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 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}. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 4 05:19 EST 2023. Contains 367541 sequences. (Running on oeis4.)