OFFSET
1,1
COMMENTS
This sequence is the union of 22*n-17, 22*n-13, 22*n-9, and 22*n-5, and A008604(22*n), for n>0.
This sequence is the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; see example.
The sequence numbers with both odd first and last digits are either palindromes or they have corresponding reversed digit numbers, e.g., 105, 501. Prime numbers in this sequence are also in A007500 (reversal primes). Some examples are 13, 17, 31, 71, 79, 97, 101.
The sequence numbers with even first digits and last digits of 2, 4, 6 or 8, are either palindromes or they have corresponding reversed digit numbers in this sequence.
The candidate Lychrel numbers, 295, 493, 691, 1677, 1765, 1857, 1945, 1997, 3493, are in this sequence.
LINKS
Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, 196 Algorithm.
Eric Weisstein's World of Mathematics, Lychrel Number
Eric Weisstein's World of Mathematics, Palindromic Number Conjecture
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
FORMULA
a(n) = a(n-1) + a(n-5) - a(n-6). - Colin Barker, Nov 20 2014
G.f.: x*(5*x^4+4*x^3+4*x^2+4*x+5) / ((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Nov 20 2014
Proof that a(n) = a(n-1) + a(n-5) - a(n-6): the sequence a(n) is a concatenation of the sequences [5+22*i, 9+22*i, 13+22*i, 17+22*i, 22+22*i] for i = 0,1,2,..., so it is clear that a(n-1) = a(n-6) + 22 and a(n) = a(n-5) + 22. - Chai Wah Wu, Jan 01 2015
EXAMPLE
Sequence consists of the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; e.g.,
for k = 5, sqrt( 20 - 2 + 3 + 1) = sqrt(22) = 4.6904;
for k = 6, sqrt( 24 - 2 + 3 + 0) = sqrt(25) = 5;
for k = 21, sqrt( 84 - 7 + 3 + 1) = sqrt(81) = 9;
for k = 44, sqrt(176 - 15 + 3 + 0) = sqrt(164) = 12.8062;
for k = 45, sqrt(180 - 15 + 3 + 1) = sqrt(169) = 13.
Of these, the only integer values are 5, 9, 13, so they are in the sequence.
MATHEMATICA
a247128[n_Integer] := Select[Range[n], MemberQ[{0, 5, 9, 13, 17}, Mod[#, 22]] &]; a247128[211] (* Michael De Vlieger, Nov 23 2014 *)
PROG
(Python)
from math import *
for n in range(0, 100001):
..if (sqrt(4*n-ceil(n/3)+3+n%2))%1==0:print(int(sqrt(4*n-ceil(n/3)+3+n%2)), end=", ")
(PARI) isok(n) = m = n % 22; (m==0) || (m==5) || (m==9) || (m==13) || (m==17);
select(x->isok(x), vector(200, i, i)) \\ Michel Marcus, Nov 28 2014
(Python)
A247128_list = [n for n in range(1, 10**5) if (n % 22) in {0, 5, 9, 13, 17}]
# Chai Wah Wu, Dec 31 2014
(Python)
A247128_list, l = [], [5, 9, 13, 17, 22]
for _ in range(10**5):
....A247128_list.extend(l)
....l = [x+22 for x in l] # Chai Wah Wu, Jan 01 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Karl V. Keller, Jr., Nov 19 2014
STATUS
approved