A369305 Number of terms in A343524 that are less than 10^n.

1, 10, 19, 55, 91, 175, 259, 385, 511, 637, 763, 847, 931, 967, 1003, 1012, 1021, 1022, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023, 1023

Offset

0,2

Comments

The tallied terms (A343524) are palindromes with digits strictly increasing up to the midpoint.

Links

Table of n, a(n) for n=0..46.

Index entries for linear recurrences with constant coefficients, signature (1).

Formula

a(n) = 1023 for n >= 18. - Michael S. Branicky, Jan 22 2024

a(n) = Sum_{k=1..n+1} binomial(9,floor(k/2)). - Andrew Howroyd, Jan 22 2024

G.f.: (-x^18 - x^17 - 9*x^16 - 9*x^15 - 36*x^14 - 36*x^13 - 84*x^12 - 84*x^11 - 126*x^10 - 126*x^9 - 126*x^8 - 126*x^7 - 84*x^6 - 84*x^5 - 36*x^4 - 36*x^3 - 9*x^2 - 9*x - 1)/(x - 1). - Chai Wah Wu, Jun 15 2024

Example

For n = 0, 10^0 = 1, there is a single A343524 term less than 1: 0.
For n = 2, 10^2 = 100, there are 19 A343524 terms less than 100: 0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99.
Examples of A343524 terms less than 100000: 1661, 28982.

Mathematica

A369305[n_] := If[n >= 18, 1023, Sum[Binomial[9, Quotient[k, 2]], {k, n+1}]];
Array[A369305, 50, 0] (* Paolo Xausa, Mar 02 2026 *)

Prog

(PARI) a(n)=sum(k=1, min(n, 18)+1, binomial(9, k\2)) \\ Andrew Howroyd, Jan 22 2024
(Python)
from math import comb
def a(n):
    if n > 18: return 1023
    return 1+sum(comb(9, (digits+1)//2) for digits in range(1, n+1))
print([a(n) for n in range(47)]) # Michael S. Branicky, Jan 22 2024

Crossrefs

Cf. A134941, A173070, A343524.

Sequence in context: A219959 A307344 A330570 * A393416 A065198 A033866

Adjacent sequences: A369302 A369303 A369304 * A369306 A369307 A369308

Keyword

nonn,base,easy

Author

James S. DeArmon, Jan 19 2024

Extensions

a(11) and beyond from Michael S. Branicky, Jan 22 2024

Status

approved