A212670 a(n) = 1/15*(128*n^5 + 320*n^4 + 80*n^3 - 200*n^2 + 92*n - 15).

27, 615, 3843, 14351, 40363, 94711, 195859, 368927, 646715, 1070727, 1692195, 2573103, 3787211, 5421079, 7575091, 10364479, 13920347, 18390695, 23941443, 30757455, 39043563, 49025591, 60951379, 75091807, 91741819, 111221447, 133876835, 160081263, 190236171

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, May 23 2012

Keywords

• nonn
• base
• easy

Comments

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).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177v2 [math.NT], 2007
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A038754, A084990, A091042, A212500, A212592, A212593, A212594, A212668, A212669.

Programs

Mathematica

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 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{27,615,3843,14351,40363,94711},30] (* Harvey P. Dale, Apr 30 2018 *)

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

Formula

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^7(Pi*i/(2*n+1))*sin(2*Pi*i/(2*n+1)).

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]