Jelly, 7 6 bytes

3rc3«i

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How it works (outdated)

R‘c3<¹S  Main link. Argument: n

R        Range; yield [1, ..., n].
 ‘       Increment; yield [2, ..., n+1].
  c3     Combinations; yield [C(2,3), ..., C(n+1,3)].
    <¹   Yield [C(2,3) < n, ..., C(n+1,3) < n].
      S  Sum; count the non-negative values of k for which C(k+2,3) < n.

Jelly, 7 bytes

R‘c3<¹S

Try it online!

How it works

R‘c3<¹S  Main link. Argument: n

R        Range; yield [1, ..., n].
 ‘       Increment; yield [2, ..., n+1].
  c3     Combinations; yield [C(2,3), ..., C(n+1,3)].
    <¹   Yield [C(2,3) < n, ..., C(n+1,3) < n].
      S  Sum; count the non-negative values of k for which C(k+2,3) < n.

Jelly, 7 bytes

R‘c3<¹S

Try it online!

How it works

R‘c3<¹S  Main link. Argument: n

R        Range; yield [1, ..., n].
 ‘       Increment; yield [2, ..., n+1].
  c3     Combinations; yield [C(2,3), ..., C(n+1,3)].
    <¹   Yield [C(2,3) < n, ..., C(n+1,3) < n].
      S  Sum; count the non-negative values of k for which C(k+2,3) < n.

Jelly, 7 6 bytes

3rc3«i

Try it online!

How it works (outdated)

R‘c3<¹S  Main link. Argument: n

R        Range; yield [1, ..., n].
 ‘       Increment; yield [2, ..., n+1].
  c3     Combinations; yield [C(2,3), ..., C(n+1,3)].
    <¹   Yield [C(2,3) < n, ..., C(n+1,3) < n].
      S  Sum; count the non-negative values of k for which C(k+2,3) < n.

Jelly, 7 bytes

R‘c3<¹S

Try it online!

Jelly, 7 bytes

R‘c3<¹S

Try it online!

How it works

R‘c3<¹S  Main link. Argument: n

R        Range; yield [1, ..., n].
 ‘       Increment; yield [2, ..., n+1].
  c3     Combinations; yield [C(2,3), ..., C(n+1,3)].
    <¹   Yield [C(2,3) < n, ..., C(n+1,3) < n].
      S  Sum; count the non-negative values of k for which C(k+2,3) < n.