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

1,2

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.

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

Example

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 .

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 *)

Prog

(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'

Crossrefs

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

Sequence in context: A356992 A099479 A120508 * A029085 A087875 A195848

Adjacent sequences: A199329 A199330 A199331 * A199333 A199334 A199335

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Nov 23 2011

Status

approved