A094763 -id:A094763 - cp's OEIS frontend

A094761 a(n) = n + (square excess of n).

0, 1, 3, 5, 4, 6, 8, 10, 12, 9, 11, 13, 15, 17, 19, 21, 16, 18, 20, 22, 24, 26, 28, 30, 32, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 64, 66, 68, 70, 72, 74, 76, 78

Offset: 0

N. J. A. Sloane, Jun 10 2004

The trajectory of n under iteration of m -> a(m) is eventually constant iff n is a perfect square.
Conjecture (verified up to 727): the numbers not in this sequence are those of A008865. - R. J. Mathar, Jan 23 2009
From Maon Wenders, Jul 01 2012: (Start)
Proof of conjecture:
(1) (n+2)^2 - n^2 = n^2 + 4n + 4 - n^2 = 4n + 4
(2) (n+1)^2 - n^2 = n^2 + 2n + 1 - n^2 = 2n + 1
(3) (n+1) + square excess of (n+1) - (n + square excess of n) = 2, except when (n+1) is a square, where a(n) collapses back to (n+1)
(4) so, cause of (2) and (3), the sequence has blocks of even and odd numbers starting with an even or odd square, m^2 and of length 2m+1:
    0,
    1, 3, 5,
    4, 6, 8, 10, 12,
    9, 11, 13, 15, 17, 19, 21,
    16, 18, 20, 22, 24, 26, 28, 30, 32,
    ...
(5) such a block of 2m+1 numbers fills in all even or odd numbers between
    n^2 and (n+2)^2
(6) but, because a block starts n^2 + 0, n^2 + 2, n^2 + 4, ..., the last number in such a block is n^2 + 2*(2n+1-1) = n^2 + 4n
(7) so the numbers n^2 + 4n + 2 = (n+2)^2 - 2 are missing.
End of proof. (End)

S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

Cf. A053186, A094763, A094764, A094765.

f[n_] := 2 n - (Floor@ Sqrt@ n)^2; Table[f@ n, {n, 0, 71}] (* Robert G. Wilson v, Jan 23 2009 *)

a(n)=2*n-sqrtint(n)^2 \\ Charles R Greathouse IV, Jul 01 2012

a(n) = n + A053186(n).

A094764 Trajectory of 7 under repeated application of the map n --> n + square excess of n.

Original entry on oeis.org

7, 10, 11, 13, 17, 18, 20, 24, 32, 39, 42, 48, 60, 71, 78, 92, 103, 106, 112, 124, 127, 133, 145, 146, 148, 152, 160, 176, 183, 197, 198, 200, 204, 212, 228, 231, 237, 249, 273, 290, 291, 293, 297, 305, 321, 353, 382, 403, 406, 412, 424, 448, 455, 469, 497, 510, 536, 543

Offset: 0

N. J. A. Sloane, Jun 10 2004

nonn

H. Brocard, Note 2837, L'Intermédiaire des Mathématiciens, 11 (1904), p. 239.

S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

Cf. A053186, A094761, A094763.

lista(nn) = {print1(n=7, ", "); for (k=2, nn, m = 2*n - sqrtint(n)^2; print1(m, ", "); n = m;);} \\ Michel Marcus, Oct 24 2015

cp's OEIS Frontend

A094761 a(n) = n + (square excess of n).

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