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

%I A101881 #64 Nov 04 2024 01:50:11
%S A101881 1,2,4,5,8,9,13,14,19,20,26,27,34,35,43,44,53,54,64,65,76,77,89,90,
%T A101881 103,104,118,119,134,135,151,152,169,170,188,189,208,209,229,230,251,
%U A101881 252,274,275,298,299,323,324,349,350,376,377,404,405,433,434,463,464,494
%N A101881 Write two numbers, skip one, write two, skip two, write two, skip three ... and so on.
%C A101881 Equals row sums of triangle A177994. - _Gary W. Adamson_, May 16 2010
%C A101881 From _Ralf Stephan_, Mar 09 2014: (Start)
%C A101881 Write the positive integers in a skewed triangle:
%C A101881   1,  2;
%C A101881   0,  3,  4,  5;
%C A101881   0,  0,  6,  7,  8,  9;
%C A101881   0,  0,  0, 10, 11, 12, 13, 14;
%C A101881   ...
%C A101881 Sequence consists of the first number in each column. (End)
%C A101881 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
%C A101881 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
%H A101881 Vincenzo Librandi, <a href="/A101881/b101881.txt">Table of n, a(n) for n = 0..1000</a>
%H A101881 Lars Blomberg, <a href="/A101881/a101881.png">Illustration for 14-polygon</a>
%H A101881 Lars Blomberg, <a href="/A101881/a101881_1.png">Illustration for 15-polygon</a>
%H A101881 Rene Marczinzik, <a href="https://arxiv.org/abs/1701.00972">Finitistic Auslander algebras</a>, arXiv:1701.00972 [math.RT], 2017. [Page 9, Conjecture]
%H A101881 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A101881 G.f.: (-1+x^3-x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
%F A101881 a(n) = (1/16)*(2*n^2 + 18*n + 15 + (2*n+1)*(-1)^n). - _Ralf Stephan_, Mar 09 2014
%F A101881 a(2*n) = A034856(n+1); a(2*n+1) = A000096(n+1). - _Reinhard Zumkeller_, Feb 20 2015
%F A101881 a(n) = n + 1 + A008805(n-2). - _Wesley Ivan Hurt_, Nov 17 2017
%F A101881 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
%t A101881 CoefficientList[Series[(-1 + x^3 - x)/((x + 1)^2 (x - 1)^3), {x, 0, 60}], x] (* _Vincenzo Librandi_, Mar 11 2014 *)
%t A101881 LinearRecurrence[{1,2,-2,-1,1},{1,2,4,5,8},60] (* _Harvey P. Dale_, Dec 07 2016 *)
%t A101881 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 *)
%o A101881 (Magma) [(1/16)*(2*n^2+18*n+15+(2*n+1)*(-1)^n): n in [0..60]]; // _Vincenzo Librandi_, Mar 11 2014
%o A101881 (Haskell)
%o A101881 import Data.List (intersperse)
%o A101881 a101881 n = a101881_list !! n
%o A101881 a101881_list = scanl1 (+) $ intersperse 1 [1..]
%o A101881 -- _Reinhard Zumkeller_, Feb 20 2015
%o A101881 (PARI) Vec((-1+x^3-x)/((x+1)^2*(x-1)^3) + O(x^60)) \\ _Iain Fox_, Nov 17 2017
%Y A101881 Cf. A000217, A101882, A101883, A177994.
%Y A101881 Cf. A000096, A034856.
%K A101881 easy,nonn
%O A101881 0,2
%A A101881 Candace Mills (scorpiocand(AT)yahoo.com), Dec 19 2004