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A101881 Write two numbers, skip one, write two, skip two, write two, skip three ... and so on. 8
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 (list; graph; refs; listen; history; text; internal format)
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

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(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

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

Cf. A000217, A101882, A101883, A177994.

Cf. A000096, A034856.

Sequence in context: A189019 A189016 A102821 * A143989 A064573 A065300

Adjacent sequences:  A101878 A101879 A101880 * A101882 A101883 A101884

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified June 2 11:35 EDT 2020. Contains 334771 sequences. (Running on oeis4.)