OFFSET
0,2
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).
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(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
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 *)
PROG
(Magma) [(1/16)*(2*n^2+18*n+15+(2*n+1)*(-1)^n): n in [0..60]]; // Vincenzo Librandi, Mar 11 2014
(Haskell)
import Data.List (intersperse)
a101881 n = a101881_list !! n
a101881_list = scanl1 (+) $ intersperse 1 [1..]
-- Reinhard Zumkeller, Feb 20 2015
(PARI) Vec((-1+x^3-x)/((x+1)^2*(x-1)^3) + O(x^60)) \\ Iain Fox, Nov 17 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Candace Mills (scorpiocand(AT)yahoo.com), Dec 19 2004
STATUS
approved