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%I #80 Nov 28 2020 15:53:28
%S 1,1,3,1,3,5,1,3,5,7,1,3,5,7,9,1,3,5,7,9,11,1,3,5,7,9,11,13,1,3,5,7,9,
%T 11,13,15,1,3,5,7,9,11,13,15,17,1,3,5,7,9,11,13,15,17,19,1,3,5,7,9,11,
%U 13,15,17,19,21,1,3,5,7,9,11,13,15,17,19,21,23
%N Triangle T(n,m) = 1+2*m of odd numbers read along rows, 0<=m<n.
%C Row sums are n^2 = A000290(n).
%C The triangle sums, see A180662 for their definitions, link this triangle of odd numbers with seventeen different sequences, see the crossrefs. The knight sums Kn14 - Kn110 have been added. - _Johannes W. Meijer_, Sep 22 2010
%C A208057 is the eigentriangle of A158405 such that as infinite lower triangular matrices, A158405 * A208057 shifts the latter, deleting the right border of 1's. - _Gary W. Adamson_, Feb 22 2012
%C T(n,k) = A099375(n-1,n-k), 1<=k<=n. [_Reinhard Zumkeller_, Mar 31 2012]
%H Seiichi Manyama, <a href="/A158405/b158405.txt">Rows n = 1..140, flattened</a>
%H Daniel Erman, <a href="https://www.youtube.com/watch?v=uCsD3ZGzMgE">The Josephus Problem</a>, Numberphile video (2016)
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%H <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%F a(n) = 2*i-1, where i = n-t(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Feb 03 2013
%F a(n) = 2*A002262(n-1) + 1. - _Eric Werley_, Sep 30 2015
%e The triangle contains the first n odd numbers in row n:
%e 1;
%e 1,3;
%e 1,3,5;
%e 1,3,5,7;
%e From _Seiichi Manyama_, Dec 02 2017: (Start)
%e | a(n) | | A000290(n)
%e -----------------------------------------------------------------
%e 0| (= 0)
%e 1| 1 = 1/3 * ( 3) (= 1)
%e 2| 1 + 3 = 1/3 * ( 5 + 7) (= 4)
%e 3| 1 + 3 + 5 = 1/3 * ( 7 + 9 + 11) (= 9)
%e 4| 1 + 3 + 5 + 7 = 1/3 * ( 9 + 11 + 13 + 15) (= 16)
%e 5| 1 + 3 + 5 + 7 + 9 = 1/3 * (11 + 13 + 15 + 17 + 19) (= 25)
%e (End)
%t Table[2 Range[1, n] - 1, {n, 12}] // Flatten (* _Michael De Vlieger_, Oct 01 2015 *)
%o (Haskell)
%o a158405 n k = a158405_row n !! (k-1)
%o a158405_row n = a158405_tabl !! (n-1)
%o a158405_tabl = map reverse a099375_tabl
%o -- _Reinhard Zumkeller_, Mar 31 2012
%o (PARI) a(n) = 2*(n-floor((-1+sqrt(8*n-7))/2)*(floor((-1+sqrt(8*n-7))/2)+1)/2)-1;
%o vector(100, n, a(n)) \\ _Altug Alkan_, Oct 01 2015
%Y Cf. A129326, A099375, A005408.
%Y Triangle sums (see the comments): A000290 (Row1; Kn11 & Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A000027 (Row2); A005563 (Kn12); A028347 (Kn13); A028560 (Kn14); A028566 (Kn15); A098603 (Kn16); A098847 (Kn17); A098848 (Kn18); A098849 (Kn19); A098850 (Kn110); A000217 (Kn21. Kn22, Kn23, Fi2, Ze2); A000384 (Kn3, Fi1, Ze3); A000212 (Ca2 & Ze4); A000567 (Ca3, Ze1); A011848 (Gi2); A001107 (Gi3). - _Johannes W. Meijer_, Sep 22 2010
%Y Cf. A063656, A063657, A208057, A292610, A292611.
%K nonn,tabl,easy
%O 1,3
%A _Paul Curtz_, Mar 18 2009
%E Edited by _R. J. Mathar_, Oct 06 2009
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