A196126 Let A = {(x,y): x, y positive natural numbers and y <= x <= y^2}. a(n) is the cardinality of the subset {(x,y) in A such that x <= n}.

1, 2, 4, 7, 10, 14, 19, 25, 32, 39, 47, 56, 66, 77, 89, 102, 115, 129, 144, 160, 177, 195, 214, 234, 255, 276, 298, 321, 345, 370, 396, 423, 451, 480, 510, 541, 572, 604, 637, 671

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).

Links

Table of n, a(n) for n=1..40.

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

Sequence in context: A025704 A025710 A023536 * A024536 A177237 A094281

Adjacent sequences: A196123 A196124 A196125 * A196127 A196128 A196129

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