A101881 Write two numbers, skip one, write two, skip two, write two, skip three ... and so on.

1, 2, 4, 5, 8, 9, 13, 14, 19, 20, 26, 27, 34, 35, 43, 44, 53, 54, 64, 65, 76, 77, 89, 90, 103, 104, 118, 119, 134, 135, 151, 152, 169, 170, 188, 189, 208, 209, 229, 230, 251, 252, 274, 275, 298, 299, 323, 324, 349, 350, 376, 377, 404, 405, 433, 434, 463, 464, 494

Offset: 0

Author

Candace Mills (scorpiocand(AT)yahoo.com), Dec 19 2004

Keywords

• easy
• nonn

Comments

Equals row sums of triangle A177994. - Gary W. Adamson, May 16 2010

From Ralf Stephan, Mar 09 2014: (Start)

Write the positive integers in a skewed triangle:

1, 2;

0, 3, 4, 5;

0, 0, 6, 7, 8, 9;

0, 0, 0, 10, 11, 12, 13, 14;

...

Sequence consists of the first number in each column. (End)

In a regular k-polygon draw lines connecting all the vertices. Select a triangle that tiles the polygon into k pieces. This triangle contains two adjacent polygon vertices. The third vertex is for even k the center of the polygon and for odd k one of the vertices of the central k-polygon (which is not included in the tiling). Count all lines connecting vertices in the original k-polygon that passes through the interior of the tiling triangle. That count is a(k-5). (See illustrations below.) - Lars Blomberg, Feb 20 2020

a(n) is the smallest number which has n+1 as a part in any of its maximally refined strict partitions. The first such are:(1),(2),(1,3),(1,4),(1,2,5),(1,2,6),(1,2,3,7),(1,2,3,8),(1,2,3,4,9) etc. - Sigurd Kittilsen, Oct 18 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Lars Blomberg, Illustration for 14-polygon
• Lars Blomberg, Illustration for 15-polygon
• Rene Marczinzik, Finitistic Auslander algebras, arXiv:1701.00972 [math.RT], 2017. [Page 9, Conjecture]
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000217, A101882, A101883, A177994.

Cf. A000096, A034856.

Programs

Haskell

import Data.List (intersperse)
a101881 n = a101881_list !! n
a101881_list = scanl1 (+) $ intersperse 1 [1..]
-- Reinhard Zumkeller, Feb 20 2015

Magma

[(1/16)*(2*n^2+18*n+15+(2*n+1)*(-1)^n): n in [0..60]]; // Vincenzo Librandi, Mar 11 2014

Mathematica

CoefficientList[Series[(-1 + x^3 - x)/((x + 1)^2 (x - 1)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 11 2014 *)
LinearRecurrence[{1,2,-2,-1,1},{1,2,4,5,8},60] (* Harvey P. Dale, Dec 07 2016 *)
With[{nn=60},Take[#,2]&/@TakeList[Range[(nn^2+nn-6)/2],Range[3,nn]]]// Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 30 2019 *)

PARI

Vec((-1+x^3-x)/((x+1)^2*(x-1)^3) + O(x^60)) \\ Iain Fox, Nov 17 2017

Formula

G.f.: (-1+x^3-x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

a(n) = (1/16)*(2*n^2 + 18*n + 15 + (2*n+1)*(-1)^n). - Ralf Stephan, Mar 09 2014

a(2*n) = A034856(n+1); a(2*n+1) = A000096(n+1). - Reinhard Zumkeller, Feb 20 2015

a(n) = n + 1 + A008805(n-2). - Wesley Ivan Hurt, Nov 17 2017

E.g.f.: (cosh(x) - sinh(x))*(1 - 2*x + (15 + 20*x + 2*x^2)*(cosh(2*x) + sinh(2*x)))/16. - Stefano Spezia, Feb 20 2020