A002262 -id:A002262 - cp's OEIS frontend

A057944 Largest triangular number less than or equal to n; write m-th triangular number m+1 times.

0, 1, 1, 3, 3, 3, 6, 6, 6, 6, 10, 10, 10, 10, 10, 15, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21, 21, 21, 28, 28, 28, 28, 28, 28, 28, 28, 36, 36, 36, 36, 36, 36, 36, 36, 36, 45, 45, 45, 45, 45, 45, 45, 45, 45, 45, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 66, 66, 66, 66, 66, 66

Offset: 0

Henry Bottomley, Oct 05 2000

			a(35) = 28 since 28 and 36 are successive triangular numbers and 28 <= 35 < 36.

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Cf. A000217, A003056, A056944, A057945, A127739.

a057944 n = a057944_list !! n          -- common flat access
a057944_list = concat a057944_tabl
a057944' n k = a057944_tabl !! n !! k  -- access when seen as a triangle
a057944_row n = a057944_tabl !! n
a057944_tabl = zipWith ($) (map replicate [1..]) a000217_list
-- Reinhard Zumkeller, Feb 03 2012

A057944 := proc(n)
        k := (-1+sqrt(1+8*n))/2 ;
        k := floor(k) ;
        k*(k+1)/2 ;
end proc; # R. J. Mathar, Nov 05 2011

f[n_] := Block[{a = Floor@ Sqrt[1 + 8 n]}, Floor[(a - 1)/2]*Floor[(a + 1)/2]/2]; Array[f, 72, 0]
t0=0; t1=1; k=1; Table[If[n < t1, t0, k++; t0=t1; t1=t1+k; t0], {n, 0, 72}]
With[{nn=15},Table[#[[1]],#[[2]]+1]&/@Thread[{Accumulate[Range[ 0,nn]],Range[ 0,nn]}]]//Flatten (* Harvey P. Dale, Mar 01 2020 *)

a(n)=my(t=(sqrtint(8*n+7)-1)\2);t*(t+1)/2 \\ Charles R Greathouse IV, Jan 26 2013

from math import comb, isqrt
def A057944(n): return comb((m:=isqrt(k:=n+1<<1))+(k>m*(m+1)),2) # Chai Wah Wu, Nov 09 2024

a(n) = floor((sqrt(1+8*n)-1)/2)*floor((sqrt(1+8*n)+1)/2)/2 = (trinv(n)*(trinv(n)-1))/2 = A000217(A003056(n)) = n - A002262(n)
a(n) = (1/2)*t*(t-1), where t = floor(sqrt(2*n+1)+1/2) = A002024(n+1). - Ridouane Oudra, Oct 20 2019
Sum_{n>=1} 1/a(n)^2 = 2*Pi^2/3 - 4. - Amiram Eldar, Aug 14 2022

Keyword tabl added by Reinhard Zumkeller, Feb 03 2012

A085207 Array A(x,y): concatenation of binary expansions of x & y, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ... Zero is expanded as an empty string.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 5, 6, 3, 4, 7, 10, 7, 4, 5, 9, 14, 11, 12, 5, 6, 11, 18, 15, 20, 13, 6, 7, 13, 22, 19, 28, 21, 14, 7, 8, 15, 26, 23, 36, 29, 22, 15, 8, 9, 17, 30, 27, 44, 37, 30, 23, 24, 9, 10, 19, 34, 31, 52, 45, 38, 31, 40, 25, 10, 11, 21, 38, 35, 60, 53, 46, 39, 56, 41, 26, 11

Offset: 0

Antti Karttunen, Jun 23 2003

			A(2,1) = 5 = '101' in binary, concatenation of 2's binary expansion '10' and 1's '1'. A(1,2) = 6 = '110' in binary, concatenation of '1' and '10'.

Same array in binary: A085209. Transpose: A085208. Variant: A085211. Can be used to compute A085201.

A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 5, 5, 1, 0, 5, 9, 7, 7, 1, 0, 6, 7, 25, 11, 11, 1, 0, 7, 15, 11, 49, 13, 13, 1, 0, 8, 11, 35, 13, 121, 17, 17, 1, 0, 9, 27, 13, 77, 17, 169, 19, 19, 1, 0, 10, 25, 125, 17, 143, 19, 289, 23, 23, 1, 0, 11, 21, 49, 343, 19, 221, 23, 361, 29, 29, 1, 0

Offset: 0

Antti Karttunen, May 12 2014

Each row i is a multiplicative function, being in essence "the i-th power" of A003961, i.e., A(i,j) = A003961^i (j). Zeroth power gives an identity function, A001477, which occurs as the row zero.
The terms in the same column have the same prime signature.
The array is read by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

			The top-left corner of the array:
  0,   1,   2,   3,   4,   5,   6,   7,   8, ...
  0,   1,   3,   5,   9,   7,  15,  11,  27, ...
  0,   1,   5,   7,  25,  11,  35,  13, 125, ...
  0,   1,   7,  11,  49,  13,  77,  17, 343, ...
  0,   1,  11,  13, 121,  17, 143,  19,1331, ...
  0,   1,  13,  17, 169,  19, 221,  23,2197, ...
...
A(2,6) = A003961(A003961(6)) = p_{1+2} * p_{2+2} = p_3 * p_4 = 5 * 7 = 35, because 6 = 2*3 = p_1 * p_2.

Antti Karttunen, Table of n, a(n) for n = 0..10439; Antidiagonals n = 0..143, flattened

Taking every second column from column 2 onward gives array A246278 which is a permutation of natural numbers larger than 1.
Transpose: A242379.
Row 0: A001477, Row 1: A003961 (from 1 onward), Row 2: A357852 (from 1 onward), Row 3: A045968 (from 7 onward), Row 4: A045970 (from 11 onward).
Column 2: A000040 (primes), Column 3: A065091 (odd primes), Column 4: A001248 (squares of primes), Column 6: A006094 (products of two successive primes), Column 8: A030078 (cubes of primes).
Excluding column 0, a subtable of A297845.
Permutations whose formulas refer to this array: A122111, A241909, A242415, A242419, A246676, A246678, A246684.

A(0,j) = j, A(i,0) = 0, A(i > 0, j > 0) = A003961(A(i-1,j)).

For j > 0, A(i,j) = A297845(A000040(i+1),j) = A297845(j,A000040(i+1)). - Peter Munn, Sep 02 2025

A079901 Triangle of powers, T(n,k) = n^k, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 9, 27, 1, 4, 16, 64, 256, 1, 5, 25, 125, 625, 3125, 1, 6, 36, 216, 1296, 7776, 46656, 1, 7, 49, 343, 2401, 16807, 117649, 823543, 1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 1, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721

Offset: 0

Reinhard Zumkeller, Feb 21 2003

Matrix inverse equals the triangle R where R(n,k) = A107045(n,k)/A107046(n,k) are coefficients with exponential-like properties. - Paul D. Hanna, May 22 2005

			Triangle begins:
  1;
  1,1;
  1,2,4;
  1,3,9,27;
  1,4,16,64,256;
  1,5,25,125,625,3125;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A107045/A107046 (matrix inverse).

Cf. A002262, A000290, A000578, A000583, A000272, A000169, A000312.

a079901 n k = a079901_tabl !! n !! k
a079901_row n = a079901_tabl !! n
a079901_tabl = zipWith (map . (^)) [0..] a002262_tabl
-- Reinhard Zumkeller, Mar 31 2015

Join[{1},Flatten[Table[n^k,{n,9},{k,0,n}]]] (* Harvey P. Dale, Feb 08 2013 *)

row(n) = vector(n+1, k, n^(k-1)); \\ Amiram Eldar, May 09 2025

T(n,k) = if k=0 then 1 else T(n,k-1)*n.
T(n,0) = 1; T(n,1) = n for n>0; T(n,2) = A000290(n) for n > 1; T(n,3) = A000578(n) for n > 2; T(n,4) = A000583(n) for n>3.
T(n,n-2) = A000272(n) for n>2; T(n,n-1) = A000169(n) for n>1; T(n,n) = A000312(n).

A163528 The X-coordinate of the n-th point in the Peano curve A163334.

Original entry on oeis.org

0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 7, 6, 5, 4, 3, 3, 4, 5, 5, 4, 3, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 7, 8, 8, 7, 6, 6, 7, 8, 9, 10, 11, 11, 10, 9, 9, 10, 11, 12, 13, 14, 14, 13, 12

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

There is a 2-state automaton that accepts exactly those pairs (n,a(n)) where n is represented in base 9 and a(n) in base 3; see accompanying file a163528.pdf - Jeffrey Shallit, Aug 10 2023

Antti Karttunen, Table of n, a(n) for n = 0..59048
Jeffrey Shallit, Automaton for A163528
Index entries for sequences related to coordinates of 2D curves

Cf. A025581, A163335, A002262, A163337, A163325, A163332.

See also: A059252, A059253, A163529, A163530, A163531, A163532.

a(n) = A025581(A163335(n)) = A002262(A163337(n)) = A163325(A163332(n)).

Name corrected by Kevin Ryde, Aug 28 2020

A225630 Array of iterated Landau-like functions, starting maximization of LCM from the partition {1+1+...+1} of n, read downwards antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 6, 2, 1, 1, 1, 6, 12, 6, 2, 1, 1, 1, 6, 30, 12, 6, 2, 1, 1, 1, 12, 30, 60, 12, 6, 2, 1, 1, 1, 15, 84, 60, 60, 12, 6, 2, 1, 1, 1, 20, 120, 420, 60, 60, 12, 6, 2, 1, 1, 1, 30, 180, 840, 420, 60, 60, 12, 6, 2, 1, 1

Offset: 0

Antti Karttunen, May 13 2013

Row 0 consists of all 1's (corresponding to lcm(1,1,...,1) computed from the {1+1+...+1} partition), after which, on each succeeding row, the entry A(row,n) is computed by finding such a partition {p1+p2+...+pk} of n that value of lcm(A(row-1,n),p1,p2,...,pk) is maximized.
This will produce the ordinary Landau's function (A000793) for row 1, the "second order Landau's function" (A225627) for row 2, etc.
For each column n, only finite number of distinct values (A225634(n)) occur, after which the fixed point A003418(n) that has been reached repeats forever.

			Table begins:
  1, 1, 1, 1,  1,  1,  1,   1,   1,  1, ...
  1, 1, 2, 3,  4,  6,  6,  12,  15, 20, ...
  1, 1, 2, 6, 12, 30, 30,  84, 120, ...
  1, 1, 2, 6, 12, 60, 60, 420, 840, ...
  ...

Index entries for sequences related to lcm's

Transpose: A225631. Cf. also A225632, A225634.
Row 0: A000012, row 1: A000793, row 2: A225627, row 3: A225628. Cf. also A225629.
Rows converge towards A003418, which is also the main diagonal of this array.
See A225640 for a variant which uses a similar process, but where the "initial seed" in column n is n instead of 1.

(define (A225630 n) (A225630bi (A025581 n) (A002262 n)))
(define (A225630bi col row) (let ((maxlcm (list 0))) (let loop ((prevmaxlcm 1) (stepsleft row)) (if (zero? stepsleft) prevmaxlcm (begin (gen_partitions col (lambda (p) (set-car! maxlcm (max (car maxlcm) (apply lcm (cons prevmaxlcm p)))))) (loop (car maxlcm) (- stepsleft 1)))))))
(define (gen_partitions m colfun) (let recurse ((m m) (b m) (n 0) (partition (list))) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (cons i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))

A265609 Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n >= 0 and k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 6, 0, 1, 4, 12, 24, 24, 0, 1, 5, 20, 60, 120, 120, 0, 1, 6, 30, 120, 360, 720, 720, 0, 1, 7, 42, 210, 840, 2520, 5040, 5040, 0, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0

Offset: 0

Peter Luschny, Dec 19 2015

The Pochhammer function is defined P(x,n) = x*(x+1)*...*(x+n-1). By convention P(0,0) = 1.
From Antti Karttunen, Dec 19 2015: (Start)
Apart from the initial row of zeros, if we discard the leftmost column and divide the rest of terms A(n,k) with (n+k) [where k is now the once-decremented column index of the new, shifted position] we get the same array back. See the given recursive formula.
When the numbers in array are viewed in factorial base (A007623), certain repeating patterns can be discerned, at least in a few of the topmost rows. See comment in A001710 and arrays A265890, A265892. (End)
A(n,k) is the k-th moment (about 0) of a gamma (Erlang) distribution with shape parameter n and rate parameter 1. - Geoffrey Critzer, Dec 24 2018

			Square array A(n,k) [where n=row, k=column] is read by ascending antidiagonals as:
A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...
Array starts:
n\k [0  1   2    3     4      5        6         7          8]
--------------------------------------------------------------
[0] [1, 0,  0,   0,    0,     0,       0,        0,         0]
[1] [1, 1,  2,   6,   24,   120,     720,     5040,     40320]
[2] [1, 2,  6,  24,  120,   720,    5040,    40320,    362880]
[3] [1, 3, 12,  60,  360,  2520,   20160,   181440,   1814400]
[4] [1, 4, 20, 120,  840,  6720,   60480,   604800,   6652800]
[5] [1, 5, 30, 210, 1680, 15120,  151200,  1663200,  19958400]
[6] [1, 6, 42, 336, 3024, 30240,  332640,  3991680,  51891840]
[7] [1, 7, 56, 504, 5040, 55440,  665280,  8648640, 121080960]
[8] [1, 8, 72, 720, 7920, 95040, 1235520, 17297280, 259459200]
.
Seen as a triangle, T(n, k) = Pochhammer(n - k, k), the first few rows are:
   [0] 1;
   [1] 1, 0;
   [2] 1, 1,  0;
   [3] 1, 2,  2,   0;
   [4] 1, 3,  6,   6,    0;
   [5] 1, 4, 12,  24,   24,    0;
   [6] 1, 5, 20,  60,  120,  120,     0;
   [7] 1, 6, 30, 120,  360,  720,   720,     0;
   [8] 1, 7, 42, 210,  840, 2520,  5040,  5040,     0;
   [9] 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 0.

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, 1994.
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 355.

NIST Digital Library of Mathematical Functions, Pochhammer's Symbol
Index entries for sequences related to factorial numbers

Triangle giving terms only up to column k=n: A124320.
Row 0: A000007, row 1: A000142, row 3: A001710 (from k=1 onward, shifted two terms left).
Column 0: A000012, column 1: A001477, column 2: A002378, columns 3-7: A007531, A052762, A052787, A053625, A159083 (shifted 2 .. 6 terms left respectively, i.e. without the extra initial zeros), column 8: A239035.
Row sums of the triangle: A000522.
A(n, n) = A000407(n-1) for n>0.
2^n*A(1/2,n) = A001147(n).
Cf. also A007623, A008279 (falling factorial), A173333, A257505, A265890, A265892.

for n from 0 to 8 do seq(pochhammer(n,k), k=0..8) od;

Table[Pochhammer[n, k], {n, 0, 8}, {k, 0, 8}]

for n in (0..8): print([rising_factorial(n,k) for k in (0..8)])

(define (A265609 n) (A265609bi (A025581 n) (A002262 n)))
(define (A265609bi row col) (if (zero? col) 1 (* (+ row col -1) (A265609bi row (- col 1)))))
;; Antti Karttunen, Dec 19 2015

A(n,k) = Gamma(n+k)/Gamma(n) for n > 0 and n^k for n=0.
A(n,k) = Sum_{j=0..k} n^j*S1(k,j), S1(n,k) the Stirling cycle numbers A132393(n,k).
A(n,k) = (k-1)!/(Sum_{j=0..k-1} (-1)^j*binomial(k-1, j)/(j+n)) for n >= 1, k >= 1.
A(n,k) = (n+k-1)*A(n,k-1) for k >= 1, A(n,0) = 1. - Antti Karttunen, Dec 19 2015
E.g.f. for row k: 1/(1-x)^k. - Geoffrey Critzer, Dec 24 2018
A(n, k) = FallingFactorial(n + k - 1, k). - Peter Luschny, Mar 22 2022
G.f. for row n as a continued fraction of Stieltjes type: 1/(1 - n*x/(1 - x/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 3*x/(1 - ... ))))))). See Wall, Chapter XVIII, equation 92.5. Cf. A226513. - Peter Bala, Aug 27 2023

A072764 Tabular N X N -> N bijection induced by Lisp/Scheme function 'cons' combining the two planar binary trees/general trees/parenthesizations encoded by A014486(X) and A014486(Y).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 8, 16, 14, 5, 17, 19, 42, 15, 9, 18, 44, 51, 43, 37, 10, 20, 47, 126, 52, 121, 38, 11, 21, 53, 135, 127, 149, 122, 39, 12, 22, 56, 154, 136, 385, 150, 123, 40, 13, 45, 60, 163, 155, 413, 386, 151, 124, 41, 23, 46, 128, 177, 164, 475, 414, 387, 152

Offset: 0

Antti Karttunen Jun 12 2002

Index entries for the sequences induced by list functions of Lisp
Index entries for sequences that are permutations of the natural numbers
A. Karttunen, Gatomorphisms (with the complete Scheme source)

Inverse permutation: A072765. a(n) = A069770(A072766(n)). Also transpose of A072766, i.e. a(n) = A072766(A038722(n)). The upper triangular region: A072773. Projection functions are A072771 ('car') & A072772 ('cdr'). The sizes of the corresponding Catalan structures: A072768. The first row: A057548, the first column: A072795, diagonal: A083938. Cf. also A080300, A025581, A002262.

a(0)=0 prepended by Sean A. Irvine, Oct 25 2024

A127324 Fourth 4-dimensional hyper-tetrahedral coordinate; 4-D analog of A056558.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4

Offset: 0

Graeme McRae, Jan 10 2007

nonn

Alternatively, write n = C(i,4)+C(j,3)+C(k,2)+C(l,1) with i>j>k>l>=0; sequence gives k values. Each n >= 0 has a unique representation as n = C(i,4)+C(j,3)+C(k,2)+C(l.1) with i>j>k>l>=0. This is the combinatorial number system of degree t = 4, where we get [A194882, A194883, A194884, A127324].
If {(W,X,Y,Z)} are 4-tuples of nonnegative integers with W>=X>=Y>=Z ordered by W, X, Y and Z, then W=A127321(n), X=A127322(n), Y=A127323(n) and Z=A127324(n). These sequences are the four-dimensional analogs of the three-dimensional A056556, A056557 and A056558.
This is a 'Matryoshka doll' sequence with alpha=0 (cf. A055462 and A000332), seq(seq(seq(seq(i,i=alpha..k),k=alpha..n),n=alpha..m),m=alpha..4). - Peter Luschny, Jul 14 2009

			See A127321 for a table of A127321, A127322, A127323, A127324
See A127327 for a table of A127324, A127325, A127326, A127327

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3, Eq. (20), p. 360.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A127321, A127322, A127323, A056556, A056557, A056558, A000332, A000292, A000217.

Cf. A002262, A056558.

import Data.List (inits)
a127324 n = a127324_list !! n
a127324_list = concatMap (concatMap concat .
               inits . inits . enumFromTo 0) $ enumFrom 0
-- Reinhard Zumkeller, Jun 01 2015

seq(seq(seq(seq(i,i=0..k),k=0..n),n=0..m),m=0..5); # Peter Luschny, Sep 22 2011

Table[i, {m, 0, 5}, {k, 0, m}, {j, 0, k}, {i, 0, j}] // Flatten  (* Robert G. Wilson v, Sep 27 2011 *)

For W>=X>=Y>=Z>=0, a(A000332(W+3)+A000292(X)+A000217(Y)+Z) = Z A127322(n+1) = A127321(n)==A127324(n) ? 0 : A127322(n)==A127324(n) ? 0 : A127323(n)==A127324(n) ? 0 : A127324(n)+1

A268038 List of y-coordinates of point moving in clockwise square spiral.

Original entry on oeis.org

0, 0, -1, -1, -1, 0, 1, 1, 1, 1, 0, -1, -2, -2, -2, -2, -2, -1, 0, 1, 2, 2, 2, 2, 2, 2, 1, 0, -1, -2, -3, -3, -3, -3, -3, -3, -3, -2, -1, 0, 1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 1, 0, -1, -2, -3, -4, -4, -4, -4, -4, -4, -4, -4, -4, -3, -2, -1, 0, 1, 2, 3, 4, 4, 4

Offset: 1

Peter Kagey, Jan 24 2016

This spiral, in either direction, is sometimes called the "Ulam spiral", but "square spiral" is a better name. (Ulam looked at the positions of the primes, but of course the spiral itself must be much older.) - N. J. A. Sloane, Jul 17 2018

			Sequence gives y-coordinate of the n-th point of the following spiral:
.
  20--21--22--23--24--25
   |                   |
  19   6---7---8---9  26
   |   |           |   |
  18   5   0---1  10  27
   |   |       |   |   |
  17   4---3---2  11  28
   |               |   |
  16--15--14--13--12  29
                       |
  35--34--33--32--31--30

Peter Kagey, Table of n, a(n) for n = 1..10000
Visualization of spiral using Plot 2. - _Hugo Pfoertner_, May 29 2018

A174344 gives sequence of x-coordinates.
This is the negative of A274923.
The (x,y) coordinates for a point sweeping a quadrant by antidiagonals are (A025581, A002262). - N. J. A. Sloane, Jul 17 2018

a[n_] := a[n] = If[n==0, 0, a[n-1] + Cos[Mod[Floor[Sqrt[4*(n-1) + 1]], 4]* Pi/2]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 11 2018, after Seppo Mustonen *)

L=1; d=-1;
for(r=1,9,d=-d;k=floor(r/2)*d;for(j=1,L++,print1(k,", "));forstep(j=k-d,-floor((r+1)/2)*d+d,-d,print1(j,", "))) \\ Hugo Pfoertner, Jul 28 2018

cp's OEIS Frontend

A057944 Largest triangular number less than or equal to n; write m-th triangular number m+1 times.

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A085207 Array A(x,y): concatenation of binary expansions of x & y, listed antidiagonalwise as A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ... Zero is expanded as an empty string.

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A242378 Square array read by antidiagonals: to obtain A(i,j), replace each prime factor prime(k) in prime factorization of j with prime(k+i).

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A079901 Triangle of powers, T(n,k) = n^k, 0 <= k <= n, read by rows.

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A163528 The X-coordinate of the n-th point in the Peano curve A163334.

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A225630 Array of iterated Landau-like functions, starting maximization of LCM from the partition {1+1+...+1} of n, read downwards antidiagonals.

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A265609 Array read by ascending antidiagonals: A(n,k) the rising factorial, also known as Pochhammer symbol, for n >= 0 and k >= 0.

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A072764 Tabular N X N -> N bijection induced by Lisp/Scheme function 'cons' combining the two planar binary trees/general trees/parenthesizations encoded by A014486(X) and A014486(Y).

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