A137180 Number of palindromes in the range [1,n] inclusive.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 0

Author

N. J. A. Sloane, Apr 21 2008

Keywords

• nonn
• base

References

• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.

Crossrefs

Cf. A002113, A136522, A136687.

Programs

Mathematica

nthPalindrome[n_] := Block[{q = n + 1 - 10^Floor[ Log10[n + 1 - 10^Floor[ Log10[ n/10]] ]], c = Sum[ Floor[ Floor[ n/(11*10^(k - 1) - 1)]/(Floor[ n/(11*10^(k - 1) - 1)] - 1/10)] - Floor[ Floor[ n/(2*10^k - 1)]/(Floor[ n/(2*10^k - 1)] - 1/10)], {k, Floor[ Log10[ n]] }]}, Mod[q, 10]*11^c*10^Floor[ Log10[ q]] + Sum[ Floor[ Mod[q, 10^(k + 1)]/10^k]*10^(Floor[ Log10[ q]] - k) (10^(2 k + c) + 1) , {k, Floor[ Log10[ q]] }]];
palindromicPi[n_] := Block[{q = Floor[ n*10^-Floor[ Log10[ 10 n]/2]]}, r = q + 10^(Floor[ Log10[ q]] + Mod[ Floor[ Log10[ n]], 2]) - 1; r + Floor[ Tanh[n - nthPalindrome[ r]] ]]; (* after the work of Eric A. Schmidt, see A002113 *)
f[n_] := If[n < 1, 0, palindromicPi@ n]; Array[f, 75, 0] (* Robert G. Wilson v, Sep 22 2014 *)

Python

def A137180(n):
    l = len(s:=str(n))
    k = l+1>>1
    return n//10**(l-k)-(int(s[k-1::-1])>n%10**k)+10**(k-1+(l&1^1))-1 # Chai Wah Wu, Jul 24 2024

Formula

a(n) = A136687(n) - 1.