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A134519
Numbers remaining when the natural numbers (A000027) are arranged into a triangle and only the beginning and end terms of each row are retained.
4
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, 92, 105, 106, 120, 121, 136, 137, 153, 154, 171, 172, 190, 191, 210, 211, 231, 232, 253, 254, 276, 277, 300, 301, 325, 326, 351, 352, 378, 379, 406, 407, 435, 436, 465, 466
OFFSET
1,2
COMMENTS
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.
FORMULA
From Colin Barker, Jul 17 2013: (Start)
a(n) = (16 + 4*n + 2*n^2)/16 for n even, a(n) = (3 + 4*n + n^2)/8 for n odd.
G.f.: -x*(x^4 - x^3 - x^2 + x + 1) / ((x - 1)^3*(x + 1)^2). (End)
a(n) = Sum_{k=0..n-1} A057979(k). - Jon Maiga, Dec 21 2018
a(n) = A000217(floor(n+1)/2) + (1 + (-1)^n)/2. - Bruno Berselli, Aug 20 2019
MAPLE
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
MATHEMATICA
Table[Sum[If[EvenQ[k], 1, (k - 1)/2], {k, 0, n}], {n, 60}] (* Jon Maiga, Dec 21 2018 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 2, 3, 4, 6}, 60] (* Harvey P. Dale, Oct 13 2024 *)
PROG
(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
(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
CROSSREFS
Cf. A084263: A000217(m) + (1 + (-1)^m)/2.
Cf. A117142: A000217(floor(m/2)+1) - (1 + (-1)^m)/2.
Sequence in context: A294848 A055802 A114220 * A101505 A259625 A068713
KEYWORD
nonn,easy
AUTHOR
Rick L. Shepherd, Oct 29 2007
STATUS
approved