3) The coefficients of the coefficients of the first fundamental form are E ...

The surface given by the parametric equations x = asinu (1) y = asinv (2) z = asin(u+v). (3) It is a sextic surface with algebraic equation (4) The coefficients of the first ...

A surface generated by the parametric equations x(u,v) = ucosv (1) y(u,v) = usinv (2) z(u,v) = vcosu. (3) The above image uses u in [-4,4] and v in [0,6.25]. The coefficients ...

Trigonometric identities which prove useful in the construction of map projections include (1) where A^' = A-C (2) B^' = 2B-4D (3) C^' = 4C (4) D^' = 8D. (5) ...

Construction of the angle pi/4=45 degrees produces an isosceles right triangle. Since the sides are equal, sin^2theta+cos^2theta=2sin^2theta=1, (1) so solving for ...