A002024 -id:A002024 - cp's OEIS frontend

A001614 Connell sequence: 1 odd, 2 even, 3 odd, ...

1, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 23, 25, 26, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 122

Offset: 1

N. J. A. Sloane

Next (2n-1) odd numbers alternating with next 2n even numbers. Squares (A000290(n)) occur at the A000217(n)-th entry. - Lekraj Beedassy, Aug 06 2004. - Comment corrected by Daniel Forgues, Jul 18 2009
a(t_n) = a(n(n+1)/2) = n^2 relates squares to triangular numbers. - Daniel Forgues
The natural numbers not included are A118011(n) = 4n - a(n) as n=1,2,3,... - Paul D. Hanna, Apr 10 2006
As a triangle with row sums = A069778 (1, 6, 21, 52, 105, ...): /Q 1;/Q 2, 4;/Q 5, 7, 9;/Q 10, 12, 14, 16;/Q ... . - Gary W. Adamson, Sep 01 2008
The triangle sums, see A180662 for their definitions, link the Connell sequence A001614 as a triangle with six sequences, see the crossrefs. - Johannes W. Meijer, May 20 2011
a(n) = A122797(n) + n - 1. - Reinhard Zumkeller, Feb 12 2012

			From _Omar E. Pol_, Aug 13 2013: (Start)
Written as a triangle the sequence begins:
   1;
   2,  4;
   5,  7,  9;
  10, 12, 14, 16;
  17, 19, 21, 23, 25;
  26, 28, 30, 32, 34, 36;
  37, 39, 41, 43, 45, 47, 49;
  50, 52, 54, 56, 58, 60, 62, 64;
  65, 67, 69, 71, 73, 75, 77, 79, 81;
  82, 84, 86, 88, 90, 92, 94, 96, 98, 100;
  ...
Right border gives A000290, n >= 1.
(End)

C. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, p. 276.
C. A. Pickover, The Mathematics of Oz, Chapter 39, Camb. Univ. Press UK 2002.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000
Ian Connell and Andrew Korsak, Problem E1382, Amer. Math. Monthly, 67 (1960), 380.
Douglas E. Iannucci and Donna Mills-Taylor, On Generalizing the Connell Sequence, J. Integer Sequences, Vol. 2, 1999, #99.1.7.
H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Gary E. Stevens, A Connell-Like Sequence, J. Integer Sequences, Vol. 1, 1998, #98.1.4.
Eric Weisstein's World of Mathematics, Connell Sequence

Cf. A117384, A118011 (complement), A118012.
Cf. A069778. - Gary W. Adamson, Sep 01 2008
From Johannes W. Meijer, May 20 2011: (Start)
Triangle columns: A002522, A117950 (n>=1), A117951 (n>=2), A117619 (n>=3), A154533 (n>=5), A000290 (n>=1), A008865 (n>=2), A028347 (n>=3), A028878 (n>=1), A028884 (n>=2), A054569 [T(2*n,n)].
Triangle sums (see the comments): A069778 (Row1), A190716 (Row2), A058187 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23, Fi1, Fi2, Ze1 and Ze2), A000292 (Related to Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4), A190717 (Related to Ca1, Ca2, Ze3, Ze4), A190718 (Related to Gi1 and Gi2). (End)
Cf. A014132, A057211, A023531.

a001614 n = a001614_list !! (n-1)
a001614_list = f 0 0 a057211_list where
   f c z (x:xs) = z' : f x z' xs where z' = z + 1 + 0 ^ abs (x - c)
-- Reinhard Zumkeller, Dec 30 2011

[2*n-Round(Sqrt(2*n)): n in [1..80]]; // Vincenzo Librandi, Apr 17 2015

A001614:=proc(n): 2*n - floor((1+sqrt(8*n-7))/2) end: seq(A001614(n),n=1..67); # Johannes W. Meijer, May 20 2011

lst={};i=0;For[j=1, j<=4!, a=i+1;b=j;k=0;For[i=a, i<=9!, k++;AppendTo[lst, i];If[k>=b, Break[]];i=i+2];j++ ];lst (* Vladimir Joseph Stephan Orlovsky, Aug 29 2008 *)
row[n_] := 2*Range[n+1]+n^2-1; Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Oct 25 2013 *)

a(n)=2*n - round(sqrt(2*n)) \\ Charles R Greathouse IV, Apr 20 2015

from math import isqrt
def A001614(n): return (m:=n<<1)-(k:=isqrt(m))-int((m<<2)>(k<<2)*(k+1)+1) # Chai Wah Wu, Jul 26 2022

a(n) = 2*n - floor( (1+ sqrt(8*n-7))/2 ).
a(n) = A005843(n) - A002024(n). - Lekraj Beedassy, Aug 06 2004
a(n) = A118012(A118011(n)). A117384( a(n) ) = n; A117384( 4*n - a(n) ) = n. - Paul D. Hanna, Apr 10 2006
a(1) = 1; then a(n) = a(n-1)+1 if a(n-1) is a square, a(n) = a(n-1)+2 otherwise. For example, a(21)=36 is a square therefore a(22)=36+1=37 which is not a square so a(23)=37+2=39 ... - Benoit Cloitre, Feb 07 2007
T(n,k) = (n-1)^2 + 2*k - 1. - Omar E. Pol, Aug 13 2013
a(n)^2 = a(n*(n+1)/2). - Ivan N. Ianakiev, Aug 15 2013
Let the sequence be written in the form of the triangle in the EXAMPLE section below and let a(n) and a(n+1) belong to the same row of the triangle. Then a(n)*a(n+1) + 1 = a(A000217(A118011(n))) = A000290(A118011(n)). - Ivan N. Ianakiev, Aug 16 2013
a(n) = 2*n-round(sqrt(2*n)). - Gerald Hillier, Apr 15 2015
From Robert Israel, Apr 20 2015 (Start):
G.f.: 2*x/(1-x)^2 - (x/(1-x))*Sum_{n>=0} x^(n*(n+1)/2) = 2*x/(1-x)^2 - (Theta2(0,x^(1/2)))*x^(7/8)/(2*(1-x)) where Theta2 is a Jacobi theta function.
a(n) = 2*n-1 - Sum_{i=0..n-2} A023531(i).  (End)
a(n) = 3*n-A014132(n). - Chai Wah Wu, Oct 19 2024

More terms from Larry Reeves (larryr(AT)acm.org), Mar 16 2001

A072643 Half of the binary width of the terms of A014486, the number of digits in A063171(n)/2.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Jun 02 2002

Paolo Xausa, Table of n, a(n) for n = 0..10000

Each value v occurs A000108(v) times. The maximum position for value v to occur is A014138(v). Permutations: A071673, A072644, A072645, A072660. Cf. also A002024, A072649.

a[n_] := Module[{i, c, a}, i = c = 0; a = 1; While[n>c, a *= (4*i+2)/(i+2); i++; c += a]; i];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 26 2017, from Sage code *)
Flatten[Array[Table[#, CatalanNumber[#]]&, 7, 0]] (* Paolo Xausa, Feb 13 2024 *)

def A072643(n) :
    i = c = 0; a = 1
    while n > c :
        a *= (4*i+2)/(2+i)
        i += 1; c += a
    return i
[A072643(n) for n in (0..100)] # Peter Luschny, Sep 07 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = Sum_{n>=1} (-1)^(n+1)/(2^n-1) = 0.76449978034844420919... . - Amiram Eldar, Feb 18 2024

A046090 Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives X+1 values.

Original entry on oeis.org

1, 4, 21, 120, 697, 4060, 23661, 137904, 803761, 4684660, 27304197, 159140520, 927538921, 5406093004, 31509019101, 183648021600, 1070379110497, 6238626641380, 36361380737781, 211929657785304, 1235216565974041, 7199369738058940, 41961001862379597, 244566641436218640

Offset: 0

Eric W. Weisstein

Solution to a*(a-1) = 2b*(b-1) in natural numbers: a = a(n), b = b(n) = A011900(n).
n such that n^2 = (1/2)*(n+floor(sqrt(2)*n*floor(sqrt(2)*n))). - Benoit Cloitre, Apr 15 2003
Place a(n) balls in an urn, of which b(n) = A011900(n) are red; draw 2 balls without replacement; 2*Probability(2 red balls) = Probability(2 balls); this is equivalent to the Pell equation A(n)^2-2*B(n)^2 = -1 with a(n) = (A(n)+1)/2; b(n) = (B(n)+1)/2; and the fundamental solution (7;5) and the solution (3;2) for the unit form. - Paul Weisenhorn, Aug 03 2010
Find base x in which repdigit yy has a square that is repdigit zzzz, corresponding to Diophantine equation zzzz_x = (yy_x)^2; then, solution z = a(n) with x = A002315(n) and y = A001653(n+1) for n >= 1 (see Maurice Protat reference). - Bernard Schott, Dec 21 2022

			For n=4: a(4)=697; b(4)=493; 2*binomial(493,2)=485112=binomial(697,2). - _Paul Weisenhorn_, Aug 03 2010

A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
Maurice Protat, Des Olympiades à l'Agrégation, De zzzz_x = (yy_x)^2 à Pell-Fermat, Problème 23, pp. 52-54, Ellipses, Paris, 1997.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
T. W. Forget and T. A. Larkin, Pythagorean triads of the form X, X+1, Z described by recurrence sequences, Fib. Quart., 6 (No. 3, 1968), 94-104.
L. J. Gerstein, Pythagorean triples and inner products, Math. Mag., 78 (2005), 205-213.
H. J. Hindin, Stars, hexes, triangular numbers and Pythagorean triples, J. Rec. Math., 16 (1983/1984), 191-193. (Annotated scanned copy)
Ron Knott, Pythagorean Triples and Online Calculators
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Other 2 sides are A001652 and A001653.
Cf. A001009, A001541, A001542, A011900, A046729, A053141, A084159, A084703, A156035.
See comments in A301383.
Cf. A001109, A001653, A002024, A002315, A029549.

a046090 n = a046090_list !! n
a046090_list = 1 : 4 : map (subtract 2)
   (zipWith (-) (map (* 6) (tail a046090_list)) a046090_list)
-- Reinhard Zumkeller, Jan 10 2012

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)/((1-6*x+x^2)*(1-x)))); // G. C. Greubel, Jul 15 2018

Digits:=100: seq(round((1+(7+5*sqrt(2))*(3+2*sqrt(2))^(n-1))/2)/2, n=0..20); # Paul Weisenhorn, Aug 03 2010

Join[{1},#+1&/@With[{c=3+2Sqrt[2]},NestList[Floor[c #]+3&,3,20]]] (* Harvey P. Dale, Aug 19 2011 *)
LinearRecurrence[{7,-7,1},{1,4,21},25] (* Harvey P. Dale, Apr 13 2012 *)
a[n_] := (2-ChebyshevT[n, 3]+ChebyshevT[n+1, 3])/4; Array[a, 21, 0] (* Jean-François Alcover, Jul 10 2016, adapted from PARI *)

a(n)=(2-subst(poltchebi(abs(n))-poltchebi(abs(n+1)),x,3))/4

x='x+O('x^30); Vec((1-3*x)/((1-6*x+x^2)*(1-x))) \\ G. C. Greubel, Jul 15 2018

a(n) = (-1+sqrt(1+8*b(n)*(b(n)+1)))/2 with b(n) = A011900(n). [corrected by Michel Marcus, Dec 23 2022]
a(n) = 6*a(n-1) - a(n-2) - 2, n >= 2, a(0) = 1, a(1) = 4.
a(n) = (A(n+1) - 3*A(n) + 2)/4 with A(n) = A001653(n).
A001652(n) = -a(-1-n).
From Barry E. Williams, May 03 2000: (Start)
G.f.: (1-3*x)/((1-6*x+x^2)*(1-x)).
a(n) = partial sums of A001541(n). (End)
From Charlie Marion, Jul 01 2003: (Start)
A001652(n)*A001652(n+1) + a(n)*a(n+1) = A001542(n+1)^2 = A084703(n+1).
Let a(n) = A001652(n), b(n) = this sequence and c(n) = A001653(n). Then for k > j, c(i)*(c(k) - c(j)) = a(k+i) + ... + a(i+j+1) + a(k-i-1) + ... + a(j-i) + k - j. For n < 0, a(n) = -b(-n-1). Also a(n)*a(n+2k+1) + b(n)*b(n+2k+1) + c(n)*c(n+2k+1) = (a(n+k+1) - a(n+k))^2; a(n)*a(n+2k) + b(n)*b(n+2k) + c(n)*c(n+2k) = 2*c(n+k)^2. (End)
a(n) = 1/2 + ((1-2^(1/2))/4)*(3 - 2^(3/2))^n + ((1+2^(1/2))/4)*(3 + 2^(3/2))^n. - Antonio Alberto Olivares, Oct 13 2003
2*a(n) = 2*A084159(n) + 1 + (-1)^(n+1) = 2*A046729(n) + 1 - (-1)^(n+1). - Lekraj Beedassy, Jul 16 2004
a(n) = A001109(n+1) - A053141(n). - Manuel Valdivia, Apr 03 2010
From Paul Weisenhorn, Aug 03 2010: (Start)
a(n+1) = round((1+(7+5*sqrt(2))*(3+2*sqrt(2))^n)/2);
b(n+1) = round((2+(10+7*sqrt(2))*(3+2*sqrt(2))^n)/4) = A011900(n+1).
(End)
a(n)*(a(n)-1)/2 = b(n)*b(n+1) and 2*a(n) - 1 = b(n) + b(n+1), where b(n) = A001109. - Kenneth J Ramsey, Apr 24 2011
T(a(n)) = A011900(n)^2 + A001109(n), where T(n) is the n-th triangular number. See also A001653. - Charlie Marion, Apr 25 2011
a(0)=1, a(1)=4, a(2)=21, a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3). - Harvey P. Dale, Apr 13 2012
Limit_{n->oo} a(n+1)/a(n) = 3 + 2*sqrt(2) = A156035. - Ilya Gutkovskiy, Jul 10 2016
a(n) = A001652(n)+1. - Dimitri Papadopoulos, Jul 06 2017
a(n) = (A002315(n) + 1)/2. - Bernard Schott, Dec 21 2022
E.g.f.: (exp(x) + exp(3*x)*(cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)))/2. - Stefano Spezia, Mar 16 2024
a(n) = A002024(A029549(n))+1. - Pontus von Brömssen, Sep 11 2024

Additional comments from Wolfdieter Lang

Comment moved to A001653 by Claude Morin, Sep 22 2023

A307730 a(n) = A307720(n) * A307720(n+1).

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6, 9, 9, 9, 9, 9, 9, 9, 9, 9, 12, 8, 8, 8, 8, 8, 8, 8, 8, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 15, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 14, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 14, 14, 14, 14, 14, 14, 14

Offset: 1

Rémy Sigrist, Apr 25 2019

For all positive integers n, n appears n times.

			The first terms in this sequence and in A307720 are:
  n   a(n)  A307720(n)
  --  ----  ----------
   1     1           1
   2     2           1
   3     2           2
   4     3           1
   5     3           3
   6     3           1
   7     6           3
   8     4           2
   9     4           2
  10     4           2

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, Scatterplot of the first 10000000 terms
Rémy Sigrist, PARI program for A307730
N. J. A. Sloane, Table of n, a(n) for n = 1..999999

Cf. A002024, A088178, A307720.

Cf. A348579 (indices of occurrence of each number), A348246 (first occurrence of each number), A348409 (last occurrence).

PARI
```
\\ See Links section.
```

from itertools import islice
from collections import Counter
def A307730(): # generator of terms. Greedy algorithm
    c, b = Counter(), 1
    while True:
        k, kb = 1, b
        while c[kb] >= kb:
            k += 1
            kb += b
        c[kb] += 1
        b = k
        yield kb
A307730_list = list(islice(A307730(),100)) # Chai Wah Wu, Oct 21 2021

A054582 Array read by antidiagonals upwards: A(m,k) = 2^m * (2k+1), m,k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 16, 24, 20, 14, 9, 32, 48, 40, 28, 18, 11, 64, 96, 80, 56, 36, 22, 13, 128, 192, 160, 112, 72, 44, 26, 15, 256, 384, 320, 224, 144, 88, 52, 30, 17, 512, 768, 640, 448, 288, 176, 104, 60, 34, 19, 1024, 1536, 1280, 896, 576, 352, 208, 120

Offset: 0

Henry Bottomley, Apr 12 2000

First column of array is powers of 2, first row is odd numbers, other cells are products of these two, so every positive integer appears exactly once. [Comment edited to match the definition. - L. Edson Jeffery, Jun 05 2015]
An analogous N X N <-> N bijection based, not on the binary, but on the Fibonacci number system, is given by the Wythoff array A035513.
As an array, this sequence (hence also A135764) is the dispersion of the even positive integers. For the definition of dispersion, see the link "Interspersions and Dispersions." The fractal sequence of this dispersion is A003602. - Clark Kimberling, Dec 03 2010

			Northwest corner of array A:
    1     3     5     7     9    11    13    15    17    19
    2     6    10    14    18    22    26    30    34    38
    4    12    20    28    36    44    52    60    68    76
    8    24    40    56    72    88   104   120   136   152
   16    48    80   112   144   176   208   240   272   304
   32    96   160   224   288   352   416   480   544   608
   64   192   320   448   576   704   832   960  1088  1216
  128   384   640   896  1152  1408  1664  1920  2176  2432
  256   768  1280  1792  2304  2816  3328  3840  4352  4864
  512  1536  2560  3584  4608  5632  6656  7680  8704  9728
[Array edited to match the definition. - _L. Edson Jeffery_, Jun 05 2015]
From _Philippe Deléham_, Dec 13 2013: (Start)
a(13-1)=20=2*10, so a(13)=10+A006519(20)=10+4=14.
a(3-1)=3=2*1+1, so a(3)=2^(1+1)=4. (End)
From _Wolfdieter Lang_, Jan 30 2019: (Start)
The triangle T begins:
   n\k   0    1    2   3   4   5   6   7  8  9 10 ...
   0:    1
   1:    2    3
   2:    4    6    5
   3:    8   12   10   7
   4:   16   24   20  14   9
   5:   32   48   40  28  18  11
   6:   64   96   80  56  36  22  13
   7:  128  192  160 112  72  44  26  15
   8:  256  384  320 224 144  88  52  30 17
   9:  512  768  640 448 288 176 104  60 34 19
  10: 1024 1536 1280 896 576 352 208 120 68 38 21
  ...
T(3, 2) = 2^1*(2*2+1) = 10. (End)

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Clark Kimberling, Interspersions and Dispersions.
Index entries for sequences that are permutations of the natural numbers

The sequence is a permutation of A000027.
Main diagonal is A014480; inverse permutation is A209268.
Cf. A001511, A002024, A003602, A006519, A025480, A035513, A075300, A135764.

a054582 n k = a054582_tabl !! n !! k
a054582_row n = a054582_tabl !! n
a054582_tabl = iterate
   (\xs@(x:_) -> (2 * x) : zipWith (+) xs (iterate (`div` 2) (2 * x))) [1]
a054582_list = concat a054582_tabl
-- Reinhard Zumkeller, Jan 22 2013

(* Array: *)
Grid[Table[2^m*(2*k + 1), {m, 0, 9}, {k, 0, 9}]] (* L. Edson Jeffery, Jun 05 2015 *)
(* Array antidiagonals flattened: *)
Flatten[Table[2^(m - k)*(2*k + 1), {m, 0, 9}, {k, 0, m}]] (* L. Edson Jeffery, Jun 05 2015 *)

T(m,k)=(2*k+1)<Charles R Greathouse IV, Jun 21 2017

As a sequence, if n is a triangular number, then a(n)=a(n-A002024(n))+2, otherwise a(n)=2*a(n-A002024(n)-1).
a(n) = A075300(n-1)+1.
Recurrence for the sequence: if a(n-1)=2*k is even, then a(n)=k+A006519(2*k); if a(n-1)=2*k+1 is odd, then a(n)=2^(k+1), a(0)=1. - Philippe Deléham, Dec 13 2013
m = A(A001511(m)-1, A003602(m)-1), for each m in A000027. - L. Edson Jeffery, Nov 22 2015
The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1), for n >= 0 and k = 0..n. - Wolfdieter Lang, Jan 30 2019

Offset corrected by Reinhard Zumkeller, Jan 22 2013

A264401 Triangle read by rows: T(n,k) is the number of partitions of n having least gap k.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 4, 2, 1, 4, 6, 4, 1, 7, 8, 5, 2, 8, 11, 8, 3, 12, 15, 10, 4, 1, 14, 20, 15, 6, 1, 21, 26, 19, 9, 2, 24, 35, 27, 12, 3, 34, 45, 34, 17, 5, 41, 58, 47, 23, 6, 1, 55, 75, 59, 31, 10, 1, 66, 96, 79, 41, 13, 2

Offset: 0

Emeric Deutsch, Nov 21 2015

The "least gap" or "mex" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
Sum of entries in row n is A000041(n).
T(n,1) = A002865(n).
Sum_{k>=1} k*T(n,k) = A022567(n).

			Row n=5 is 2,3,2; indeed, the least gaps of [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1] are 1, 2, 1, 2, 3, 3, and 2, respectively (i.e., two 1s, three 2s, and two 3s).
Triangle begins:
   1
   0   1
   1   1
   1   1   1
   2   2   1
   2   3   2
   4   4   2   1
   4   6   4   1
   7   8   5   2
   8  11   8   3
  12  15  10   4   1
  14  20  15   6   1
  21  26  19   9   2

Alois P. Heinz, Rows n = 0..1000, flattened
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
P. J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Row sums are A000041.
Row lengths are A002024.
Column k = 1 is A002865.
Column k = 2 is A027336.
The strict case is A343348.
A000009 counts strict partitions.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A257993 gives the least gap of the partition with Heinz number n.
A339564 counts factorizations with a selected factor.
A342050 ranks partitions with even least gap.
A342051 ranks partitions with odd least gap.
Cf. A003242, A022567, A048004, A083710, A098743, A130689, A338470, A343341.

g := (sum(t^j*x^((1/2)*j*(j-1))*(1-x^j), j = 1 .. 80))/(product(1-x^i, i = 1 .. 80)): gser := simplify(series(g, x = 0, 23)): for n from 0 to 30 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, `if`(i=0, [1, 0],
      [0, x]), `if`(i<1, 0, (p-> [0, p[2] +p[1]*x^i])(
      b(n, i-1)) +add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=1..degree(p)))(b(n, n+1)[2]):
seq(T(n), n=0..20);  # Alois P. Heinz, Nov 29 2015

Needs["Combinatorica`"]; {1, 0}~Join~Flatten[Table[Count[Map[If[# == {}, 0, First@ #] &@ Complement[Range@ n, #] &, Combinatorica`Partitions@ n], n_ /; n == k], {n, 17}, {k, n}] /. 0 -> Nothing] (* Michael De Vlieger, Nov 21 2015 *)
mingap[q_]:=Min@@Complement[Range[If[q=={},0,Max[q]]+1],q];Table[Length[Select[IntegerPartitions[n],mingap[#]==k&]],{n,0,15},{k,Round[Sqrt[2*(n+1)]]}] (* Gus Wiseman, Apr 19 2021 *)
b[n_, i_] := b[n, i] = If[n == 0, If[i == 0, {1, 0}, {0, x}], If[i<1, {0, 0}, {0, #[[2]] + #[[1]]*x^i}&[b[n, i-1]] + Sum[b[n-i*j, i - 1], {j, 1, n/i}]]];
T[n_] := CoefficientList[b[n, n + 1], x][[2]] // Rest;
T /@ Range[0, 20] // Flatten (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

G.f.: G(t,x) = Sum_{j>=1} (t^j*x^{j(j-1)/2}*(1-x^j))/Product_{i>=1}(1-x^i).

A064866 Write numbers 1, then 1 up to 2^2, then 1 up to 3^2, then 1 up to 4^2 and so on.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28

Offset: 1

Floor van Lamoen, Oct 08 2001

This is a fractal sequence: if the first instance of each number is deleted, the original sequence is recovered. - Franklin T. Adams-Watters, Dec 14 2013
Subsequences start at indices A000330 + 1. - Ralf Stephan, Dec 17 2013
When sequence fills a triangular array by rows, the main diagonal is A064865:
This triangle begins:
     1
    1 2
   3 4 1
  2 3 4 5
 6 7 8 9 1
From Antti Karttunen, Feb 17 2014: (Start)
A more natural way of organizing this sequence is as an irregular table consisting of successively larger square matrices:
   1;
   1, 2;
   3, 4;
   1, 2, 3;
   4, 5, 6;
   7, 8, 9;
   1, 2, 3, 4;
   5, 6, 7, 8;
   9,10,11,12;
  13,14,15,16;
  etc.
(End)

Cf. A002260, A002262, A002024.
Cf. A074279, A121997, A238013, A237451, A237452.
Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.

Table[Range[n^2],{n,10}]//Flatten (* Harvey P. Dale, Mar 05 2018 *)

A064866_vec(N=9)=concat(vector(N, i, vector(i^2, j, j))) \\ Note: This creates a vector; use A064866_vec()[n] to get the n-th term. - M. F. Hasler, Feb 17 2014

from sympy import integer_nthroot
def A064866(n): return n-(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k-1)*((k<<1)-1)//6 # Chai Wah Wu, Nov 04 2024

a(n) = A237451(n) + (A237452(n)*A074279(n)) + 1. - M. F. Hasler, Feb 17 2014
For 1 <= n <= 650, a(n) = n - t(t-1)(2t-1)/6, where t = floor((3*n)^(1/3)+1/2). - Mikael Aaltonen, Jan 17 2015
a(n) = n-k(k-1)(2k-1)/6 where k = m+1 if n>m(m+1)(2m+1)/6 and k = m otherwise and m = floor((3n)^(1/3)). - Chai Wah Wu, Nov 05 2024

Edited by Ralf Stephan, Dec 17 2013

A068009 Square array T(m,n) with m (row) >= 1 and n (column) >= 0 read by antidiagonals: number of subsets of {1,2,3,...n} that sum to 0 mod m (including the empty set, whose sum is 0).

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 8, 2, 1, 1, 16, 4, 2, 1, 1, 32, 8, 4, 1, 1, 1, 64, 16, 6, 2, 1, 1, 1, 128, 32, 12, 4, 2, 1, 1, 1, 256, 64, 24, 8, 4, 2, 1, 1, 1, 512, 128, 44, 16, 8, 3, 1, 1, 1, 1, 1024, 256, 88, 32, 14, 6, 3, 1, 1, 1, 1, 2048, 512, 176, 64, 26, 12, 5, 2, 1, 1, 1, 1, 4096, 1024, 344, 128, 52, 22, 10, 4, 2, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Feb 11 2002

When p is an odd prime, T(p,k+p) = 2*T(p,k) + (2^k * ((2^p) - 2)/p) for all k >= 0. [Sophie LeBlanc]
When m divides n (with n >= m), T(m,n) = (1/m) Sum_{d | m and d is odd} phi(d) * 2^(n/d). [N. Kitchloo and L. Pachter; D. Rusin]
A068009(C(i+1,2), i) = 2, A068009(C(i,2)+1, i) = A000009(i-1) + 1. [AK, cf. A068049]

			Table for T(m,n) (with rows m >= 1 and columns n >= 0) begins as follows:
  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, ...
  1, 1, 2, 4,  8, 16, 32,  64, 128, 256,  512, 1024, ...
  1, 1, 2, 4,  6, 12, 24,  44,  88, 176,  344, ...
  1, 1, 1, 2,  4,  8, 16,  32,  64, 128,  ...
  1, 1, 1, 2,  4,  8, 14,  26,  52, ...
  1, 1, 1, 2,  3,  6, 12,  22, ...
  1, 1, 1, 1,  3,  5, 10, ...
  1, 1, 1, 1,  2,  4, ...
  1, 1, 1, 1,  2, ...
  1, 1, 1, 1, ...
  1, 1, 1, ...
  1, 1, ...
  1, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Antti Karttunen, Scheme code for computing this table and its rows.
N. Kitchloo and L. Pachter, An interesting result about subset sums, Nov 27 1993.
William Kuszmaul, A new approach to enumerating statistics modulo n, arXiv:1402.3839 [math.CO], 2014.
Lior Pachter, Subset sums, 2015.
Bill Pet, Sophie LeBlanc, Will Self et al., Subsets of {1,2,3,...,n} (discussion in sci.math).
R. P. Stanley and M. F. Yoder, A study of Varshamov codes for asymmetric channels, JPL Technical Report 32-1526, Vol. XIV (1972), 117-123.
Index entries for sequences related to subset sums mod m

Main diagonal: A000016, superdiagonal: A063776. The first term greater than one occurs on each row m in the position A002024(m) and these are given in A068049.

Row 1: A000079, row 2: A011782, row 3: A068010, row 5: A068011, row 6: A068012, row 7: A068013, row 9: A068030, row 10: A068031, row 11: A068032, row 12: A068033, row 13: A068034, row 14: A068035, row 15: A068036, row 16: A068037, row 17: A068038, row 18: A068039, row 19: A068040, row 20: A068041, row 21: A068042, row 25: A068043, row 32: A068044, row 64: A068045.

b:= proc(n, m, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
       b(n-1, m, t)+ b(n-1, m, irem(t+n,m)))
    end:
T:= (m, n)-> b(n, m, 0):
seq(seq(T(1+m, d-m), m=0..d), d=0..12);  # Alois P. Heinz, Jan 18 2014

max = 13; row[m_] := (ClearAll[t]; im = IdentityMatrix[m]; v = Join[ {Last[im]}, Most[im] ]; t[0] = im[[1]]; t[k_] := t[k] = (im + MatrixPower[v, k]) . t[k-1]; Table[ t[k][[1]], {k, 0, max}]); rows = Table[ row[m], {m, 1, max}]; A068009 = Flatten[ Table[ rows[[m-n+1, n]], {m, 1, max, 1}, {n, m, 1, -1}]] (* Jean-François Alcover, Apr 02 2012, after Will Self *)
b[n_, m_, t_] := b[n, m, t] = If[n == 0, If[t == 0, 1, 0], b[n-1, m, t]+b[n-1, m, Mod[t+n, m]]]; T[m_, n_] := b[n, m, 0]; Table[Table[T[1+m, d-m], {m, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 13 2015, after Alois P. Heinz *)

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

A069537 Multiples of 2 whose digit sum is 2.

Original entry on oeis.org

2, 20, 110, 200, 1010, 1100, 2000, 10010, 10100, 11000, 20000, 100010, 100100, 101000, 110000, 200000, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000, 20000000, 100000010, 100000100, 100001000

Offset: 1

Amarnath Murthy, Apr 01 2002

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A002024, A002260, A088404 (half).
Cf. A069521 to A069530, A069532, A069534, A069536.
Subsequence of A005349.
Row n=2 of A245062.

a(n) = my(r,s=sqrtint((n-1)<<1,&r), x=s+(r>s), y=if(r>s,r-s,r+s)>>1); 10^x + 10^y; \\ Kevin Ryde, Jul 17 2025

from itertools import product
def agen():
  digits = 1
  while True:
    for i in range(digits-2): yield int("1"+"0"*(digits-3-i)+"1"+"0"*i+"0")
    yield int("2"+"0"*(digits-1))
    digits += 1
g = agen()
print([next(g) for i in range(32)]) # Michael S. Branicky, Feb 20 2021

a(n) = 10^A002024(n-1) + 10^A002260(n-1) for n >= 2. - Kevin Ryde, Jul 17 2025

Corrected and extended by Ray Chandler, Sep 28 2003

cp's OEIS Frontend

A001614 Connell sequence: 1 odd, 2 even, 3 odd, ...

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A072643 Half of the binary width of the terms of A014486, the number of digits in A063171(n)/2.

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A046090 Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives X+1 values.

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A307730 a(n) = A307720(n) * A307720(n+1).

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A054582 Array read by antidiagonals upwards: A(m,k) = 2^m * (2k+1), m,k >= 0.

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A264401 Triangle read by rows: T(n,k) is the number of partitions of n having least gap k.

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