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%I #31 Sep 08 2022 08:45:32
%S 1,2,3,4,6,7,10,11,15,16,21,22,28,29,36,37,45,46,55,56,66,67,78,79,91,
%T 92,105,106,120,121,136,137,153,154,171,172,190,191,210,211,231,232,
%U 253,254,276,277,300,301,325,326,351,352,378,379,406,407,435,436,465,466
%N Numbers remaining when the natural numbers (A000027) are arranged into a triangle and only the beginning and end terms of each row are retained.
%C Equivalently, this is TriRet(A000027,{1}) = TriRem(A000027,{2,3,4,...}), using the operations defined in A134509. Bisections are A000217-{0} and A000124-{1}. A055802 and A114220 appear to be this sequence with two and three additional leading terms, respectively.
%H Muniru A Asiru, <a href="/A134519/b134519.txt">Table of n, a(n) for n = 1..10000</a>
%H Bruno Berselli, <a href="/A134519/a134519.jpg">An interpretation of initial terms</a>.
%F From _Colin Barker_, Jul 17 2013: (Start)
%F a(n) = (16 + 4*n + 2*n^2)/16 for n even, a(n) = (3 + 4*n + n^2)/8 for n odd.
%F G.f.: -x*(x^4 - x^3 - x^2 + x + 1) / ((x - 1)^3*(x + 1)^2). (End)
%F a(n) = Sum_{k=0..n-1} A057979(k). - _Jon Maiga_, Dec 21 2018
%F a(n) = A000217(floor(n+1)/2) + (1 + (-1)^n)/2. - _Bruno Berselli_, Aug 20 2019
%p seq(coeff(series(-x*(x^4-x^3-x^2+x+1)/((x-1)^3*(x+1)^2),x,n+1), x, n), n = 1 .. 60); # _Muniru A Asiru_, Dec 21 2018
%t Table[Sum[If[EvenQ[k], 1, (k - 1)/2], {k, 0, n}], {n, 60}] (* _Jon Maiga_, Dec 21 2018 *)
%o (GAP) a:=[];; for n in [1..60] do if n mod 2=0 then Add(a,(16+4*n+2*n^2)/16); else Add(a,(3+4*n+n^2)/8); fi; od; a; # _Muniru A Asiru_, Dec 21 2018
%o (Magma) T:=func<i | i*(i+1)/2>; [T(Floor((n+1)/2))+(1+(-1)^n)/2: n in [1..60]]; // _Bruno Berselli_, Aug 20 2019
%Y Cf. A000027, A000124, A000217, A055802, A057979, A114220, A134509.
%Y Cf. A084263: A000217(m) + (1 + (-1)^m)/2.
%Y Cf. A117142: A000217(floor(m/2)+1) - (1 + (-1)^m)/2.
%K nonn,easy
%O 1,2
%A _Rick L. Shepherd_, Oct 29 2007
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