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%I #54 Aug 20 2022 18:24:46
%S 1,4,10,13,19,22,28,31,37,40,46,49,55,58,64,67,73,76,82,85,91,94,100,
%T 103,109,112,118,121,127,130,136,139,145,148,154,157,163,166,172,175,
%U 181,184,190,193,199,202,208,211,217,220,226,229,235,238,244,247,253
%N Numbers that are congruent to {1, 4} mod 9.
%C 3*a(n) is conjectured to be the total number of sides (straight double bonds (long side) and single bond (short side)) of a certain equilateral triangle expansion shown in one of the links. The pattern is supposed to become the planar Archimedean net 3.3.3.3.6 when n -> infinity. 3*a(n) is also conjectured to be the total number of sided (bonds) in another version of an equilateral triangle expansion that is supposed to become the planar Archimedean net 3.6.3.6. See the illustrations in the links. - _Kival Ngaokrajang_, Nov 30 2014
%H David Lovler, <a href="/A087444/b087444.txt">Table of n, a(n) for n = 1..10000</a>
%H Kival Ngaokrajang, <a href="/A087444/a087444.pdf">Illustration of initial terms (3.3.3.3.6)</a>, <a href="/A087444/a087444_1.pdf">(3.6.3.6)</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*(1+3*x+5*x^2)/((1+x)*(1-x)^2).
%F E.g.f.: 5 + ((9*x - 17/2)*exp(x) - (3/2)*exp(-x))/2.
%F a(n) = (18*n - 17 - 3*(-1)^n)/4.
%F a(n) = 9*n - a(n-1) - 13 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010
%t Select[Range[300],MemberQ[{1,4},Mod[#,9]]&] (* or *) LinearRecurrence[ {1,1,-1},{1,4,10},60] (* _Harvey P. Dale_, Jan 22 2019 *)
%o (PARI) a(n) = (18*n - 17 - 3*(-1)^n)/4 \\ _David Lovler_, Aug 20 2022
%Y Cf. A001651, A047241, A087445, A087446.
%K easy,nonn
%O 1,2
%A _Paul Barry_, Sep 04 2003
%E _Kival Ngaokrajang_'s comment reworded by _Wolfdieter Lang_, Dec 05 2014
%E E.g.f. and formula adapted to offset by _David Lovler_, Aug 20 2022