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A101881
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Write two numbers, skip one, write two, skip two, write two, skip three ... and so on.
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9
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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
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MATHEMATICA
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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 *)
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PROG
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(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..]
(PARI) Vec((-1+x^3-x)/((x+1)^2*(x-1)^3) + O(x^60)) \\ Iain Fox, Nov 17 2017
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Candace Mills (scorpiocand(AT)yahoo.com), Dec 19 2004
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STATUS
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approved
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