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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. 8

%I

%S 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 9,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,

%U 13,13,13,13,13,13,13,13,13,13,13,13,13,14,14,14,14,14,14,14

%N 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 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 Note that an arrangement of the blocks of this sequence shows the growth of the generalized pentagonal numbers A001318 (see example).

%C The sums of each block give the positive integers of A129194: 1, 2, 9, 8, 25, 18, 49,...

%C Partial sums of A080995. - Paolo P. Lava, Aug 23 2011.

%C 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 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 Reinhard Zumkeller, <a href="/A193832/b193832.txt">Rows n = 1..150 of triangle, flattened</a>

%H 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 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 a(n) = sqrt(8n/3) plus or minus 1 [Granville] - _N. J. A. Sloane_, Dec 13 2017

%F If 8 <= n <= 52, then a(n-1) < a(n) if and only if n is in A221672. - _Jonathan Sondow_, Dec 14 2017

%e a) If written as a triangle the initial rows are

%e 1, 2,

%e 3, 3, 3, 4, 4,

%e 5, 5, 5, 5, 5, 6, 6, 6,

%e 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8,

%e 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10,

%e ...

%e Row sums give A126587.

%e 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 .

%e . 9

%e . 8 9

%e . 8 7 9

%e . 8 6 7 9

%e . 8 6 5 7 9

%e . 6 4 5 7 9

%e . 4 3 5 7 9

%e . 2 3 5 7 9

%e . 1 3 5 7 9

%e ...

%t Array[Join @@ MapIndexed[ConstantArray[#, #/(1 + Boole[First@ #2 == 2])] &, {2 # - 1, 2 #}] &, 7] // Flatten (* or *)

%t 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 (Haskell)

%o a193832 n k = a193832_tabf !! (n-1) !! (k-1)

%o a193832_row n = a193832_tabf !! (n-1)

%o a193832_tabf = zipWith (++) a001650_tabf a111650_tabl

%o a193832' n = a193832_list !! (n - 1)

%o a193832_list = concat a193832_tabf

%o -- _Reinhard Zumkeller_, Nov 15 2015

%Y Cf. A001318, A080995, A126587, A129194, A221671, A221672.

%Y Cf. A001650, A111650, A260672, A260706.

%K nonn,easy,tabf

%O 1,2

%A _Omar E. Pol_, Aug 22 2011

%E Edited by _N. J. A. Sloane_, Dec 13 2017

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Last modified February 24 12:52 EST 2021. Contains 341569 sequences. (Running on oeis4.)