A343524 -id:A343524 - cp's OEIS frontend

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

James S. DeArmon, Jan 19 2024

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

			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.

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

Cf. A134941, A173070, A343524.

a(n)=sum(k=1, min(n,18)+1, binomial(9,k\2)) \\ Andrew Howroyd, Jan 22 2024

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

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

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

A062351 Palindromic primes with strictly increasing digits up to the middle and then strictly decreasing.

Original entry on oeis.org

2, 3, 5, 7, 11, 131, 151, 181, 191, 353, 373, 383, 787, 797, 12421, 12721, 12821, 13831, 13931, 14741, 17971, 34543, 34843, 35753, 1235321, 1245421, 1257521, 1268621, 1278721, 1456541, 1469641, 1489841, 1579751, 1589851, 3479743

Offset: 1

Amarnath Murthy, Jun 23 2001

The last term of the finite series is a(63) = 123467898764321.

			13831 belongs to the sequence as it is a palindromic prime in which the digits are increasing up to the middle digit 8 and then decreasing.

Nathaniel Johnston, Table of n, a(n) for n = 1..63 (complete sequence)
Patrick De Geest, World of Palindromic Primes

Cf. A343524 (strictly increasing palindromes), A062352, A084836.

from sympy import isprime
from itertools import combinations
def agen():
  for digits in range(1, 19):
    for left in combinations("123456789", (digits+1)//2):
      left = "".join(left)
      yield int(left + (left[:digits//2])[::-1])
print(list(filter(isprime, agen()))) # Michael S. Branicky, Apr 25 2021

Corrected and edited by Patrick De Geest, Jun 07 2003

cp's OEIS Frontend

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

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