%I #23 Feb 16 2025 08:33:16

%S 1,2,3,4,4,4,5,6,7,8,9,9,9,9,9,10,11,12,13,14,15,16,16,16,16,16,16,16,

%T 17,18,19,20,21,22,23,24,25,25,25,25,25,25,25,25,25,26,27,28,29,30,31,

%U 32,33,34,35,36,36,36,36,36,36,36,36,36,36,36,37,38

%N Triangle read by rows, where even numbered rows contain the nonsquares (cf. A000037) and odd rows contain replicated squares.

%C An approximation of the Euler-Mascheroni constant by rational numbers: the sum ((-1)^(n+1) * Sum_{k=1..n} (1/T(n,k))) converges to gamma, cf. Pólya-Szegő reference.

%D G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Two, Chap. 1, sec. 2, Problem 19.2., page 51.

%H Reinhard Zumkeller, <a href="/A199332/b199332.txt">Rows n = 1..150 of triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a>

%e 1: 1 1

%e 2: 2 3 2 .. 3

%e 3: 4 4 4 4

%e 4: 5 6 7 8 5 .. 8

%e 5: 9 9 9 9 9 9

%e 6: 10 11 12 13 14 15 10 .. 15

%e 7: 16 16 16 16 16 16 16 16

%e 8: 17 18 19 20 21 22 23 24 17 .. 24

%e 9: 25 25 25 25 25 25 25 25 25 25 .

%t t[n_, k_] := If[OddQ[n], (n+1)^2/4, n^2/4 + k]; Flatten[ Table[ t[n, k], {n, 1, 12}, {k, 1, n}]](* _Jean-François Alcover_, Dec 05 2011 *)

%t Flatten[Table[If[IntegerQ[Sqrt[n]],Table[n,{2*Sqrt[n]-1}],n],{n,40}]] (* _Harvey P. Dale_, Nov 11 2013 *)

%o (Haskell)

%o a199332 n k = a199332_tabl !! (n-1) !! (k-1)

%o a199332_row n = a199332_tabl !! (n-1)

%o a199332_list = concat a199332_tabl

%o a199332_tabl = f [1..] [1..] where

%o f (x:xs) ys'@(y:ys) | odd x = (replicate x y) : f xs ys

%o | even x = us : f xs vs

%o where (us,vs) = splitAt x ys'

%Y Cf. A000037, A000290 & A002620 (central terms), A199771 (row sums).

%K nonn,tabl,changed

%O 1,2

%A _Reinhard Zumkeller_, Nov 23 2011