A083834 Palindromes of the form 5p + 1 where p is also a palindrome. Palindromes arising in A083833.

6, 11, 111, 606, 656, 1111, 11111, 60106, 60606, 65156, 65656, 111111, 1111111, 6010106, 6015106, 6060606, 6065606, 6510156, 6515156, 6560656, 6565656, 11111111, 111111111, 601010106, 601060106, 601515106, 601565106, 606010606

Offset

1,1

Comments

From Michael S. Branicky, Jun 13 2021: (Start)

All terms start and end with the digit 6, except those consisting of d+1 1's, which arise from palindromes of d 2's.

If a(n) > 6 starts with a 6, then its second and second-to-last digits are either both 0 or both 5; and if it has 5 or more digits, its third and third-to-last digits are either both 1 or both 6; thus, terms in the latter case only start with 111, 601, 606, 651, 656.

Using ^ to denote repeated concatenation, contains terms of the forms 1^(d+1), arising from palindromes of the form 2^d; (60)^d 6, arising from (12)^d 1; and (65)^d 6, arising from (13)^d 1; among other patterns.

(Conjectures)

All terms contain only the digits {0, 1, 5, 6}.

For d odd, the only term with d digits is 1^d; equivalently, all terms starting with 6 have odd length. (End)

Links

Michael S. Branicky, Table of n, a(n) for n = 1..8217 (all terms where p has <= 26 digits)

Mathematica

Select[5Select[Range@100000, PalindromeQ]+1, PalindromeQ] (* Giorgos Kalogeropoulos, Jun 11 2021 *)

Prog

(Python)
from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all positive d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]:
            t = int(left + mid + right)
            if t > 0: yield t
def ok(pal): return ispal(5*pal+1)
print([5*p+1 for d in range(1, 10) for p in pals(d) if ok(p)]) # Michael S. Branicky, Jun 11 2021
(PARI) is(n) = my(x=(n-1)/5, dn=digits(n), dx); if(x!=ceil(x), return(0)); dx=digits(x); dn==Vecrev(dn) && dx==Vecrev(dx) && n>1 \\ Felix Fröhlich, Jun 11 2021

Crossrefs

Cf. A083829, A083830, A083832, A083833.

Sequence in context: A351991 A318858 A136980 * A136978 A145746 A024267

Adjacent sequences: A083831 A083832 A083833 * A083835 A083836 A083837

Keyword

base,nonn

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 09 2003

Extensions

Corrected and extended by Ray Chandler, May 21 2003

Status

approved