A022846 -id:A022846 - cp's OEIS frontend

A063957 Numbers not of the form round(m*sqrt(2)) for any integer m, i.e., complement of A022846.

2, 5, 9, 12, 15, 19, 22, 26, 29, 32, 36, 39, 43, 46, 50, 53, 56, 60, 63, 67, 70, 73, 77, 80, 84, 87, 90, 94, 97, 101, 104, 108, 111, 114, 118, 121, 125, 128, 131, 135, 138, 142, 145, 149, 152, 155, 159, 162, 166, 169, 172, 176, 179, 183, 186, 189, 193, 196, 200

Offset: 1

Henry Bottomley, Sep 04 2001

nonn

Consider natural numbers A000027 as a triangle 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc., then the a(n) indicate rows without a square.
Similar to Beatty sequences: where a pair of complementary Beatty sequences are floor(n*c) and floor(n*c/(c-1)) for c an irrational constant > 1, these pairs of complementary sequences are in general round(n*c) and round((n-1/2)*c/(c-1)) for c an irrational constant > 1.
This sequence is an inhomogeneous Beatty sequence s(alpha,rho) with slope alpha = 2 + sqrt(2), and intercept rho = -1/2 - sqrt(2)/2. - Michel Dekking, Sep 15 2022
Let D := 3,4,3,3,4,3,4,3,3,4,3,4,3,4,3,3,4,3,... be the sequence of first differences of (a(n)). It follows from Yasutomi's criterion that D is NOT the fixed point of a morphism. - Michel Dekking, Sep 20 2022

			round(m*sqrt(2)) starts 1,3,4,6,7,8,10,11,13,... so this sequence must start 2,5,9,12,...

Harry J. Smith, Table of n, a(n) for n = 1..1000
A. S. Fraenkel, The bracket function and complementary sets of integers, Canadian J. of Math. 21 (1969) 6-27. (Theorem XI)
Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.
S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, on ResearchGate.

Cf. A001951, A001952, A007064, A022846.

{ f=2 + sqrt(2); t=f/2; for (n=1, 1000, write("b063957.txt", n, " ", round(n*f - t)) ) } \\ Harry J. Smith, Sep 03 2009

from math import isqrt
def A063957(n): return (a:=(n<<1)-1)+(m:=isqrt(k:=a**2<<1)>>1)+int(((m<<1)+1)**2Chai Wah Wu, Feb 11 2025

a(n) = round((n - 1/2)*(2 + sqrt(2))) = round(n*3.4142...-1.7071...).

A214848 First difference of A022846.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1

Offset: 0

Philippe Deléham, Mar 08 2013

Number of triangular numbers in interval [n^2, (n+1)^2).
From Michel Dekking, Sep 20 2022: (Start)
(a(n)) is an inhomogeneous Sturmian sequence s(alpha, rho) with slope alpha = sqrt(2) and intercept 1/2, since A022846(n) = floor(n*sqrt(2) + 1/2).
(a(n)) is the fixed point of the morphism 1->12121, 2->1212121.
This is proved by writing the 0-1 version psi: 0->01010, 1->0101010 of this morphism as a composition
     psi = psi_1 psi_3 psi_1 psi_4,
where the psi_i are the three elementary Sturmian morphisms
     psi_1: 0->01, 1->0,   psi_3: 0->0, 1->01,   psi_4: 0->0, 1->10.
By Lemma 2.2.18 in Lothaire it then follows that the 0-1 word (a(n)-1) = A214848 is fixed by the morphism psi (note that in Lothaire psi_1 is phi, psi_3 is G, and psi_4 is G^~). (End)

			28 is in [25, 36), a(5) = 1.
36 and 45 are in [36, 49), a(6) = 2.

S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, in Number theory and its applications (Kyoto, 1997), pp. 347-373, Kluwer Acad. Publ., Dordrecht, 1999.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Svetlana Jitomirskaya, Small denominators and multiplicative Jensen's formula, ICM 2022. See the initial slides "Playing with numbers".
M. Lothaire, Algebraic combinatorics on words, Cambridge University Press. Online publication date: April 2013; Print publication year: 2002.
S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, on ResearchGate.

Cf. A022846, A006337, A006338.

a214848 n = a214848_list !! n
a214848_list = zipWith (-) (tail a022846_list) a022846_list
-- Reinhard Zumkeller, Mar 03 2014

Differences[Round[Sqrt[2]Range[0,100]]] (* Harvey P. Dale, Jun 14 2020 *)

For n > 0: a(n) = A006338(n). - Reinhard Zumkeller, Mar 03 2014

A002024 k appears k times; a(n) = floor(sqrt(2n) + 1/2).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13

Offset: 1

N. J. A. Sloane

Integer inverse function of the triangular numbers A000217. The function trinv(n) = floor((1+sqrt(1+8n))/2), n >= 0, gives the values 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, ..., that is, the same sequence with offset 0. - N. J. A. Sloane, Jun 21 2009
Array T(k,n) = n+k-1 read by antidiagonals.
Eigensequence of the triangle = A001563. - Gary W. Adamson, Dec 29 2008
Can apparently also be defined via a(n+1)=b(n) for n >= 2 where b(0)=b(1)=1 and b(n) = b(n-b(n-2))+1. Tested to be correct for all n <= 150000. - José María Grau Ribas, Jun 10 2011
For any n >= 0, a(n+1) is the least integer m such that A000217(m)=m(m+1)/2 is larger than n. This is useful when enumerating representations of n as difference of triangular numbers; see also A234813. - M. F. Hasler, Apr 19 2014
Number of binary digits of A023758, i.e., a(n) = ceiling(log_2(A023758(n+2))). - Andres Cicuttin, Apr 29 2016
a(n) and A002260(n) give respectively the x(n) and y(n) coordinates of the sorted sequence of points in the integer lattice such that x(n) > 0, 0 < y(n) <= x(n), and min(x(n), y(n)) < max(x(n+1), y(n+1)) for n > 0. - Andres Cicuttin, Dec 25 2016
Partial sums (A060432) are given by S(n) = (-a(n)^3 + a(n)*(1+6n))/6. - Daniel Cieslinski, Oct 23 2017
As an array, T(k,n) is the number of digits columns used in carryless multiplication between a k-digit number and an n-digit number. - Stefano Spezia, Sep 24 2022
a(n) is the maximum number of possible solutions to an n-statement Knights and Knaves Puzzle, where each statement is of the form "x of us are knights" for some 1 <= x <= n, knights can only tell the truth and knaves can only lie. - Taisha Charles and Brittany Ohlinger, Jul 29 2023

			From _Clark Kimberling_, Sep 16 2008: (Start)
As a rectangular array, a northwest corner:
  1 2 3 4 5 6
  2 3 4 5 6 7
  3 4 5 6 7 8
  4 5 6 7 8 9
This is the weight array (cf. A144112) of A107985 (formatted as a rectangular array). (End)
G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 4*x^9 + 4*x^9 + 4*x^10 + ...

Edward S. Barbeau, Murray S. Klamkin, and William O. J. Moser, Five Hundred Mathematical Challenges, Prob. 441, pp. 41, 194. MAA 1995.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 97.
K. Hardy and K. S. Williams, The Green Book of Mathematical Problems, p. 59, Solution to Prob. 14, Dover NY, 1985
R. Honsberger, Mathematical Morsels, pp. 133-134, MAA 1978.
J. F. Hurley, Litton's Problematical Recreations, pp. 152; 313-4 Prob. 22, VNR Co., NY, 1971.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 43.
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).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 129.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5050
Jaegug Bae and Sungjin Choi, A generalization of a subset-sum-distinct sequence, J. Korean Math. Soc. 40 (2003), no. 5, 757--768. MR1996839 (2004d:05198). See b(n).
Jonathan H. B. Deane and Guido Gentile, A diluted version of the problem of the existence of the Hofstadter sequence, arXiv:2311.13854 [math.NT], 2023. See p. 10.
Nathan Fox, Connecting Slow Solutions to Nested Recurrences with Linear Recurrent Sequences, arXiv:2203.09340 [math.CO], 2022.
H. T. Freitag and H. W. Gould, Solution to Problem 571, Math. Mag., 38 (1965), 185-187.
H. T. Freitag (Proposer) and H. W. Gould (Solver), Problem 571: An Ordering of the Rationals, Math. Mag., 38 (1965), 185-187 [Annotated scanned copy]
Mikel Garcia-de-Andoin, Álvaro Saiz, Pedro Pérez-Fernández, Lucas Lamata, Izaskun Oregi, and Mikel Sanz, Digital-Analog Quantum Computation with Arbitrary Two-Body Hamiltonians, arXiv:2307.00966 [quant-ph], 2023.
S. W. Golomb, Discrete chaos: sequences satisfying "strange" recursions, unpublished manuscript, circa 1990 [cached copy, with permission (annotated)]
Henry W. Gould, Letters to N. J. A. Sloane, Oct 1973 and Jan 1974.
Abraham Isgur, Vitaly Kuznetsov, and Stephen Tanny, A combinatorial approach for solving certain nested recursions with non-slow solutions, arXiv:1202.0276 [math.CO], 2012.
Stanley Wu-Wei Liu, Closed form expressions for A002024(n).
M. A. Nyblom, Some curious sequences involving floor and ceiling functions, Am. Math. Monthly 109 (#6, 200), 559-564.
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Raphael Schumacher, Extension of Summation Formulas involving Stirling series, arXiv:1605.09204 [math.NT], 2016.
Raphael Schumacher, The self-counting identity, Fib. Quart., 55 (No. 2 2017), 157-167.
Raphael Schumacher, The Self-Counting Flow, Fibonacci Quart. 60 (2022), no. 5, 324-343.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970. (note that A1148 has now become A005282)
Michael Somos, Sequences used for indexing triangular or square arrays.
L. J. Upton, Letter to N. J. A. Sloane, May 22 1991.
Eric Weisstein's World of Mathematics, Self-Counting Sequence.
Index entries for Hofstadter-type sequences

a(n+1) = 1+A003056(n), A022846(n)=a(n^2), a(n+1)=A002260(n)+A025581(n).
Cf. A001462, A002262, A003881, A004736, A127899, A107985, A001563, A014132, A000194, A005145, A131507, A093995, A060432 (partial sums).
A123578 is an essentially identical sequence.

a002024 n k = a002024_tabl !! (n-1) !! (k-1)
a002024_row n = a002024_tabl !! (n-1)
a002024_tabl = iterate (\xs@(x:_) -> map (+ 1) (x : xs)) [1]
a002024_list = concat a002024_tabl
a002024' = round . sqrt . (* 2) . fromIntegral
-- Reinhard Zumkeller, Jul 05 2015, Feb 12 2012, Mar 18 2011

a002024_list = [1..] >>= \n -> replicate n n

a002024 = (!!) $ [1..] >>= \n -> replicate n n
-- Sascha Mücke, May 10 2016

[Floor(Sqrt(2*n) + 1/2): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014

A002024 := n-> ceil((sqrt(1+8*n)-1)/2); seq(A002024(n), n=1..100);

a[1] = 1; a[n_] := a[n] = a[n - a[n - 1]] + 1 (* Branko Curgus, May 12 2009 *)
Table[n, {n, 13}, {n}] // Flatten (* Robert G. Wilson v, May 11 2010 *)
Table[PadRight[{},n,n],{n,15}]//Flatten (* Harvey P. Dale, Jan 13 2019 *)

t1(n)=floor(1/2+sqrt(2*n)) /* A002024 = this sequence */

t2(n)=n-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1) */

t3(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1 /* A004736 */

t4(n)=n-1-binomial(floor(1/2+sqrt(2*n)),2) /* A002260(n-1)-1 */

A002024(n)=(sqrtint(n*8)+1)\2 \\ M. F. Hasler, Apr 19 2014

PARI
```
a(n)=(sqrtint(8*n-7)+1)\2
```

a(n)=my(k=1);while(binomial(k+1,2)+1<=n,k++);k \\ R. J. Cano, Mar 17 2014

from math import isqrt
def A002024(n): return (isqrt(8*n)+1)//2 # Chai Wah Wu, Feb 02 2022

[floor(sqrt(2*n) +1/2) for n in (1..80)] # G. C. Greubel, Dec 10 2018

a(n) = floor(1/2 + sqrt(2n)). Also a(n) = ceiling((sqrt(1+8n)-1)/2). [See the Liu link for a large collection of explicit formulas. - N. J. A. Sloane, Oct 30 2019]
a((k-1)*k/2 + i) = k for k > 0 and 0 < i <= k. - Reinhard Zumkeller, Aug 30 2001
a(n) = a(n - a(n-1)) + 1, with a(1)=1. - Ian M. Levitt (ilevitt(AT)duke.poly.edu), Aug 18 2002
a(n) = round(sqrt(2n)). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
T(n,k) = A003602(A118413(n,k)); = T(n,k) = A001511(A118416(n,k)). - Reinhard Zumkeller, Apr 27 2006
G.f.: (x/(1-x))*Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic, Oct 06 2003
Equals A127899 * A004736. - Gary W. Adamson, Feb 09 2007
Sum_{i=1..n} Sum_{j=i..n+i-1} T(j,i) = A000578(n); Sum_{i=1..n} T(n,i) = A000290(n). - Reinhard Zumkeller, Jun 24 2007
a(n) + n = A014132(n). - Vincenzo Librandi, Jul 08 2010
a(n) = ceiling(-1/2 + sqrt(2n)). - Branko Curgus, May 12 2009
a(A169581(n)) = A038567(n). - Reinhard Zumkeller, Dec 02 2009
a(n) = round(sqrt(2*n)) = round(sqrt(2*n-1)); there exist a and b greater than zero such that 2*n = 2+(a+b)^2 -(a+3*b) and a(n)=(a+b-1). - Fabio Civolani (civox(AT)tiscali.it), Feb 23 2010
A005318(n+1) = 2*A005318(n) - A205744(n), A205744(n) = A005318(A083920(n)), A083920(n) = n - a(n). - N. J. A. Sloane, Feb 11 2012
Expansion of psi(x) * x / (1 - x) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Mar 19 2014
G.f.: (x/(1-x)) * Product_{n>=1} (1 + x^n) * (1 - x^(2*n)). - Paul D. Hanna, Feb 27 2016
a(n) = 1 + Sum_{i=1..n/2} ceiling(floor(2(n-1)/(i^2+i))/(2n)). - José de Jesús Camacho Medina, Jan 07 2017
a(n) = floor((sqrt(8*n-7)+1)/2). - Néstor Jofré, Apr 24 2017
a(n) = floor((A000196(8*n)+1)/2). - Pontus von Brömssen, Dec 10 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 (A003881). - Amiram Eldar, Oct 01 2022
G.f. as array: (x^2*(1 - y)^2 + y^2 + x*y*(1 - 2*y))/((1 - x)^2*(1 - y)^2). - Stefano Spezia, Apr 22 2024

A006338 An "eta-sequence": floor((n+1)sqrt(2) + 1/2) - floor(nsqrt(2) + 1/2).

Original entry on oeis.org

2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2

Offset: 1

D. R. Hofstadter, Jul 15 1977

Equals its own "second derivative" (cf. A006337).
Presumably this is the same as the following sequence from Hofstadter's book: the number of triangular numbers between each successive pair of squares. More precisely, a(n) is the number of triangular numbers T such that n^2 <= T < (n+1)^2. E.g., a(3) = 2 because 3^2 <= T < 4^2 permits T(4) = 10 and T(5) = 15 and no other triangular number. - Hugo van der Sanden, May 03 2005.
a(n) = A214848(n) = A022846(n+1) - A022846(n). - Reinhard Zumkeller, Mar 03 2014

Douglas Hofstadter, "Fluid Concepts and Creative Analogies", Chapter 1: "To seek whence cometh a sequence".
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991

Cf. A006337, A022846, A214848.

a006338 n = a006338_list !! (n-1)
a006338_list = tail a214848_list
-- Reinhard Zumkeller, Mar 03 2014

[Floor((n+1)*Sqrt(2)+1/2) - Floor(n*Sqrt(2)+1/2): n in [1..30]]; // G. C. Greubel, Nov 18 2017

a[n_] := Floor[(n+1)*Sqrt[2]+1/2] - Floor[n*Sqrt[2]+1/2]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Nov 24 2015 *)
Differences[Table[Floor[n Sqrt[2]+1/2],{n,120}]] (* Harvey P. Dale, Dec 10 2021 *)

for(n=1,30, print1(floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2), ", ")) \\ G. C. Greubel, Nov 18 2017

a(n) = floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2). - G. C. Greubel, Nov 18 2017

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003

A057062 Let R(i,j) be the infinite square array with antidiagonals 1; 2,3; 4,5,6; ...; the n-th prime is in antidiagonal a(n).

Original entry on oeis.org

2, 2, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25

Offset: 1

Clark Kimberling, Jul 30 2000

nonn

The smallest integer in the j-th antidiagonal is A000124(j-1). So a(n) is the index j such that A000124(j-1) <= prime(n) < A000124(j). - R. J. Mathar, Dec 02 2011

			The array begins
   1  3  6 10 15 ...
   2  5  9 14 ...
   4  8 13 ...
   7 12 ...
  11 ...
  ...
The third prime, 5, is in the 3rd antidiagonal, so a(3) = 3.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A057045, A057048, A022846, A057057, A057054. A066888 counts how many times each positive integer appears in this sequence.

Cf. A010051.

a057062 n = a057062_list !! (n-1)
a057062_list = f 1 [1..] where
   f j xs = (replicate (sum $ map a010051 dia) j) ++ f (j + 1) xs'
     where (dia, xs') = splitAt j xs
-- Reinhard Zumkeller, Jul 26 2012

Table[Round[Sqrt[2*Prime[n]]], {n, 100}] (* T. D. Noe, Dec 03 2011 *)

a(n)=(sqrtint(8*prime(n))+1)\2 \\ Charles R Greathouse IV, Jul 26 2012

from math import isqrt
from sympy import prime
def A057062(n): return isqrt(prime(n)<<3)+1>>1 # Chai Wah Wu, Jun 19 2024

a(n) = round(sqrt(2*prime(n))). - Vladeta Jovovic, Jun 14 2003

A214857 Number of triangular numbers in interval [0, n^2].

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 33, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 68, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 95, 96, 98, 99, 100

Offset: 0

Philippe Deléham, Mar 09 2013

Partial sums of A214856.

			0, 1, 3, 6 are in interval [0, 9], a(3) = 4.
0, 1, 3, 6, 10, 15 are in interval [0, 16], a(4) = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A022846, A214848, A214856.

nn = 100; tri = Table[n (n + 1)/2, {n, 0, nn}]; Table[Count[tri, ?(# <= n^2 &)], {n, 0, Sqrt[tri[[-1]]]}] (* _T. D. Noe, Mar 11 2013 *)
Table[Floor[(Sqrt[8*n^2+1]-1)/2]+1,{n,0,80}] (* Harvey P. Dale, Oct 14 2014 *)

from math import isqrt
def A214857(n): return isqrt(n**2+1<<3)+1>>1 # Chai Wah Wu, Dec 09 2024

a(n) = floor((1 + sqrt(1+8*n^2))/2). - Ralf Stephan, Jan 30 2014

A057045 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; the n-th Lucas number is in antidiagonal a(n).

Original entry on oeis.org

2, 1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 41, 52, 66, 85, 107, 137, 174, 221, 281, 358, 455, 579, 737, 937, 1192, 1516, 1929, 2454, 3121, 3970, 5050, 6424, 8171, 10394, 13221, 16818, 21393, 27212

Offset: 1

Clark Kimberling, Jul 30 2000

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A057062, A057048, A022846, A057057, A057054.

from gmpy2 import isqrt_rem, lucas
def A057045(n):
    i, j = isqrt_rem(2*lucas(n-1))
    return int(i + int(4*(j-i) >= 1)) # Chai Wah Wu, Aug 16 2016

Round(sqrt(2*A000032(n-1))). - Vladeta Jovovic, Jun 14 2003

A184867 Numbers m such that prime(m) is of the form floor(1/2+k*sqrt(2)). Complement of A184870.

Original entry on oeis.org

2, 4, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 20, 22, 23, 24, 27, 28, 29, 30, 31, 33, 34, 36, 37, 38, 39, 40, 42, 43, 45, 46, 47, 48, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 86, 88, 89, 91, 92, 94, 95, 96, 97, 99, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 113, 114, 115, 116, 117, 119, 120, 122, 123, 124, 126, 127, 129, 130, 133, 134, 135, 136, 140, 143, 144, 145, 146

Offset: 1

Clark Kimberling, Jan 23 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A022846, A184865, A184866, A184870.

r=2^(1/2); h=1/2; a[n_]:=Floor[n*r+h];
Table[a[n], {n, 1, 120}] (* A022846, int. nearest 2^(1/2) *)
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1, a[n]]], {n, 1, 600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2, n]], {n, 1, 600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3, n]], {n, 1, 300}]; t3
(* Lists t1, t2, t3 match A184865, A184866, A184867. *)

A057054 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; n^3 is in antidiagonal a(n).

Original entry on oeis.org

1, 4, 7, 11, 16, 21, 26, 32, 38, 45, 52, 59, 66, 74, 82, 91, 99, 108, 117, 126, 136, 146, 156, 166, 177, 187, 198, 210, 221, 232, 244, 256, 268, 280, 293, 305, 318, 331, 344, 358, 371, 385, 399, 413, 427, 441, 456, 470, 485, 500

Offset: 1

Clark Kimberling, Jul 30 2000

nonn

Cf. A057045, A057062, A057048, A022846, A057057.

Table[Round[n Sqrt[2n]],{n,50}] (* Harvey P. Dale, Feb 10 2020 *)

Round(n*(sqrt(2*n))). - Vladeta Jovovic, Jun 14 2003

A057057 Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; C(n,3) is in antidiagonal a(n).

Original entry on oeis.org

1, 3, 4, 6, 8, 11, 13, 15, 18, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 55, 60, 64, 68, 72, 76, 81, 85, 90, 95, 100, 104, 109, 114, 119, 125, 130, 135, 141, 146, 152, 157, 163, 168, 174, 180, 186, 192, 198, 204, 210, 216, 223, 229

Offset: 3

Clark Kimberling, Jul 30 2000

nonn

Cf. A057045, A057062, A057048, A022846, A057054.

Table[Round[Sqrt[2*Binomial[n,3]]],{n,3,60}] (* Harvey P. Dale, Nov 16 2021 *)

Round(sqrt(2*binomial(n, 3))). - Vladeta Jovovic, Jun 14 2003

cp's OEIS Frontend

A063957 Numbers not of the form round(m*sqrt(2)) for any integer m, i.e., complement of A022846.

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A214848 First difference of A022846.

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A002024 k appears k times; a(n) = floor(sqrt(2n) + 1/2).

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A006338 An "eta-sequence": floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2).

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A057062 Let R(i,j) be the infinite square array with antidiagonals 1; 2,3; 4,5,6; ...; the n-th prime is in antidiagonal a(n).

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A006338 An "eta-sequence": floor((n+1)sqrt(2) + 1/2) - floor(nsqrt(2) + 1/2).