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

Author

Rick L. Shepherd, Oct 29 2007

Keywords

• nonn
• easy

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.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

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.

Programs

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; [T(Floor((n+1)/2))+(1+(-1)^n)/2: n in [1..60]]; // 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 *)

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