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A193832 Irregular triangle read by rows in which row n lists 2n-1 copies of 2n-1 and n copies of 2n, for n >= 1.

%I A193832 #52 Dec 15 2017 04:27:58
%S A193832 1,2,3,3,3,4,4,5,5,5,5,5,6,6,6,7,7,7,7,7,7,7,8,8,8,8,9,9,9,9,9,9,9,9,
%T A193832 9,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,
%U A193832 13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,14,14,14,14,14
%N A193832 Irregular triangle read by rows in which row n lists 2n-1 copies of 2n-1 and n copies of 2n, for n >= 1.
%C A193832 Sequence of successive positive integers k in which if k is odd then k appears k times, otherwise if k is even then k appears k/2 times.
%C A193832 Note that an arrangement of the blocks of this sequence shows the growth of the generalized pentagonal numbers A001318 (see example).
%C A193832 The sums of each block give the positive integers of A129194: 1, 2, 9, 8, 25, 18, 49,...
%C A193832 Partial sums of A080995. - Paolo P. Lava, Aug 23 2011.
%C A193832 Concatenations of rows of triangles A001650 and A111650; also, seen as a flat list, the row lengths of triangle A260672 and the first differences of its row sums (cf. A260706). - _Reinhard Zumkeller_, Nov 17 2015
%C A193832 Also a(n) = number of squares in the arithmetic progression {24k + 1: 0 <= k <= n-1} [Granville]. - _N. J. A. Sloane_, Dec 13 2017
%H A193832 Reinhard Zumkeller, <a href="/A193832/b193832.txt">Rows n = 1..150 of triangle, flattened</a>
%H A193832 Andrew Granville, <a href="https://arxiv.org/abs/1708.06951">Squares in arithmetic progressions and infinitely many primes</a>, arXiv:1708.06951 [math.NT], 2017.
%H A193832 Andrew Granville, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.124.10.951">Squares in arithmetic progressions and infinitely many primes</a>, The American Mathematical Monthly, 124.10 (2017): 951-954. See p. 952.
%F A193832 a(n) = sqrt(8n/3) plus or minus 1 [Granville] - _N. J. A. Sloane_, Dec 13 2017
%F A193832 If 8 <= n <= 52, then a(n-1) < a(n) if and only if n is in A221672. - _Jonathan Sondow_, Dec 14 2017
%e A193832 a) If written as a triangle the initial rows are
%e A193832   1, 2,
%e A193832   3, 3, 3, 4, 4,
%e A193832   5, 5, 5, 5, 5, 6, 6, 6,
%e A193832   7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8,
%e A193832   9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10,
%e A193832   ...
%e A193832 Row sums give A126587.
%e A193832 b) An application using the blocks of this sequence: the illustration of the growth of an arrangement which represents the generalized pentagonal numbers A001318. For example; the first 9 positive initial terms: 1, 2, 5, 7, 12, 15, 22, 26, 35.
%e A193832 .
%e A193832 .         9
%e A193832 .       8 9
%e A193832 .     8 7 9
%e A193832 .   8 6 7 9
%e A193832 . 8 6 5 7 9
%e A193832 . 6 4 5 7 9
%e A193832 . 4 3 5 7 9
%e A193832 . 2 3 5 7 9
%e A193832 . 1 3 5 7 9
%e A193832 ...
%t A193832 Array[Join @@ MapIndexed[ConstantArray[#, #/(1 + Boole[First@ #2 == 2])] &, {2 # - 1, 2 #}] &, 7] // Flatten (* or *)
%t A193832 Table[If[k <= 2 n - 1, 2 n - 1, 2 n], {n, 7}, {k, 3 n - 1}] // Flatten (* _Michael De Vlieger_, Dec 14 2017 *)
%o A193832 (Haskell)
%o A193832 a193832 n k = a193832_tabf !! (n-1) !! (k-1)
%o A193832 a193832_row n = a193832_tabf !! (n-1)
%o A193832 a193832_tabf = zipWith (++) a001650_tabf a111650_tabl
%o A193832 a193832' n = a193832_list !! (n - 1)
%o A193832 a193832_list = concat a193832_tabf
%o A193832 -- _Reinhard Zumkeller_, Nov 15 2015
%Y A193832 Cf. A001318, A080995, A126587, A129194, A221671, A221672.
%Y A193832 Cf. A001650, A111650, A260672, A260706.
%K A193832 nonn,easy,tabf
%O A193832 1,2
%A A193832 _Omar E. Pol_, Aug 22 2011
%E A193832 Edited by _N. J. A. Sloane_, Dec 13 2017