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%I #55 Sep 10 2022 07:33:12
%S 6,31,44,69,82,107,120,145,158,183,196,221,234,259,272,297,310,335,
%T 348,373,386,411,424,449,462,487,500,525,538,563,576,601,614,639,652,
%U 677,690,715,728,753,766,791,804,829,842,867,880,905
%N Numbers congruent to 6 or 31 mod 38.
%C A007310(a(n) + 1) is always a multiple of 19.
%C A020639(A007310(a(n) + 1)) = 5, 7, 11, 13, 17, or 19.
%C It equals 5 when n is a term in A273669.
%C It equals 7 when n is congruent to 3 or 12 (mod 14) but not a term in A273669.
%C It equals 11 when n is congruent to 4 or 19 (mod 22) but not a case where it equals 5 or 7.
%C It equals 13 when n is congruent to 5 or 22 (mod 26) (one more than a term in A306285) but not a case where it equals 5, 7, or 11.
%C It equals 17 when n is congruent to 6 or 29 (mod 34) but not a case where it equals 5, 7, 11, or 13.
%C For all other cases, it equals 19.
%C a(n) and (n - 1) have the same remainder (mod 6) (see A010875).
%H Colin Barker, <a href="/A306331/b306331.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F G.f.: x*(6 + 25*x + 7*x^2) / ((1 - x)^2*(1 + x)). - _Colin Barker_, Feb 09 2019
%F a(n) = a(n - 1) + a(n - 2) - a(n - 3) for n > 3.
%F a(n) = 19*n - 10 + 3*(-1)^n. - _Wesley Ivan Hurt_, Mar 10 2019
%F a(n) = 19*n - 13 when n is odd and 19*n - 7 when n is even.
%F a(n) = 19*n - (A040031(n + 1) + 1).
%F E.g.f.: 7 + (19*x - 10)*exp(x) + 3*exp(-x). - _David Lovler_, Sep 10 2022
%p seq(seq(38*i+j, j=[6, 31]), i=0..200);
%t Select[Range[200], MemberQ[{6, 31}, Mod[#, 38]] &]
%t Union[38Range[30] - 32, 38Range[30] - 7] (* _Alonso del Arte_, Feb 08 2019 *)
%o (PARI) for(n=6, 905, if((n%38==6) || (n%38==31), print1(n, ", ")))
%o (PARI) Vec(x*(6 + 25*x + 7*x^2) / ((1 - x)^2*(1 + x)) + O(x^40)) \\ _Colin Barker_, Feb 09 2019
%o (Scala) (6 to 1108 by 38).union(31 to 1133 by 38).sorted // _Alonso del Arte_, Feb 08 2019
%Y Cf. A005408, A005843, A007310, A010875, A020639, A040031, A273669, A306285.
%K nonn,easy
%O 1,1
%A _Davis Smith_, Feb 07 2019
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