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%I #40 May 22 2023 11:37:35
%S 0,2,5,8,9,11,14,17,18,20,23,26,27,29,32,35,36,38,41,44,45,47,50,53,
%T 54,56,59,62,63,65,68,71,72,74,77,80,81,83,86,89,90,92,95,98,99,101,
%U 104,107,108,110,113,116,117,119,122,125,126,128,131,134,135,137
%N Numbers that are congruent to {0, 2, 5, 8} mod 9.
%C Also the set of nonnegative integers expressible as (x + 2y)(2x + y) for integer x and y, where integers of the form 3k + 2 are given by x = 2k + 1, y = -k, and integers of the form 9k are given by x = 2k - 1, y = 2 - k. - _Drake Thomas_, Nov 10 2022
%C From _Klaus Purath_, Feb 21 2023: (Start)
%C The sum of digits of any term belongs to the sequence. Also the products of an odd number of terms as well as products of one term each of this sequence and one term of A056991 are members. The products of an even number of terms belong to A056991.
%C Nonnegative integers of the forms 2*x^2 + (2*m+1)*x*y + ((m^2+m-2)/2)*y^2, for integers m. This includes the formula given by Drake Thomas above.
%C The union of this sequence and A056991 is closed under multiplication. (End)
%H Harvey P. Dale, <a href="/A174438/b174438.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F a(n) = 3*(n-floor(n/4)) - (3 - i^n - (-i)^n - (-1)^n)/4 where i=sqrt(-1), offset=0.
%F From _Wesley Ivan Hurt_, Jun 07 2016: (Start)
%F G.f.: x^2*(2 + 3*x + 3*x^2 + x^3)/((x-1)^2*(1 + x + x^2 + x^3)).
%F a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
%F a(n) = (18*n - 15 + i^(2*n) + (3-i)*i^(-n) + (3+i)*i^n)/8 where i=sqrt(-1). (End)
%p seq(3*(n-floor(n/4))-(3-I^n-(-I)^n-(-1)^n)/4, n=0..100);
%t Table[(18n-15+I^(2n)+(3-I)*I^(-n)+(3+I)*I^n)/8, {n, 80}] (* _Wesley Ivan Hurt_, Jun 07 2016 *)
%t Select[Range[0,150],MemberQ[{0,2,5,8},Mod[#,9]]&] (* _Harvey P. Dale_, Jan 02 2019 *)
%t LinearRecurrence[{1,0,0,1,-1},{0,2,5,8,9},70] (* _Harvey P. Dale_, Jan 15 2022 *)
%o (Magma) [n : n in [0..150] | n mod 9 in [0, 2, 5, 8]]; // _Wesley Ivan Hurt_, Jun 07 2016
%o (Python)
%o def A174438(n): return (0,2,5,8)[n&3]+9*(n>>2) # _Chai Wah Wu_, Jan 30 2023
%K nonn,easy
%O 1,2
%A _Gary Detlefs_, Mar 19 2010
%E a(23) corrected by _Chai Wah Wu_, Jun 10 2016
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