A002262 -id:A002262 - cp's OEIS frontend

A056559 Tetrahedron with T(t,n,k) = t - n; succession of growing finite triangles with declining values per row.

0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2

Offset: 0

Henry Bottomley, Jun 26 2000

			First triangle: [0]; second triangle: [1; 0 0]; third triangle: [2; 1 1; 0 0 0]; ...

Peter Luschny, Table of n, a(n) for n = 0..10000 (first 105 terms by Henry Bottomley).

Together with A056558 and A056560 might enable reading "by antidiagonals" of cube arrays as 3-dimensional analog of A002262 and A025581 with square arrays.

Bisection (y-coordinates) of A332662.

function a_list(N)
    a = Int[]
    for n in 1:N
        for j in ((k:-1:1) for k in 1:n)
            t = n - j[1]
            for m in j
                push!(a, t)
end end end; a end
A = a_list(10) # Peter Luschny, Feb 19 2020

from math import isqrt, comb
from sympy import integer_nthroot
def A056559(n): return (m:=integer_nthroot(6*(n+1),3)[0])-(a:=nChai Wah Wu, Dec 11 2024

a(n) = A056556(n) - A056557(n).

A196199 Count up from -n to n for n = 0, 1, 2, ... .

Original entry on oeis.org

0, -1, 0, 1, -2, -1, 0, 1, 2, -3, -2, -1, 0, 1, 2, 3, -4, -3, -2, -1, 0, 1, 2, 3, 4, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8

Offset: 0

Franklin T. Adams-Watters, Sep 29 2011

This sequence contains every integer infinitely often, hence all integer sequences are subsequences.
This is a fractal sequence.
Indeed, if all terms (a(n),a(n+1)) such that a(n+1) does NOT equal a(n)+1 (<=> a(n+1) < a(n)) are deleted, the same sequence is recovered again. See A253580 for an "opposite" yet similar "fractal tree" construction. - M. F. Hasler, Jan 04 2015

			Table starts:
            0,
        -1, 0, 1,
    -2, -1, 0, 1, 2,
-3, -2, -1, 0, 1, 2, 3,
...
The sequence of fractions A196199/A004737 = 0/1, -1/1, 0/2, 1/1, -2/1, -1/2, 0/3, 1/2, 2/1, -3/1, -2/2, -1/3, 0/4, 1/3, 2/2, 3/1, -4/4. -3/2, ... contains every rational number (infinitely often) [Laczkovich]. - _N. J. A. Sloane_, Oct 09 2013

Miklós Laczkovich, Conjecture and Proof, TypoTex, Budapest, 1998. See Chapter 10.

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. absolute values A053615, A002262, A002260, row lengths A005408, row sums A000004, A071797.

Cf. A004737, A381232, A381233.

a196199 n k = a196199_row n !! k
a196199_tabf = map a196199_row [0..]
a196199_row n = [-n..n]
b196199 = bFile' "A196199" (concat $ take 101 a196199_tabf) 0
-- Reinhard Zumkeller, Sep 30 2011

seq(seq(j-k-k^2, j=k^2 .. (k+1)^2-1), k = 0 .. 10); # Robert Israel, Jan 05 2015
# Alternatively, as a table with rows -n<=k<=n (compare A257564):
r := n -> (n-(n mod 2))/2: T := (n, k) -> r(n+k) - r(n-k):
seq(print(seq(T(n, k), k=-n..n)), n=0..6); # Peter Luschny, May 28 2015

Table[Range[-n, n], {n, 0, 9}] // Flatten
(* or *)
a[n_] := With[{t = Floor[Sqrt[n]]}, n - t (t + 1)];
Table[a[n], {n, 0, 99}] (* Jean-François Alcover, Jul 13 2018, after Boris Putievskiy *)

r=[];for(k=0,8,r=concat(r,vector(2*k+1,j,j-k-1)));r

from math import isqrt
def A196199(n): return n-(t:=isqrt(n))*(t+1) # Chai Wah Wu, Aug 04 2022

a(n) = n - t*t - t - 1, where t = floor(sqrt(n-1)). - Boris Putievskiy, Jan 28 2013

G.f.: x/(x-1)^2 + 1/(x-1)*sum(k >= 1, 2*k*x^(k^2)). The series is related to Jacobi theta functions. - Robert Israel, Jan 05 2015

A265345 Square array A(row,col): For row=0, A(0,col) = A265341(col), for row > 0, A(row,col) = A265342(A(row-1,col)).

Original entry on oeis.org

1, 3, 2, 7, 6, 4, 5, 10, 12, 8, 9, 22, 20, 24, 16, 21, 18, 28, 40, 48, 64, 13, 30, 36, 56, 80, 192, 32, 19, 26, 60, 72, 112, 160, 96, 184, 25, 14, 52, 120, 144, 224, 640, 552, 352, 11, 46, 76, 208, 240, 576, 448, 320, 1056, 704, 15, 58, 68, 136, 104, 480, 288, 1720, 1600, 2112, 1408

Offset: 1

Antti Karttunen, Dec 18 2015

Square array A(row,col) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...
All the terms in the same column are either all divisible by 3, or none of them are.
Reducing A265342 to its constituent sequences gives A265342(n) = A263273(2*A263273(n)). Iterating this function k times starting from n reduces to (because A263273 is an involution, so pairs of them are canceled) to A263273((2^k)*A263273(n)).

			The top left corner of the array:
    1,    3,    7,    5,    9,   21,   13,   19,   25,   11,   15,    39, .
    2,    6,   10,   22,   18,   30,   26,   14,   46,   58,   66,    78, .
    4,   12,   20,   28,   36,   60,   52,   76,   68,   44,   84,   156, .
    8,   24,   40,   56,   72,  120,  208,  136,   88,  232,  168,   624, .
   16,   48,   80,  112,  144,  240,  104,  200,  496,  424,  336,   312, .
   64,  192,  160,  224,  576,  480,  520,  256,  344,  608,  672,  1560, .
   32,   96,  640,  448,  288, 1920, 1144,  512, 1984,  736, 1344,  3432, .
  184,  552,  320, 1720, 1656,  960, 2072, 1024, 1376, 4384, 5160,  6216, .
  352, 1056, 1600,  824, 3168, 4800, 3712, 6040, 5344, 2936, 2472, 11136, .
  ...

Inverse: A265346.
Transpose: A265347.
Leftmost column: A264980.
Topmost row: A265341.
Row index: A265330 (zero-based), A265331 (one-based).
Column index: A265910 (zero-based), A265911 (one-based).
Cf. also A265342.
Related permutations: A263273, A265895.

(define (A265345 n) (A265345bi (A002262 (+ -1 n)) (A025581 (+ -1 n)))) ;; o=1.
(define (A265345bi row col) (A263273 (* (A000079 row) (A263273 (A265341 col))))) ;; Faster than below.
(define (A265345bi row col) (if (= 0 row) (A265341 col) (A265342 (A265345bi (- row 1) col)))) ;; row>=0, col>=0.

For row=0, A(0,col) = A265341(col), for row>0, A(row,col) = A265342(A(row-1,col)).

A(row, col) = A263273((2^row) * A263273(A265341(col))). [The above reduces to this.]

A056560 Tetrahedron with T(t,n,k)=n-k; succession of growing finite triangles with increasing values towards bottom left.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Jun 26 2000

nonn

			First triangle: [0]; second triangle: [0; 1 0]; third triangle: [0; 1 0; 2 1 0]; ...

Together with A056558 and A056559 might enable reading "by antidiagonals" of cube arrays as 3-dimensional analog of A002262 and A025581 with square arrays.

a(n) = A056557(n) - A056558(n).

A072766 Transpose of A072764, 'cons' with arguments swapped.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 14, 16, 8, 9, 15, 42, 19, 17, 10, 37, 43, 51, 44, 18, 11, 38, 121, 52, 126, 47, 20, 12, 39, 122, 149, 127, 135, 53, 21, 13, 40, 123, 150, 385, 136, 154, 56, 22, 23, 41, 124, 151, 386, 413, 155, 163, 60, 45, 24, 107, 125, 152, 387, 414, 475, 164

Offset: 0

Antti Karttunen, Jun 12 2002

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

Inverse permutation: A072767. a(n) = A069770(A072764(n)). Also transpose of A072764, i.e. a(n) = A072764(A038722(n)). Projection functions are A072772 & A072771. The sizes of the corresponding Catalan structures: A072768. The first column: A057548, the first row: A072795. Cf. also A025581, A002262.

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

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n(2k+1) - 1 with n,k >= 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 7, 11, 9, 6, 15, 23, 19, 13, 8, 31, 47, 39, 27, 17, 10, 63, 95, 79, 55, 35, 21, 12, 127, 191, 159, 111, 71, 43, 25, 14, 255, 383, 319, 223, 143, 87, 51, 29, 16, 511, 767, 639, 447, 287, 175, 103, 59, 33, 18, 1023, 1535, 1279, 895, 575, 351, 207, 119

Offset: 0

Antti Karttunen, Sep 12 2002

From Philippe Deléham, Feb 19 2014: (Start)
A(0,k)  = 2*k = A005843(k),
A(1,k)  = 4*k + 1 = A016813(k),
A(2,k)  = 8*k + 3 = A017101(k),
A(n,0)  = A000225(n),
A(n,1)  = A153893(n),
A(n,2)  = A153894(n),
A(n,3)  = A086224(n),
A(n,4)  = A052996(n+2),
A(n,5)  = A086225(n),
A(n,6)  = A198274(n),
A(n,7)  = A238087(n),
A(n,8)  = A198275(n),
A(n,9)  = A198276(n),
A(n,10) = A171389(n). (End)
A permutation of the nonnegative integers. - Alzhekeyev Ascar M, Jun 05 2016
The values in array row n, when expressed in binary, have n trailing 1-bits. - Ruud H.G. van Tol, Mar 18 2025

			The array A begins:
   0    2    4    6    8   10   12   14   16   18 ...
   1    5    9   13   17   21   25   29   33   37 ...
   3   11   19   27   35   43   51   59   67   75 ...
   7   23   39   55   71   87  103  119  135  151 ...
  15   47   79  111  143  175  207  239  271  303 ...
  31   95  159  223  287  351  415  479  543  607 ...
  ... - _Philippe Deléham_, Feb 19 2014
From _Wolfdieter Lang_, Jan 31 2019: (Start)
The triangle T begins:
   n\k   0    1    2   3   4   5   6   7  8  9 10 ...
   0:    0
   1:    1    2
   2:    3    5    4
   3:    7   11    9   6
   4:   15   23   19  13   8
   5    31   47   39  27  17  10
   6:   63   95   79  55  35  21  12
   7:  127  191  159 111  71  43  25  14
   8:  255  383  319 223 143  87  51  29 16
   9:  511  767  639 447 287 175 103  59 33 18
  10: 1023 1535 1279 895 575 351 207 119 67 37 20
  ...
T(3, 1) = 2^2*(2*1+1) - 1 = 12 - 1 = 11.  (End)

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A075301. Transpose: A075302. The X-projection is given by A007814(n+1) and the Y-projection A025480.

Cf. A002262, A025581, A054582, A241957.

A075300bi := (x,y) -> (2^x * (2*y + 1))-1;
A075300 := n -> A075300bi(A025581(n), A002262(n));
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);

Table[(2^# (2 k + 1)) - 1 &[m - k], {m, 0, 10}, {k, 0, m}] (* Michael De Vlieger, Jun 05 2016 *)

From Wolfdieter Lang, Jan 31 2019: (Start)
Array A(n, k) = 2^n*(2*k+1) - 1, for n >= 0 and m >= 0.
The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1) - 1, n >= 0, k=0..n.
See also A054582 after subtracting 1. (End)
From Ruud H.G. van Tol, Mar 17 2025: (Start)
A(0, k) is even. For n > 0, A(n, k) is odd and (3 * A(n, k) + 1) / 2 = A(n-1, 3*k+1).
A(n, k) = 2^n - 1 (mod 2^(n+1)) (equivalent to the comment about trailing 1-bits). (End)

A138612 Permutation of natural numbers generated with the sieve algorithm described in the comment lines.

Original entry on oeis.org

1, 2, 4, 3, 7, 12, 5, 11, 19, 28, 6, 15, 26, 39, 53, 8, 20, 35, 52, 71, 91, 9, 23, 42, 64, 88, 114, 141, 10, 27, 49, 76, 106, 138, 172, 207, 13, 33, 60, 93, 129, 168, 210, 253, 297, 14, 37, 68, 105, 148, 194, 243, 294, 347, 401, 16, 43, 79, 122, 171, 225, 282, 342

Offset: 1

Ctibor O. Zizka, May 14 2008

Sieve proceeds as:
1) take the 1st element from natural numbers (A000027): 1; remaining set is 2,3,4,5,6,7,8,9,10,...; S1={1}
2) take the 1st element from the remaining set: 2; remaining set is 3,4,5,6,7,8,9,10,...; take the 2nd element from the remaining set: 4; remaining set is 3,5,6,7,8,9,10,...; S2={2,4}
3) take the 1st element from the remaining set: 3; remaining set is 5,6,7,8,9,10,...; take the 3rd element from the remaining set: 7; remaining set is 5,6,8,9,10,11,12,...; take the 7th element from the remaining set: 12; remaining set is 5,6,8,9,10,11,13,14,15,16,17,18,19,20,..; S3={3,7,12}
4) take the 1st element from the remaining set: 5; remaining set is 6,8,9,10,11,13,14,15,16,17,18,19,20,..; take the 5th element from the remaining set: 11; remaining set is 6,8,9,10,13,14,15,16,17,18,19,20,..; take the 11th element from the remaining set: 19; remaining set is 6,8,9,10,13,14,15,16,17,18,20,..; take the 19th element from the remaining set: 28; remaining set is 6,8,9,10,13,14,15,16,17,18,20,21,22,23,24,25,26,27,29,30,31,...;
thus S4={5,11,19,28}.
The sequence is concatenation of such subsequences S1,S2,S3,S4,S5,...,Sn, ..., where each subsequence consists of n nondecreasing terms. Alternatively, these can be viewed as rows of a triangular table.

Inverse: A166017. Left edge A166018, Right edge: A166019, Row sums: A166020. Cf. A138606-A138609.

Edited, extended, keyword tabl and Scheme-code added by Antti Karttunen, Oct 05 2009

A187769 Triangle read by rows: equivalence classes of natural numbers, where numbers are equivalent when having equal numbers of zeros and ones in binary representation, respectively.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 20, 24, 19, 21, 22, 25, 26, 28, 23, 27, 29, 30, 31, 32, 33, 34, 36, 40, 48, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 47, 55, 59, 61, 62, 63, 64, 65, 66

Offset: 0

Reinhard Zumkeller, Jan 05 2013

Row lengths are given by Pascal's triangle (cf. A007318), seen as flattened sequence, or for n > 0: length of n-th row = A007318(A003056(n-1),A002262(n-1));
1 <= i < j <= length of n-th row: A023416(T(n,i)) = A023416(T(n,j)), A000120(T(n,i)) = A000120(T(n,j)) and A070939(T(n,i)) = A070939(T(n,j));
the table provides a permutation of the natural numbers when seen as flattened sequence.
This sequence can be seen as an irregular triangle S(i,k) where row 0 = {1}, row n = { m = 2^(n-1)..2^n - 1 } sorted according to omega(A019565(m)), where omega = A001221. Under this arrangement, the rows can be further subdivided into segments of m with the same omega(m), which align with the original definition's triangle T. - Michael De Vlieger, Jan 03 2025

			See link.

Reinhard Zumkeller, Rows n = 0..78 of triangle, flattened - all terms < 2^12
Reinhard Zumkeller, Illustration of initial terms
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^21-1.
Michael De Vlieger, Fan style binary tree of a(n), n = 0..8192, i.e., rows 0..12, with a color function associated with (a(n) mod 2) / 2^floor(log_2 n) that illustrates the relationship with Pascal's triangle.
Index entries for sequences related to binary expansion of n
Index entries for triangles and arrays related to Pascal's triangle
Index entries for sequences that are permutations of the nonnegative integers

Rows of A187786, duplicates removed;
Cf. A099627 (left edge), A023758 (right edge).
Cf. A019565, A294648, A344084.

import List (elemIndices)
a187769 n k = a187769_tabf !! n !! k
a187769_row n = a187769_tabf !! n
a187769_tabf = [0] : [elemIndices (b, len - b) $
   takeWhile ((<= len) . uncurry (+)) $ zip a000120_list a023416_list |
   len <- [1 ..], b <- [1 .. len]]
a187769_list = concat a187769_tabf

{{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten (* Michael De Vlieger, Jan 03 2025 *)

A226314 Triangle read by rows: T(i,j) = j+(i-j)/gcd(i,j) (1<=i<=j).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 09 2013

The triangle of fractions A226314(i,j)/A054531(i,j) is an efficient way to enumerate the rationals [Fortnow].

Sum(A226314(n,k)/A054531(n,k): 1<=k<=n) = A226555(n)/A040001(n). - Reinhard Zumkeller, Jun 10 2013

			Triangle begins:
[1]
[1, 2]
[1, 2, 3]
[1, 3, 3, 4]
[1, 2, 3, 4, 5]
[1, 4, 5, 5, 5, 6]
[1, 2, 3, 4, 5, 6, 7]
[1, 5, 3, 7, 5, 7, 7, 8]
[1, 2, 7, 4, 5, 8, 7, 8, 9]
[1, 6, 3, 7, 9, 8, 7, 9, 9, 10]
...
The resulting triangle of fractions begins:
1,
1/2, 2,
1/3, 2/3, 3,
1/4, 3/2, 3/4, 4,
1/5, 2/5, 3/5, 4/5, 5,
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Lance Fortnow, Counting the Rationals Quickly, Computational Complexity Weblog, Monday, March 01, 2004.
Yoram Sagher, Counting the rationals, Amer. Math. Monthly, 96 (1989), p. 823. Math. Rev. 90i:04001.

Cf. A002262.

Cf. A037161, A037162, A066657, A066658.

a226314 n k = n - (n - k) `div` gcd n k
a226314_row n = a226314_tabl !! (n-1)
a226314_tabl = map f $ tail a002262_tabl where
   f us'@(_:us) = map (v -) $ zipWith div vs (map (gcd v) us)
     where (v:vs) = reverse us'
-- Reinhard Zumkeller, Jun 10 2013

f:=(i,j) -> j+(i-j)/gcd(i,j);
g:=n->[seq(f(i,n),i=1..n)];
for n from 1 to 20 do lprint(g(n)); od:

A227189 Square array A(n>=0,k>=0) where A(n,k) gives the (k+1)-th part of the unordered partition which has been encoded in the binary expansion of n, as explained in A227183. The array is scanned antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), etc.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jul 06 2013

Discarding the trailing zero terms, on each row n there is a unique partition of integer A227183(n). All possible partitions of finite natural numbers eventually occur. The first partition that sums to n occurs at row A227368(n).

Irregular table A227739 lists only the nonzero terms.

			The top-left corner of the array:
row #  row starts as
    0  0, 0, 0, 0, 0, ...
    1  1, 0, 0, 0, 0, ...
    2  1, 1, 0, 0, 0, ...
    3  2, 0, 0, 0, 0, ...
    4  2, 2, 0, 0, 0, ...
    5  1, 1, 1, 0, 0, ...
    6  1, 2, 0, 0, 0, ...
    7  3, 0, 0, 0, 0, ...
    8  3, 3, 0, 0, 0, ...
    9  1, 2, 2, 0, 0, ...
   10  1, 1, 1, 1, 0, ...
   11  2, 2, 2, 0, 0, ...
   12  2, 3, 0, 0, 0, ...
   13  1, 1, 2, 0, 0, ...
   14  1, 3, 0, 0, 0, ...
   15  4, 0, 0, 0, 0, ...
   16  4, 4, 0, 0, 0, ...
   17  1, 3, 3, 0, 0, ...
etc.
8 has binary expansion "1000", whose runlengths are [3,1] (the length of the run in the least significant end comes first) which maps to nonordered partition {3+3} as explained in A227183, thus row 8 begins as 3, 3, 0, 0, ...
17 has binary expansion "10001", whose runlengths are [1,3,1] which maps to nonordered partition {1,3,3}, thus row 17 begins as 1, 3, 3, ...

Antti Karttunen, The first 141 antidiagonals of the table, flattened
Index entries for sequences related to partitions

Only nonzero terms: A227739. Row sums: A227183. The product of nonzero terms on row n>0 is A227184(n). Number of nonzero terms on each row: A005811. The leftmost column, after n>0: A136480. The rightmost nonzero term: A227185.

Cf. A227368 and also arrays A227186 and A227188.

(define (A227189 n) (A227189bi (A002262 n) (A025581 n)))
(define (A227189bi n k) (cond ((< (A005811 n) (+ 1 k)) 0) ((zero? k) (A136480 n)) (else (+ (- (A136480 n) 1) (A227189bi (A163575 n) (- k 1))))))

cp's OEIS Frontend

A056559 Tetrahedron with T(t,n,k) = t - n; succession of growing finite triangles with declining values per row.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Julia

Python

Formula

A196199 Count up from -n to n for n = 0, 1, 2, ... .

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A265345 Square array A(row,col): For row=0, A(0,col) = A265341(col), for row > 0, A(row,col) = A265342(A(row-1,col)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

A056560 Tetrahedron with T(t,n,k)=n-k; succession of growing finite triangles with increasing values towards bottom left.

Views

Author

Keywords

Examples

Crossrefs

Formula

A072766 Transpose of A072764, 'cons' with arguments swapped.

Views

Author

Keywords

Links

Crossrefs

Extensions

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n*(2*k+1) - 1 with n,k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A138612 Permutation of natural numbers generated with the sieve algorithm described in the comment lines.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A187769 Triangle read by rows: equivalence classes of natural numbers, where numbers are equivalent when having equal numbers of zeros and ones in binary representation, respectively.

Views

Author

Keywords

Comments

Examples

Links

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n(2k+1) - 1 with n,k >= 0.