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A247128 Positive numbers that are congruent to {0,5,9,13,17} mod 22.

%I A247128 #57 May 22 2025 10:21:40
%S A247128 5,9,13,17,22,27,31,35,39,44,49,53,57,61,66,71,75,79,83,88,93,97,101,
%T A247128 105,110,115,119,123,127,132,137,141,145,149,154,159,163,167,171,176,
%U A247128 181,185,189,193,198,203,207,211
%N A247128 Positive numbers that are congruent to {0,5,9,13,17} mod 22.
%C A247128 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.
%C A247128 This sequence is the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; see example.
%C A247128 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.
%C A247128 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.
%C A247128 The candidate Lychrel numbers, 295, 493, 691, 1677, 1765, 1857, 1945, 1997, 3493, are in this sequence.
%H A247128 Karl V. Keller, Jr., <a href="/A247128/b247128.txt">Table of n, a(n) for n = 1..100000</a>
%H A247128 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/196-Algorithm.html">196 Algorithm.</a>
%H A247128 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LychrelNumber.html">Lychrel Number</a>
%H A247128 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PalindromicNumberConjecture.html">Palindromic Number Conjecture</a>
%H A247128 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A247128 a(n) = a(n-1) + a(n-5) - a(n-6). - _Colin Barker_, Nov 20 2014
%F A247128 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
%F A247128 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
%e A247128 Sequence consists of the integer values of sqrt(4*k - ceiling(k/3) + 3 + k mod 2), for k>0; e.g.,
%e A247128 for k =  5, sqrt( 20 -  2 + 3 + 1) = sqrt(22)  =  4.6904;
%e A247128 for k =  6, sqrt( 24 -  2 + 3 + 0) = sqrt(25)  =  5;
%e A247128 for k = 21, sqrt( 84 -  7 + 3 + 1) = sqrt(81)  =  9;
%e A247128 for k = 44, sqrt(176 - 15 + 3 + 0) = sqrt(164) = 12.8062;
%e A247128 for k = 45, sqrt(180 - 15 + 3 + 1) = sqrt(169) = 13.
%e A247128 Of these, the only integer values are 5, 9, 13, so they are in the sequence.
%t A247128 a247128[n_Integer] := Select[Range[n], MemberQ[{0, 5, 9, 13, 17}, Mod[#, 22]] &]; a247128[211] (* _Michael De Vlieger_, Nov 23 2014 *)
%o A247128 (Python)
%o A247128 from math import *
%o A247128 for n in range(0,100001):
%o A247128   if (sqrt(4*n-ceil(n/3)+3+n%2))%1==0:print(int(sqrt(4*n-ceil(n/3)+3+n%2)),end=",")
%o A247128 (PARI) isok(n) = m = n % 22; (m==0) || (m==5) || (m==9) || (m==13) || (m==17);
%o A247128 select(x->isok(x), vector(200, i, i)) \\ _Michel Marcus_, Nov 28 2014
%o A247128 (Python)
%o A247128 A247128_list = [n for n in range(1,10**5) if (n % 22) in {0,5,9,13,17}]
%o A247128 # _Chai Wah Wu_, Dec 31 2014
%o A247128 (Python)
%o A247128 A247128_list, l = [], [5,9,13,17,22]
%o A247128 for _ in range(10**5):
%o A247128     A247128_list.extend(l)
%o A247128     l = [x+22 for x in l] # _Chai Wah Wu_, Jan 01 2015
%Y A247128 Cf. A008604, A002113 (palindromes), A007500 (reversible primes).
%Y A247128 Cf. A023108.
%K A247128 nonn,easy
%O A247128 1,1
%A A247128 _Karl V. Keller, Jr._, Nov 19 2014