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.

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.

Links

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Bruno Berselli, An interpretation of initial terms.

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 *)

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. A000027, A000124, A000217, A055802, A057979, A114220, A134509.

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

Adjacent sequences: A134516 A134517 A134518 * A134520 A134521 A134522

Keyword

nonn,easy

Author

Rick L. Shepherd, Oct 29 2007

Status

approved