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%I #22 Apr 30 2018 09:07:13
%S 27,615,3843,14351,40363,94711,195859,368927,646715,1070727,1692195,
%T 2573103,3787211,5421079,7575091,10364479,13920347,18390695,23941443,
%U 30757455,39043563,49025591,60951379,75091807,91741819,111221447,133876835,160081263,190236171
%N a(n) = 1/15*(128*n^5 + 320*n^4 + 80*n^3 - 200*n^2 + 92*n - 15).
%C a(n) is the difference between numbers of nonnegative multiples of 2*n+1 with even and odd digit sum in base 2*n in interval [0, 128*n^7).
%H Colin Barker, <a href="/A212670/b212670.txt">Table of n, a(n) for n = 1..1000</a>
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/0710.3177">On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m</a>, arXiv:0710.3177v2 [math.NT], 2007
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^7(Pi*i/(2*n+1))*sin(2*Pi*i/(2*n+1)).
%F G.f.: x*(27+453*x+558*x^2-22*x^3+7*x^4+x^5)/(1-x)^6. [_Bruno Berselli_, May 24 2012]
%t Table[(1/15) (8 n^2 - 4 n + 1) (16 n^3 + 48 n^2 + 32 n - 15), {n, 29}] (* _Bruno Berselli_, May 24 2012 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{27,615,3843,14351,40363,94711},30] (* _Harvey P. Dale_, Apr 30 2018 *)
%o (PARI) Vec(x*(27+453*x+558*x^2-22*x^3+7*x^4+x^5)/(1-x)^6 + O(x^50)) \\ _Colin Barker_, Dec 01 2015
%Y Cf. A038754, A084990, A091042, A212500, A212592, A212593, A212594, A212668, A212669.
%K nonn,base,easy
%O 1,1
%A _Vladimir Shevelev_ and _Peter J. C. Moses_, May 23 2012
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