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Form a sequence from an alphabet of letters [1,n] such that there are no consecutive letters and no alternating subsequences of length greater than d. Then the sequence is a ...

The evolute of the cardioid x = cost(1+cost) (1) y = sint(1+cost) (2) is the curve x_e = 2/3a+1/3acostheta(1-costheta) (3) y_e = 1/3asintheta(1-costheta), (4) which is a ...

For the cardioid given parametrically as x = a(1+cost)cost (1) y = a(1+cost)sint, (2) the involute is given by x_i = 2a+3acostheta(1-costheta) (3) y_i = ...

A concordance between knots K_0 and K_1 in S^3 is a locally flat cylinder C=S^1×[0,1] embedded in S^3×[0,1] in such a way that the ends S^1×{1} are embedded in S^3×{i} as ...

The Elsasser function is defined by the integral E(y,u)=int_(-1/2)^(1/2)exp[-(2piyusinh(2piy))/(cosh(2piy)-cos(2pix))]dx. (1) Special values include E(0,u) = 1 (2) E(y,0) = ...

The number of alternating permutations for n elements is sometimes called an Euler zigzag number. Denote the number of alternating permutations on n elements for which the ...

The inverse of the Laplace transform F(t) = L^(-1)[f(s)] (1) = 1/(2pii)int_(gamma-iinfty)^(gamma+iinfty)e^(st)f(s)ds (2) f(s) = L[F(t)] (3) = int_0^inftyF(t)e^(-st)dt. (4)

The gear curve is a curve resembling a gear with n teeth given by the parametric equations x = rcost (1) y = rsint, (2) where r=a+1/btanh[bsin(nt)], (3) where tanhx is the ...

A pyramidal number of the form n(n+1)(4n-1)/6, The first few are 1, 7, 22, 50, 95, ... (OEIS A002412). The generating function of the hexagonal pyramidal numbers is ...

If xsinalpha=sin(2beta-alpha), then (1+x)int_0^alpha(dphi)/(sqrt(1-x^2sin^2phi))=2int_0^beta(dphi)/(sqrt(1-(4x)/((1+x)^2)sin^2phi)).