A080037 a(0)=2; for n > 0, a(n) = n + floor(sqrt(4n-3)) + 2.

2, 4, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 89, 90, 92

Offset: 0

Author

N. J. A. Sloane, Mar 14 2003

Keywords

• nonn

Comments

a(0)=2, a(1)=4; for n > 2, a(n) = a(n-1) + 1 if n is already in the sequence, a(n) = a(n-1) + 2 otherwise. [corrected by Jon E. Schoenfield, Jun 24 2018]

For n > 0, a(n) = ceiling((1 + sqrt(n))^2). Empirical observation. - Ronald S. Tiberio, Jun 24 2018

For n > 0, a(n) is the minimal number of thumbtacks needed to secure n square sheets of paper on a bulletin board - one thumbtack in each corner of each sheet - slight overlap allowed so that one thumbtack can secure up to four sheets. - Ronald S. Tiberio, Jun 24 2018

From Ya-Ping Lu, Mar 17 2022: (Start)

Integers m such that connecting the external lattice points of a square spiral with m lattice points forms a polyomino (m is indicated by * in the figure below).

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37--36*-35*-34*-33*-32*-31

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38* 17--16*-15*-14*-13 30*

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39* 18* 5---4*--3 12* 29*

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40* 19* 6* 1---2* 11* 28*

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41* 20* 7---8*--9*-10 27*

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42* 21--22*-23*-24*-25*-26

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43--44*-45*-46*-47*-48*-49* (End)

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
• B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.

Crossrefs

Cf. A000267, A080036, A033638 (complement, except 2).

Programs

Maple

f:= n -> n + floor(sqrt(4*n-3)) + 2:
f(0):= 2:
map(f, [$0..100]); # Robert Israel, Jul 27 2023

Mathematica

Join[{2},Table[n+Floor[Sqrt[4n-3]]+2,{n,100}]] (* Harvey P. Dale, Jul 22 2025 *)

Python

from math import isqrt
def A080037(n): return n+2+isqrt((n<<2)-1) if n else 2 # Chai Wah Wu, Jul 27 2022

Formula

G.f.: (2-z)/(1-z)^2 + Sum_{k >= 1} z^(k^2+1)/(1-z) + Sum_{k >= 0} z^(k^2+k+1)/(1-z) = 1/2 + 1/(z-1)^2 - 1/(2*(z-1)) + z^(3/4)*JacobiTheta2(0,z)/(2*(1-z)) + z*JacobiTheta3(0,z)/(2*(1-z)). - Robert Israel, Jul 27 2023