A205744 -id:A205744 - cp's OEIS frontend

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

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

A005318 Conway-Guy sequence: a(n + 1) = 2a(n) - a(n - floor( 1/2 + sqrt(2n) )).

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 24, 44, 84, 161, 309, 594, 1164, 2284, 4484, 8807, 17305, 34301, 68008, 134852, 267420, 530356, 1051905, 2095003, 4172701, 8311101, 16554194, 32973536, 65679652, 130828948, 261127540, 521203175, 1040311347, 2076449993, 4144588885, 8272623576

Offset: 0

N. J. A. Sloane

Conway and Guy conjecture that the set of k numbers {s_i = a(k) - a(k-i) : 1 <= i <= k} has the property that all its subsets have distinct sums - see Guy's book. These k-sets are the rows of A096858. [This conjecture has apparently now been proved by Bohman. - I. Halupczok (integerSequences(AT)karimmi.de), Feb 20 2006]

J. H. Conway and R. K. Guy, Solution of a problem of Erdos, Colloq. Math. 20 (1969), p. 307.
R. K. Guy, Unsolved Problems in Number Theory, C8.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Wald, Problem 1192, Unequal sums, J. Rec. Math., 15 (No. 2, 1983-1984), pp. 148-149.

Alois P. Heinz, Table of n, a(n) for n = 0..3324 (first 301 terms from T. D. Noe)
Tom Bohman, A sum packing problem of Erdos and the Conway-Guy sequence, Proc. AMS 124, (No. 12, 1996), pp. 3627-3636.
P. Borwein and M. J. Mossinghoff, Newman Polynomials with Prescribed Vanishing and Integer Sets with Distinct Subset Sums, Math. Comp., 72 (2003), 787-800.
J. H. Conway & R. K. Guy, Sets of natural numbers with distinct sums, Manuscript.
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982.
R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982. (Annotated scanned copy)
G. Kreweras, Sur quelques problèmes relatifs au vote pondéré [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), 45-63.
G. Kreweras, Alvarez Rodriguez, Miguel-Angel, Pondération entière minimale de N telle que pour tout k toutes les k-parties de N aient des poids distincts, [Minimal integer weighting of N such that for any k all the k-subsets of N have unequal weights] C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 8, 345-347.
G. Kreweras, Alvarez Rodriguez, Miguel-Angel, Pondération entière minimale de N telle que pour tout k toutes les k-parties de N aient des poids distincts, [Minimal integer weighting of N such that for any k all the k-subsets of N have unequal weights], C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 8, 345-347. (Annotated scanned copy)
W. F. Lunnon, Integer sets with distinct subset-sums, Math. Comp. 50 (1988), 297-320.
M. Wald & N. J. A. Sloane, Correspondence and Attachment, 1987

A276661 is the main entry for the distinct subset sums problem.

Cf. A037254, A096858, A096796, A096824, A205744, A206239, A083920, A003056.

a005318 n = a005318_list !! n
a005318_list = 0 : 1 : zipWith (-)
   (map (* 2) $ tail a005318_list) (map a005318 a083920_list)
-- Reinhard Zumkeller, Feb 12 2012

a[n_] := a[n] = 2*a[n-1] - a[n - Floor[Sqrt[2]*Sqrt[n-1] + 1/2] - 1]; a[0]=0; a[1]=1; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, May 15 2013 *)

a(n)=if(n<=1,n==1,2*a(n-1)-a(n-1-(sqrtint(8*n-15)+1)\2))

A=[]; /* This is the program above with memoization. */
a(n)=if(n<3, return(n)); if(n>#A, A=concat(A,vector(n-#A)), if(A[n], return(A[n]))); A[n]=2*a(n-1)-a(n-1-(sqrtint(8*n-15)+1)\2) \\ Charles R Greathouse IV, Sep 09 2016

from sympy import sqrt, floor
def a(n): return n if n<2 else 2*a(n - 1) - a(n - floor(sqrt(2)*sqrt(n - 1) + 1/2) - 1) # Indranil Ghosh, Jun 03 2017

a(n+1) = 2*a(n)-A205744(n), A205744(n) = a(A083920(n)), A083920(n) = n - A002024(n). - N. J. A. Sloane, Feb 11 2012

More terms from Larry Reeves (larryr(AT)acm.org), Sep 21 2000

A083920 Number of nontriangular numbers <= n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62

Offset: 0

Clark Kimberling, May 08 2003

nonn

An alternative description: the sequence of nonnegative integers with the triangular numbers repeated.

a(t(n)) = t(n+1), where t(n)=A000217(n)=n(n+1)/2, the n-th triangular number. For n>=1, a(n)=a(n-1) if and only if n is a triangular number; otherwise, a(n)=1+a(n-1).

			a(7)=4 counts the nontriangular numbers, 2,4,5,7, that are <=7.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
R. K. Guy, Sets of integers whose subsets have distinct sums, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982. See Table 1, column (4).

Cf. A000217, A002024, A003056, A005318, A205744.
Essentially partial sums of A023532.
Number of nonzero terms in row n+1 of A342557.

a083920 n = a083920_list !! n
a083920_list = scanl1 (+) $ map (1 -) a010054_list
-- Reinhard Zumkeller, Feb 12 2012

[n-Floor((Sqrt(8*n+1)-1)/2):n in [1..75]]; // Marius A. Burtea, Jun 19 2019

f[n_] := n - Floor[(Sqrt[8n + 1] - 1)/2]; Table[ f[n], {n, 0, 73}] (* Robert G. Wilson v, Oct 22 2005 *)
Accumulate[Table[If[OddQ[Sqrt[8n+1]],0,1],{n,0,120}]] (* Harvey P. Dale, Oct 14 2014 *)

a(n)=n-(sqrtint(8*n+1)-1)\2 \\ Charles R Greathouse IV, Sep 02 2015

from math import isqrt
def A083920(n): return n-(k:=isqrt(m:=n+1<<1))+((m>=k*(k+1)+1)^1) # Chai Wah Wu, Jun 07 2025

a(n) = n-floor((x-1)/2) = n-A003056(n), where x = sqrt(8*n+1).
A005318(n+1) = 2*A005318(n)-A205744(n), A205744(n) = A005318(a(n)), a(n) = n - A002024(n). - N. J. A. Sloane, Feb 11 2012
G.f.: 1/(1 - x)^2 - (1/(1 - x))*Product_{k>=1} (1 - x^(2*k))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, May 30 2017
a(n) = n - floor(sqrt(2*n + 1) - 1/2). - Ridouane Oudra, Jun 19 2019

Added alternative definition and Guy reference. - N. J. A. Sloane, Feb 09 2012

cp's OEIS Frontend

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

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A005318 Conway-Guy sequence: a(n + 1) = 2a(n) - a(n - floor( 1/2 + sqrt(2n) )).

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A083920 Number of nontriangular numbers <= n.

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A206239 Subsequence A005318(t_n), where t_n is the n-th triangular number (A000217).

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