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 A215929 Laminated lattice kissing numbers in A002336 of the form (2^k - 2)*(2*k - 2) for some k. 5
 0, 2, 24, 240, 196560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The corresponding values of k = 1, 0, 3, 5, 13 = A215795(n) + (-1)^A215795(n) = (A060728(n) - 3) + (-1)^(A060728(n)) The average number of spheres/dimension for this sequence maps 1:1 to the "Ramanujan-Nagell pronic numbers" {0, 2, 6, 30, 8190}, all of which are 2 less than a power of 2 and have the form n^2 + n [kissing numbers for the A_n series of lattices (see Conway and Sloane p. 109)], equal to twice a triangular number (= 2*A076046(n)). Dimension # * Ramanujan-Nagell Pronic Number = {0, 1, 4, 8, 24} * {0, 2, 6, 30, 8190} = Lambda_{0, 1, 4, 8, 24}, where Lambda_n is the n-dimensional laminated lattice kissing number. As is demonstrated in A060728, there is a close relationship between the Ramanujan-Nagell pronic numbers and the complex solutions to the quadratic equation n^2 - n + 2 = 0; n = (1 + i*sqrt(7))/2 and (1 - i*sqrt(7))/2, perhaps interesting since Kleinian lattices are lattices over Z[(1+sqrt(-7))/2]. Also, laminated lattice kissing numbers in dimension (2*m - 3 + m(mod 2)) = A042948_(m - 1), where m == Fibonacci(c + 1) == ceiling[e^((c - 1)/2)] = {1, 2, 3, 5, 13} = (A215795(n) + 1) = (A060728(n) - 2), and c is a positive solution to {n in N | 2 cos (2*Pi/n) is in Z} = {1,2,3,4,6} (see A217290). {c}, therefore, consists of the complete set of positive solutions to the short proof of the Crystallographic Restriction Theorem. See A060728 for a possible relationship between this sequence and the number of conjugacy classes in Clifford Group CL(m); m = {1, 2, 3, 5, 13}. See A216162 for a possible relationship between this sequence, the Pell Numbers (A000129), n such that n^2-1 is a triangular number (A006452), and the indices of the Sophie Germain triangular numbers (A216134). Comment from N. J. A. Sloane, Jan 24 2016: The above comments should not be taken too seriously. There is no such thing as "the laminated lattice kissing number in dimension n"! See Sphere Packings, Lattices and Groups, Chapter 6. LINKS J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 3rd, 1999. See Chapter 6. G. Nebe and N. J. A. Sloane, A Catalogue of Lattices Pfender and Ziegler, Kissing Numbers, Sphere Packings, and Some Unexpected Proofs Eric W. Weisstein, MathWorld: Ramanujan's Square Equation Wikipedia, Crystallographic Restriction Theorem FORMULA Where m = {1, 2, 3, 5, 13} =  (A060728(n) - 2), and n = {0, 2, 24, 240, 196560}, then: n == (2^(m - 1 - (-1)^m) - 2)*(2*(m - 1 - (-1)^m) - 2) n == (A049332(m) - 4)*A005843(m - 1) = A101622(m)*A005843(m - 1)  = (m - 1)*(2^(m + 1) - (-1)^m  - 5) n == A000918(m) * A042948_(m - 1) =  (2^m - 2)*(2*m - 3 + m(mod 2)) n == (A002249(m)^2 - 1)/4* A042948_(m - 1)  =  ((((1 + i*sqrt(7))/2)^m + ((1 - i*sqrt(7))/2)^m)^2 - 1)/4 * (2*m - 3 + m(mod 2)) n == ((A107920(2*m))^2 - 1)/4* A042948_(m - 1)  =  (((((1 + i*sqrt(7))/2)^(2*m) - ((1 - i*sqrt(7))/2)^(2*m))/(i*sqrt(7)))^2 - 1)/4 * (2*m - 3 + m(mod 2)) EXAMPLE For x = {1, 0, 3, 5, 13} = (A060728(n) - 3)-(-1)^A060728(n), then: (2^1 - 2)*(2*1 - 2) = 0*0 = 0 (2^0 - 2)*(2*0 - 2) = -1*-2 = 2 (2^3 - 2)*(2*3 - 2) = 6*4 = 24 (2^5 - 2)*(2*5 - 2) = 30*8 = 240 (2^13 - 2)*(2*13 - 2) = 196560 For m = 13, then: (2^(13 - 1 - (-1)^13) - 2)*(2*(13 - 1 - (-1)^13) - 2) = 196560 (13 - 1)*(2^(13 + 1) - (-1)^13  - 5) = 196560 (2^13 - 2)*(2*13 - 3 + 13(mod 2)) = 196560 ((((1 + i*sqrt(7))/2)^13 + ((1 - i*sqrt(7))/2)^13)^2 - 1)/4 * (2*13 - 3 + 13(mod 2)) = 196560 (((((1 + i*sqrt(7))/2)^(2*13) - ((1 - i*sqrt(7))/2)^(2*13))/(i*sqrt(7)))^2 - 1)/4 * (2*13 - 3 + 13(mod 2)) = 196560 CROSSREFS Cf. A002336, A076046, A038198, A215797, A180445, A215795, A060728, A049332, A042948 where A076046  = {0, 1, 3, 15, 4095} A038198 = {1, 3, 5, 11, 181} A215797 = {0, 1, 2, 5, 90} A180445 = {1, 2, 3, 6, 91} A215795  = {0, 1, 2, 4, 12} A060728  = {3, 4, 5, 7, 15} A049332(A060728(n) - 2) = {4, 5, 10, 34, 8194} A042948(A060728(n) - 3) = {0, 1, 4, 8, 24} Then a(n) = 2* A042948(A060728(n) - 3)*A076046(n) a(n) = A042948(A060728(n) - 3)*(((A038198(n) - 1)/2)^2 + ((A038198(n) - 1)/2)) a(n) = A042948(A060728(n) - 3)*(A215797(n)^2 + A215797(n)) a(n) = A042948(A060728(n) - 3)*(A180445(n)^2 - A180445(n)) a(n) = A042948(A060728(n) - 3)*(2^(A215795(n) + 1) - 2) a(n) = A042948(A060728(n) - 3)*(2^(A060728(n) - 2) - 2) a(n) = (2*A060728(n) - 6)*(A049332(A060728(n) - 2) - 4) Cf. A217290, A002249 Sequence in context: A143407 A228619 A252764 * A132596 A099669 A019520 Adjacent sequences:  A215926 A215927 A215928 * A215930 A215931 A215932 KEYWORD nonn,more AUTHOR Raphie Frank, Aug 27 2012 EXTENSIONS Sequence renamed and comments, formula and example sections revised by Raphie Frank, Dec 11 2015 STATUS approved

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Last modified April 2 18:43 EDT 2020. Contains 333189 sequences. (Running on oeis4.)