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A199332 Triangle read by rows, where even numbered rows contain the nonsquares (cf. A000037) and odd rows contain replicated squares.

%I A199332 #23 Feb 16 2025 08:33:16
%S A199332 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 A199332 17,18,19,20,21,22,23,24,25,25,25,25,25,25,25,25,25,26,27,28,29,30,31,
%U A199332 32,33,34,35,36,36,36,36,36,36,36,36,36,36,36,37,38
%N A199332 Triangle read by rows, where even numbered rows contain the nonsquares (cf. A000037) and odd rows contain replicated squares.
%C A199332 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 A199332 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 A199332 Reinhard Zumkeller, <a href="/A199332/b199332.txt">Rows n = 1..150 of triangle, flattened</a>
%H A199332 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>
%H A199332 Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a>
%e A199332 1:                    1                           1
%e A199332 2:                  2   3                      2 .. 3
%e A199332 3:                4   4   4                       4
%e A199332 4:              5   6   7   8                  5 .. 8
%e A199332 5:            9   9   9   9   9                   9
%e A199332 6:         10  11  12  13  14  15             10 .. 15
%e A199332 7:       16  16  16  16  16  16  16              16
%e A199332 8:     17  18  19  20  21  22  23  24         17 .. 24
%e A199332 9:   25  25  25  25  25  25  25  25  25          25 .
%t A199332 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 A199332 Flatten[Table[If[IntegerQ[Sqrt[n]],Table[n,{2*Sqrt[n]-1}],n],{n,40}]] (* _Harvey P. Dale_, Nov 11 2013 *)
%o A199332 (Haskell)
%o A199332 a199332 n k = a199332_tabl !! (n-1) !! (k-1)
%o A199332 a199332_row n = a199332_tabl !! (n-1)
%o A199332 a199332_list = concat a199332_tabl
%o A199332 a199332_tabl = f [1..] [1..] where
%o A199332    f (x:xs) ys'@(y:ys) | odd x  = (replicate x y) : f xs ys
%o A199332                        | even x = us : f xs vs
%o A199332                        where (us,vs) = splitAt x ys'
%Y A199332 Cf. A000037, A000290 & A002620 (central terms),  A199771 (row sums).
%K A199332 nonn,tabl
%O A199332 1,2
%A A199332 _Reinhard Zumkeller_, Nov 23 2011