OFFSET
1,2
COMMENTS
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.
Note that an arrangement of the blocks of this sequence shows the growth of the generalized pentagonal numbers A001318 (see example).
The sums of each block give the positive integers of A129194: 1, 2, 9, 8, 25, 18, 49,...
Partial sums of A080995. - Paolo P. Lava, Aug 23 2011.
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
Also a(n) = number of squares in the arithmetic progression {24k + 1: 0 <= k <= n-1} [Granville]. - N. J. A. Sloane, Dec 13 2017
LINKS
Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Andrew Granville, Squares in arithmetic progressions and infinitely many primes, arXiv:1708.06951 [math.NT], 2017.
Andrew Granville, Squares in arithmetic progressions and infinitely many primes, The American Mathematical Monthly, 124.10 (2017): 951-954. See p. 952.
FORMULA
a(n) = sqrt(8n/3) plus or minus 1 [Granville] - N. J. A. Sloane, Dec 13 2017
If 8 <= n <= 52, then a(n-1) < a(n) if and only if n is in A221672. - Jonathan Sondow, Dec 14 2017
EXAMPLE
a) If written as a triangle the initial rows are
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, 9, 10, 10, 10, 10, 10,
...
Row sums give A126587.
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.
.
. 9
. 8 9
. 8 7 9
. 8 6 7 9
. 8 6 5 7 9
. 6 4 5 7 9
. 4 3 5 7 9
. 2 3 5 7 9
. 1 3 5 7 9
...
MATHEMATICA
Array[Join @@ MapIndexed[ConstantArray[#, #/(1 + Boole[First@ #2 == 2])] &, {2 # - 1, 2 #}] &, 7] // Flatten (* or *)
Table[If[k <= 2 n - 1, 2 n - 1, 2 n], {n, 7}, {k, 3 n - 1}] // Flatten (* Michael De Vlieger, Dec 14 2017 *)
PROG
(Haskell)
a193832 n k = a193832_tabf !! (n-1) !! (k-1)
a193832_row n = a193832_tabf !! (n-1)
a193832_tabf = zipWith (++) a001650_tabf a111650_tabl
a193832' n = a193832_list !! (n - 1)
a193832_list = concat a193832_tabf
-- Reinhard Zumkeller, Nov 15 2015
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Omar E. Pol, Aug 22 2011
EXTENSIONS
Edited by N. J. A. Sloane, Dec 13 2017
STATUS
approved