OFFSET
1,2
COMMENTS
The set A locates integer points in the first quadrant above the parabola y=sqrt(x) up to the diagonal y=x. a(n) counts them up to a sliding right margin.
The first differences of the sequence are 1, 2, 3, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 31, ....
In that way the sequence is constructed from first differences which are the natural numbers and repetitions for 3, 7, 13, 21, 31, 43, 57, 73, 91,...., (apparently the elements of A002061 starting at 3).
FORMULA
a(n) = u*(u+1)*(2*u+1)/6 - u*(u-1)/2 + (n-u)*(n-u+1)/2, where u = floor(sqrt(n)) = A000196(n).
EXAMPLE
The set is A = {(1,1),(2,2),(3,2),(4,2),(3,3),(4,3),(5,3),(6,3),(7,3),(8,3),(9,3),(4,4),(5,4),...}.
a(1) = 1 that is the number of elements in {(1,1)},
a(2) = 2 that is the number of elements in {(1,1),(2,2)} and
a(3) = 4 that is the number of elements in {(1,1),(2,2),(3,2),(3,3)}, ...
MATHEMATICA
(* Calculates a(n) using the definition of the sequence. *)
data = Flatten[Table[Table[{k, n}, {k, n, n^2}], {n, 1, 40}], 1];
Table[Length[Select[data, #[[1]] <= m &]], {m, 1, 40}]
(* Calculates a(n) using a formula. *)
ff[t_] := Block[{u}, u = Floor[Sqrt[t]]; u (u + 1) (2 u + 1)/6 - u (u - 1)/2 + (t - u) (t - u + 1)/2]; Table[ff[t], {t, 1, 40}]
PROG
(PARI) a(n)=my(u=sqrtint(n)); u*(u^2+2)/3+(n-u)*(n-u+1)/2 \\ Charles R Greathouse IV, Oct 05 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Taishi Inoue, Hiroshi Matsui, and Ryohei Miyadera, Sep 27 2011
EXTENSIONS
Entry rewritten by R. J. Mathar, Jan 28 2012
STATUS
approved