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%I #24 Sep 25 2023 07:51:19
%S 0,1,3,6,16,33,91,190,528,1105,3075,6438,17920,37521,104443,218686,
%T 608736,1274593,3547971,7428870,20679088,43298625,120526555,252362878,
%U 702480240,1470878641,4094354883,8572908966,23863649056,49966575153,139087539451
%N a(n) = (A077446(n) + 1)/2 for n >= 0.
%C a(n) and b(n) = A006452(n+1), for n >= 0, give the nonnegative solution of the equation binomial(a(n), 2) = b(n)^2 - 1.
%C This shows that the number of independent elements of an antisymmetric a(n) X a(n) matrix coincides with the number of independent elements of a traceless b(n) X b(n) matrix. The n = 0 case is trivial: 0 = 0. (The question about this coincidence was posed to W. L. by Martin Bordemann, Mar 03 1991.)
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-1,1).
%F a(n) = 6*a(n-2) - a(n-4) - 2, for n >= 0, with a(-4) = -32, a(-3) = -15, a(-2) = -5, a(-1) = -2.
%F O.g.f.: G(x) = x*(1 + 2*x - 3*x^2 - 2*x^3)/((1 - x)*(1 - 6*x^2 + x^4)) = x*(1 + 2*x - 3*x^2 - 2*x^3)/((1 - x)*(1 - 2*x - x^2)*(1 + 2*x - x^2)).
%F Bisection: a(2*k) = (5*S(k-1, 6) + S(k-2, 6) + 1)/2 and a(2*k+1) = (S(k, 6) + 5*S(k-1, 6) + 1)/2, for k >= 0, with the Chebyshev polynomials S(n, x) (A049310) with S(-2, x) = -1, S(-1, x) = 0, evaluated at x = 6. S(n, 6) = A001109(n+1).
%F Bisection: a(2*k) = (1 + 8*q(k) - p(k))/2 and a(2*k+1) = (1 + 8*q(k) + p(k))/2, for k >= 0, with p(k) = A001541(k) = S(k, 6) - 3*S(k-1, 6) and q(k) = A001109(k) = S(k-1, 6).
%F E.g.f.: (cosh(x) - cosh(sqrt(2)*x)*(cosh(x) - 3*sinh(x)) + sinh(x) - sqrt(2)*(cosh(x) - 2*sinh(x))*sinh(sqrt(2)*x))/2. - _Stefano Spezia_, Aug 29 2023
%e The solutions (a(n), b(n)) begin:
%e n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e ------------------------------------------------------------------------
%e a: 0 1 3 6 16 33 91 190 528 1105 3075 6438 17920 37521 104443 218686 ...
%e b: 1 1 2 4 11 23 64 134 373 781 2174 4552 12671 26531 73852 154634 ...
%t LinearRecurrence[{1, 6, -6, -1, 1}, {0, 1, 3, 6, 16}, 31] (* _Robert P. P. McKone_, Aug 29 2023 *)
%Y Cf. A001109, A001541, A006452, A049310, A077446.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Aug 15 2023
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