A001951 - OEIS (original) (raw)

0, 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 24, 25, 26, 28, 29, 31, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 50, 52, 53, 55, 56, 57, 59, 60, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 76, 77, 79, 80, 82, 83, 84, 86, 87, 89, 90, 91, 93, 94, 96, 97, 98, 100

COMMENTS

Earliest monotonic sequence greater than 0 satisfying the condition: "a(n) + 2n is not in the sequence". - Benoit Cloitre, Mar 25 2004

Also the integer part of the hypotenuse of isosceles right triangles. The real part of these numbers is irrational. For proof see Jones and Jones.

First differences are 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, ... ([A006337](/A006337 "An "eta-sequence": a(n) = floor( (n+1)sqrt(2) ) - floor( nsqrt(2) ).") with a 1 in front). - Philippe Deléham, May 29 2006

It appears that the distance between the a(n)-th triangular number and the nearest square is not greater than floor(a(n)/2). - Ralf Stephan, Sep 14 2013

These are the nonnegative integers m satisfying sin(m*Pi/r)*sin((m+1)*Pi/r) <= 0, where r = sqrt(2). In general, the Beatty sequence of an irrational number r > 1 consists of the numbers m satisfying sin(m*x)*sin((m+1)*x) <= 0, where x = Pi/r. Thus the numbers m satisfying sin(m*x)*sin((m+1)*x) > 0 form the Beatty sequence of r/(1-r). - Clark Kimberling, Aug 21 2014

We can generate A001951 and A001952 without using sqrt(2).

First write the even positive integers in a row:

2 4 6 8 10 12 14 . . .

Then put 1 under 2 and add:

2 4 6 8 10 12 14 . . .

1

3

Next, under 4, put the least positive integer that is not yet in rows 2 and 3;

it is 2; and add:

2 4 6 8 10 12 14 . . .

1 2

3 6

Next, under the 6 in row 1, put the least positive integer not yet in rows 2 and 3;

it is 4, and add:

2 4 6 8 10 12 14 . . .

1 2 4

3 6 10

Continue in this manner. (End)

This sequence contains an infinite number of powers of 2 (proof in Crux Mathematicorum link). See A103341. - Bernard Schott, Mar 08 2019

The terms of this sequence generate the multiplicative group of positive rational numbers (observation by Stephen M. Gagola, Jr.; see References). - Allen Stenger, Aug 05 2023

a(n) is also the number of distinct straight cylinders with integer radius and height having the same surface as a sphere with radius n. - Felix Huber, Sep 20 2024

Let P(x,y) be the condition x^2 + y^2 <= n^2, then 4*a(n) is the number of integer points (x,y) such that P(x,y) is true and at least one of P(x+1,y), P(x-1,y), P(x,y+1), P(x,y-1) is false. See LINKS for examples. - Bob de Boisvilliers, May 14 2025

REFERENCES

Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5; MSRI Publications, Vol. 70 (2017), pages 101-153.

Stephen M. Gagola Jr., Solution of Problem 12282, Am. Math. Monthly, 130 (2023), pp. 682-683.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 77.

Gareth A. Jones and J. Mary Jones, Elementary Number Theory, Springer, 1998; pp. 221-222.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Roland Sprague, Recreations in Mathematics, Blackie and Son, (1963).

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, Revised edition (1997), Entry sqrt(2), p. 18.

LINKS

L. Carlitz, Richard Scoville and Verner E. Hoggatt, Jr., Pellian representatives, Fib. Quart., 10 (1972), 449-488.

Ed Doolittle, Problem 19, 26th I.M.O. Finland proposed by Romania, Crux Mathematicorum, p. 70, Vol. 14, Mar. 88.

FORMULA

a(n) = floor(csc(1/(sqrt(2)*n))) for n > 0, since sqrt(2)*n < csc(1/(sqrt(2)*n)) < sqrt(2)*n + 1/(3*sqrt(2)*n) < floor(sqrt(2)*n) + 1 for n > 0. - Jianing Song, Sep 07 2021

MAPLE

a:=n->floor(n*sqrt(2)): seq(a(n), n=0..80); # Muniru A Asiru, Mar 09 2019

PROG

(PARI) f(n) = for(j=1, n, print1(floor(sqrt(2*j^2))", "))

(Magma) [Floor(n*Sqrt(2)): n in [0..60]]; // Vincenzo Librandi, Oct 22 2011

(Magma) [Isqrt(2*n^2):n in[0..60]]; // Jason Kimberley, Oct 28 2016

(Maxima) makelist(floor(n*sqrt(2)), n, 0, 100); /* Martin Ettl, Oct 17 2012 */

(Haskell)

a001951 = floor . (* sqrt 2) . fromIntegral

(Python)

from sympy import integer_nthroot

def A001951(n): return integer_nthroot(2*n**2, 2)[0] # Chai Wah Wu, Mar 16 2021

CROSSREFS

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, [A006337](/A006337 "An "eta-sequence": a(n) = floor( (n+1)sqrt(2) ) - floor( nsqrt(2) )."), A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021