A212668 a(n) = (16/3)*(n+1)*n*(n-1) + 8*n^2 + 1.

9, 65, 201, 449, 841, 1409, 2185, 3201, 4489, 6081, 8009, 10305, 13001, 16129, 19721, 23809, 28425, 33601, 39369, 45761, 52809, 60545, 69001, 78209, 88201, 99009, 110665, 123201, 136649, 151041, 166409, 182785, 200201, 218689, 238281, 259009, 280905, 304001

Offset: 1

Author

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

Keywords

• nonn
• 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, 32*n^5).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A038754, A084990, A091042, A212500, A212592.

Programs

Magma

[(16/3)*(n+1)*n*(n-1)+8*n^2+1: n in [1..40]]; // Vincenzo Librandi, Dec 01 2015

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {9, 65, 201, 449}, 40] (* Vincenzo Librandi, Dec 01 2015 *)
CoefficientList[Series[x (9+29x-5x^2-x^3)/(1-x)^4,{x,0,40}],x] (* Harvey P. Dale, Mar 29 2023 *)

PARI

a(n)=16*(n+1)*n*(n-1)/3+8*n^2+1 \\ Charles R Greathouse IV, Oct 07 2015

PARI

Vec(x*(9+29*x-5*x^2-x^3)/(1-x)^4 + O(x^100)) \\ Colin Barker, Nov 30 2015

Formula

a(n) = 2/(2*n+1)*Sum_{i=1..n} tan^5(Pi*i/(2*n+1)) * sin(2*Pi*i/(2*n+1)).

G.f.: x*(9+29*x-5*x^2-x^3) / (1-x)^4. - Colin Barker, Nov 30 2015