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

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, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 37, 38

Offset: 1

Author

Reinhard Zumkeller, Nov 23 2011

Keywords

• nonn
• tabl

Comments

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.

Examples

			1:                    1                           1
2:                  2   3                      2 .. 3
3:                4   4   4                       4
4:              5   6   7   8                  5 .. 8
5:            9   9   9   9   9                   9
6:         10  11  12  13  14  15             10 .. 15
7:       16  16  16  16  16  16  16              16
8:     17  18  19  20  21  22  23  24         17 .. 24
9:   25  25  25  25  25  25  25  25  25          25 .
		

References

• 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.

Links

• Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
• Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant
• Wikipedia, Euler-Mascheroni constant

Crossrefs

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

Programs

Haskell

a199332 n k = a199332_tabl !! (n-1) !! (k-1)
a199332_row n = a199332_tabl !! (n-1)
a199332_list = concat a199332_tabl
a199332_tabl = f [1..] [1..] where
   f (x:xs) ys'@(y:ys) | odd x  = (replicate x y) : f xs ys
                       | even x = us : f xs vs
                       where (us,vs) = splitAt x ys'

Mathematica

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 *)
Flatten[Table[If[IntegerQ[Sqrt[n]],Table[n,{2*Sqrt[n]-1}],n],{n,40}]] (* Harvey P. Dale, Nov 11 2013 *)