A303050 Number of 2n-digit decimal numbers without leading zeros where each digit appears an even number of times.

9, 252, 10944, 617472, 41457024, 3141499392, 259346018304, 22749987520512, 2082519895670784, 196336888097144832, 18885576885968830464, 1841197721072909156352, 181107406524439376953344, 17918621160090649673859072, 1779590652288735614991335424

Offset

1,1

Links

T. V. Raziman, Table of n, a(n) for n = 1..30

Quora, How many 2n digit numbers are there such that each digit is repeated even (possibly 0) times?

Index entries for linear recurrences with constant coefficients, signature (220,-16368,489280,-5395456,14745600).

Formula

a(n) = 9/10 * 1/(2^10) * (2 * 10^(2n) + 20 * 8^(2n) + 90 * 6^(2n) + 240 * 4^(2n) + 420 * 2^(2n)) (proved in the Quora answers).

G.f.: -9 *(1111680*x^4 -229888*x^3 +11424*x^2 -192*x +1) *x / ((100*x-1) *(4*x-1) *(36*x-1) *(64*x-1) *(16*x-1)). - Alois P. Heinz, Apr 17 2018

Example

For n=1, the a(1)=9 numbers are 11, 22, 33, ..., 88, 99.
For n=2, the a(2)=252 numbers are 1001, 1010, 1100, 1111, 1122, 1212, 1221, ..., 9988, 9999.

Mathematica

LinearRecurrence[{220, -16368, 489280, -5395456, 14745600}, {9, 252, 10944, 617472, 41457024}, 20] (* Paolo Xausa, Mar 10 2024 *)

Crossrefs

Sequence in context: A075987 A135099 A073427 * A194790 A370357 A158621

Adjacent sequences: A303047 A303048 A303049 * A303051 A303052 A303053

Keyword

nonn,base,easy

Author

T. V. Raziman, Apr 17 2018

Status

approved