A001953 a(n) = floor((n + 1/2) * sqrt(2)). (Formerly M0543 N0193)

0, 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 30, 31, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 53, 54, 55, 57, 58, 60, 61, 62, 64, 65, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 88, 89, 91, 92, 94, 95

Offset

0,2

Comments

Let s(n) = zeta(3) - Sum_{k = 1..n} 1/k^3. Conjecture: for n >= 1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1's, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - Clark Kimberling, Oct 05 2014

References

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).

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Ian G. Connell, A generalization of Wythoff's game, Canad. Math. Bull. 2 (1959) 181-190.

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Formula

From Ralf Steiner, Oct 23 2019: (Start)

a(n) = floor(2*sqrt(A000217(n))).

a(n) = A136119(n + 1) - 1.

a(n + 1) - a(n) is in {1,2}.

a(n + 3) - a(n) is in {4,5}. (End)

Maple

seq( floor((2*n+1)/sqrt(2)), n=0..100); # G. C. Greubel, Nov 14 2019

Mathematica

Table[Floor[(n + 1/2) Sqrt[2]], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)

Prog

(PARI) a(n)=floor((n+1/2)*sqrt(2))
(PARI) a(n)={sqrtint(2*n*(n+1))} \\ Andrew Howroyd, Oct 24 2019
(Magma) [Floor((2*n+1)/Sqrt(2)): n in [0..100]]; // G. C. Greubel, Nov 14 2019
(Sage) [floor((2*n+1)/sqrt(2)) for n in (0..100)] # G. C. Greubel, Nov 14 2019

Crossrefs

Complement of A001954.

Cf. A000217 (T), A136119, A001108.

Sequence in context: A362443 A175320 A325597 * A230748 A078607 A292043

Adjacent sequences: A001950 A001951 A001952 * A001954 A001955 A001956

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Michael Somos, Apr 26 2000.

Status

approved