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A094761 a(n) = n + (square excess of n). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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)

LINKS

Table of n, a(n) for n=0..71.

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

FORMULA

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A053186, A094763, A094764, A094765.

Sequence in context: A063259 A333717 A064425 * A114748 A182733 A107431

Adjacent sequences: A094758 A094759 A094760 * A094762 A094763 A094764

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jun 10 2004

STATUS

approved

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Last modified December 1 23:44 EST 2022. Contains 358485 sequences. (Running on oeis4.)