A212592 a(n) is the difference between multiples of 7 with even and odd digit sum in base 6 in interval [0,6^n).

1, 6, 11, 106, 201, 2022, 3843, 38794, 73745, 744646, 1415547, 14293930, 27172313, 274381478, 521590643, 5266936010, 10012281377, 101102361990, 192192442603, 1940727511786, 3689262580969, 37253563629926, 70817864678883, 715107089849866

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, May 22 2012

Keywords

• nonn
• base
• easy

Comments

In general for all z, given a sequence of the form: a(n) is the difference between multiples of 2z+1 with even and odd digit sum in base 2z in interval [0,(2z)^n); then a(n) = (a(n+1) + a(n-1))/2 when n is even. The equation applies here where z=3. - Bob Selcoe, Jun 10 2014

Links

• Vladimir 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 (0,21,0,-35,0,7).

Crossrefs

Cf. A038754, A212500, A091042.

Programs

Mathematica

LinearRecurrence[{0, 21, 0, -35, 0, 7}, {1, 6, 11, 106, 201, 2022}, 24] (* Bruno Berselli, May 22 2012 *)

Formula

For n>=7, a(n) = 21*a(n-2)-35*a(n-4)+7*a(n-6).

G.f.: x*(1+6*x-10*x^2-20*x^3+5*x^4+6*x^5)/(1-21*x^2+35*x^4-7*x^6). [Bruno Berselli, May 22 2012]

a(n) = 2a(n-1) - a(n-2) when n is odd; a(n) = (a(n+1) + a(n-1))/2 when n is even. - Bob Selcoe, Jun 10 2014