A002262 -id:A002262 - cp's OEIS frontend

A268820 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = A003188(1+A006068(A(r-1,c-1))) = A268717(1+A(r-1,c-1)), read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 6, 3, 1, 0, 5, 2, 2, 3, 1, 0, 6, 12, 7, 2, 3, 1, 0, 7, 4, 6, 6, 2, 3, 1, 0, 8, 7, 13, 5, 6, 2, 3, 1, 0, 9, 5, 12, 7, 7, 6, 2, 3, 1, 0, 10, 24, 5, 15, 4, 7, 6, 2, 3, 1, 0, 11, 8, 4, 13, 5, 5, 7, 6, 2, 3, 1, 0, 12, 11, 25, 4, 14, 12, 5, 7, 6, 2, 3, 1, 0, 13, 9, 24, 12, 15, 4, 4, 5, 7, 6, 2, 3, 1, 0, 14, 13, 9, 27, 12, 10, 13, 4, 5, 7, 6, 2, 3, 1, 0

Offset: 0

Antti Karttunen, Feb 14 2016

			The top left [0 .. 16] x [0 .. 19] section of the array:
0, 1, 2, 3, 4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
0, 1, 3, 6, 2, 12,  4,  7,  5, 24,  8, 11,  9, 13, 15, 10, 14, 48, 16, 19
0, 1, 3, 2, 7,  6, 13, 12,  5,  4, 25, 24,  9,  8, 15, 14, 11, 10, 49, 48
0, 1, 3, 2, 6,  5,  7, 15, 13,  4, 12, 27, 25,  8, 24, 14, 10,  9, 11, 51
0, 1, 3, 2, 6,  7,  4,  5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11,  8,  9
0, 1, 3, 2, 6,  7,  5, 12,  4, 10, 14, 13, 15, 30, 26, 25, 27, 11,  9, 24
0, 1, 3, 2, 6,  7,  5,  4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26,  9,  8
0, 1, 3, 2, 6,  7,  5,  4, 12, 15, 13,  9, 11, 14, 10, 29, 31, 26, 30,  8
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 14, 15,  8,  9, 10, 11, 28, 29, 30, 31
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 10, 14, 24,  8, 11,  9, 20, 28, 31
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 11, 10, 25, 24,  9,  8, 21, 20
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10,  9, 11, 27, 25,  8, 24, 23
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  8,  9, 26, 27, 24, 25
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  9, 24,  8, 30, 26, 25
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  9,  8, 25, 24, 31, 30
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  9,  8, 24, 27, 25, 29
0, 1, 3, 2, 6,  7,  5,  4, 12, 13, 15, 14, 10, 11,  9,  8, 24, 25, 26, 27

Antti Karttunen, Table of n, a(n) for n = 0..32895; the first 256 antidiagonals of array

Cf. A003188, A006068.
Inverses of these permutations can be found in table A268830.
Row 0: A001477, Row 1: A268717, Row 2: A268821, Row 3: A268823, Row 4: A268825, Row 5: A268827, Row 6: A268831, Row 7: A268933.
Rows converge towards A003188, which is also the main diagonal.
Cf. array A268715 (can be extracted from this one).
Cf. array A268833 (shows related Hamming distances with regular patterns).

A003188[n_]:=BitXor[n, Floor[n/2]]; A006068[n_]:=If[n<2, n, Block[{m=A006068[Floor[n/2]]}, 2m + Mod[Mod[n,2] + Mod[m, 2], 2]]]; a[r_, 0]:= 0; a[0, c_]:=c; a[r_, c_]:= A003188[1 + A006068[a[r - 1, c - 1]]]; Table[a[c, r - c], {r, 0, 15}, {c, 0, r}] //Flatten (* Indranil Ghosh, Apr 02 2017 *)

A003188(n) = bitxor(n, n\2);
A006068(n) = if(n<2, n, {my(m = A006068(n\2)); 2*m + (n%2 + m%2)%2});
a(r, c) = if(r==0, c, if(c==0, 0, A003188(1 + A006068(a(r - 1, c - 1)))));
for(r=0, 15, for(c=0, r, print1(a(c, r - c),", "); ); print(); ); \\ Indranil Ghosh, Apr 02 2017

def A003188(n): return n^(n//2)
def A006068(n):
    if n<2: return n
    else:
        m=A006068(n//2)
        return 2*m + (n%2 + m%2)%2
def a(r, c): return c if r<1 else 0 if c<1 else A003188(1 + A006068(a(r - 1, c - 1)))
for r in range(16):
    print([a(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Apr 02 2017

(define (A268820 n) (A268820bi (A002262 n) (A025581 n)))
(define (A268820bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (A268717 (+ 1 (A268820bi (- row 1) (- col 1)))))))
(define (A268820bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (A003188 (+ 1 (A006068 (A268820bi (- row 1) (- col 1))))))))

For row zero: A(0,k) = k, for column zero: A(n,0) = 0, and in other cases: A(n,k) = A003188(1+A006068(A(n-1,k-1)))
Other identities. For all n >= 0:
A(n,n) = A003188(n).
A(A006068(n),A006068(n)) = n.

A071673 Sequence a(n) obtained by setting a(0) = 0; then reading the table T(x,y)=a(x)+a(y)+1 in antidiagonal fashion.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 30 2002

The fixed point of RASTxx transformation. The repeated applications of RASTxx starting from A072643 seem to converge toward this sequence. Compare to A072768 from which this differs first time at the position n=37, where A072768(37) = 4, while A071673(37) = 5.
Each term k occurs A000108(k) times, and maximal position where k occurs is A072638(k).
The size of each Catalan structure encoded by the corresponding terms in triangles A071671 & A071672 (i.e., the number of digits / 2), as obtained with the global ranking/unranking scheme presented in A071651-A071654.

			The first 15 rows of this irregular triangular table:
               0,
               1,
              2, 2,
             3, 3, 3,
            3, 4, 4, 3,
           4, 4, 5, 4, 4,
          4, 5, 5, 5, 5, 4,
         4, 5, 6, 5, 6, 5, 4,
        4, 5, 6, 6, 6, 6, 5, 4,
       5, 5, 6, 6, 7, 6, 6, 5, 5,
      5, 6, 6, 6, 7, 7, 6, 6, 6, 5,
     4, 6, 7, 6, 7, 7, 7, 6, 7, 6, 4,
    5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5,
   5, 6, 6, 7, 8, 7, 7, 7, 8, 7, 6, 6, 5,
  6, 6, 7, 6, 8, 8, 7, 7, 8, 8, 6, 7, 6, 6
etc.
E.g., we have
  a(1) = T(0,0) = a(0) + a(0) + 1 = 1,
  a(2) = T(1,0) = a(1) + a(0) + 1 = 2,
  a(3) = T(0,1) = a(0) + a(1) + 1 = 2,
  a(4) = T(2,0) = a(2) + a(0) + 1 = 3, etc.

Antti Karttunen, Table of n, a(n) for n = 0..10440 (rows 0..144 of the triangle, flattened)
N. J. A. Sloane, Transforms (Maple code for RASTxx transform)

Same triangle computed modulo 2: A071674.
Permutations of this sequence include: A072643, A072644, A072645, A072660, A072768, A072789, A075167.
Cf. also A000108, A002260, A004736, A002262, A025581, A072638.

up_to = 105;
A002260(n) = (n-binomial((sqrtint(8*n)+1)\2, 2)); \\ From A002260
A004736(n) = (1-n+(n=sqrtint(8*n)\/2)*(n+1)\2); \\ From A004736
A071673list(up_to) = { my(v=vector(1+up_to)); v[1] = 0; for(n=1,up_to,v[1+n] = 1 + v[A004736(n)] + v[A002260(n)]); (v); };
v071673 = A071673list(up_to);
A071673(n) = v071673[1+n]; \\ Antti Karttunen, Aug 17 2021

(define (A071673 n) (cond ((zero? n) n) (else (+ 1 (A071673 (A025581 (-1+ n))) (A071673 (A002262 (-1+ n)))))))

a(0) = 0, a(n) = 1 + a(A025581(n-1)) + a(A002262(n-1)) = 1 + a(A004736(n)) + a(A002260(n)).

Self-referential definition added Jun 03 2002

Term a(0) = 0 prepended and the Example-section amended by Antti Karttunen, Aug 17 2021

A073345 Table T(n,k), read by ascending antidiagonals, giving the number of rooted plane binary trees of size n and height k.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 6, 8, 0, 0, 0, 0, 0, 0, 0, 4, 20, 0, 0, 0, 0, 0, 0, 0, 0, 1, 40, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 68, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 94, 152, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 114, 376, 144, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Jul 31 2002

			The top-left corner of this square array is
  1 0 0 0 0 0 0 0 0 ...
  0 1 0 0 0 0 0 0 0 ...
  0 0 2 1 0 0 0 0 0 ...
  0 0 0 4 6 6 4 1 0 ...
  0 0 0 0 8 20 40 68 94 ...
E.g. we have A000108(3) = 5 binary trees built from 3 non-leaf (i.e. branching) nodes:
_______________________________3
___\/__\/____\/__\/____________2
__\/____\/__\/____\/____\/_\/__1
_\/____\/____\/____\/____\./___0
The first four have height 3 and the last one has height 2, thus T(3,3) = 4, T(3,2) = 1 and T(3,any other value of k) = 0.

Luo Jian-Jin, Catalan numbers in the history of mathematics in China, in Combinatorics and Graph Theory, (Yap, Ku, Lloyd, Wang, Editors), World Scientific, River Edge, NJ, 1995.

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Henry Bottomley and Antti Karttunen, Notes concerning diagonals of the square arrays A073345 and A073346.
Andrew Odlyzko, Analytic methods in asymptotic enumeration.

Variant: A073346. Column sums: A000108. Row sums: A001699.
Diagonals: A073345(n, n) = A011782(n), A073345(n+3, n+2) = A014480(n), A073345(n+2, n) = A073773(n), A073345(n+3, n) = A073774(n) - Henry Bottomley and AK, see the attached notes.
A073429 gives the upper triangular region of this array. Cf. also A065329, A001263.

A073345 := n -> A073345bi(A025581(n), A002262(n));
A073345bi := proc(n,k) option remember; local i,j; if(0 = n) then if(0 = k) then RETURN(1); else RETURN(0); fi; fi; if(0 = k) then RETURN(0); fi; 2 * add(A073345bi(n-i-1,k-1) * add(A073345bi(i,j),j=0..(k-1)),i=0..floor((n-1)/2)) + 2 * add(A073345bi(n-i-1,k-1) * add(A073345bi(i,j),j=0..(k-2)),i=(floor((n-1)/2)+1)..(n-1)) - (`mod`(n,2))*(A073345bi(floor((n-1)/2),k-1)^2); end;
A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);

a[0, 0] = 1; a[n_, k_]/;k2^n-1 := 0; a[n_, k_]/;1 <= n <= k <= 2^n-1 := a[n, k] = Sum[a[n-1, k-1-i](2Sum[ a[j, i], {j, 0, n-2}]+a[n-1, i]), {i, 0, k-1}]; Table[a[n, k], {n, 0, 9}, {k, 0, 9}]
(* or *) a[0] = 0; a[1] = 1; a[n_]/;n>=2 := a[n] = Expand[1 + x a[n-1]^2]; gfT[n_] := a[n]-a[n-1]; Map[CoefficientList[ #, x, 8]&, Table[gfT[n], {n, 9}]/.{x^i_/;i>=9 ->0}] (Callan)

(See the Maple code below. Is there a nicer formula?)
This table was known to the Chinese mathematician Ming An-Tu, who gave the following recurrence in the 1730s. a(0, 0) = 1, a(n, k) = Sum[a(n-1, k-1-i)( 2*Sum[ a(j, i), {j, 0, n-2}]+a(n-1, i) ), {i, 0, k-1}]. - David Callan, Aug 17 2004
The generating function for row n, T_n(x):=Sum[T(n, k)x^k, k>=0], is given by T_n = a(n)-a(n-1) where a(n) is defined by the recurrence a(0)=0, a(1)=1, a(n) = 1 + x a(n-1)^2 for n>=2. - David Callan, Oct 08 2005

A073346 Table T(n,k) (listed antidiagonalwise in order T(0,0), T(1,0), T(0,1), T(2,0), T(1,1), ...) giving the number of rooted plane binary trees of size n and "contracted height" k.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 1, 0, 8, 8, 0, 0, 0, 0, 0, 0, 12, 16, 0, 0, 0, 0, 0, 0, 2, 12, 40, 16, 0, 0, 0, 0, 0, 0, 2, 12, 80, 48, 0, 0, 0, 0, 0, 0, 0, 0, 12, 136, 144, 32, 0, 0, 0, 0, 0, 0, 0, 2, 20, 224, 384, 128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16

Offset: 0

Antti Karttunen, Jul 31 2002

The height of binary trees is computed here in the same way as in A073345, except that whenever a complete binary tree of (2^k)-1 nodes with all its leaves at the same level, i.e., one of the following trees:
__________________\/\/\/\/_
___________\/\/\/\/
______________\/____\ /__
____.__\/__\/____\/__ etc.
is encountered as a terminating subtree, it is regarded just a variant of . (an empty tree, a single leaf) and contributes nothing to the height of the tree.

			The top-left corner of this square array:
1 1 0 1 0 0 0 1 ...
0 0 2 0 2 2 0 0 ...
0 0 0 4 4 8 12 12 ...
0 0 0 0 8 16 40 80 ...

H. Bottomley and A. Karttunen, Notes concerning diagonals of the square arrays A073345 and A073346.

Variant: A073345. The first row: A036987. Column sums: A000108. Diagonals: T(n, n) = A000007(n), T(n+1, n) = A000079(n), T(n+2, n) = A058922(n), T(n+3, n) = A074092(n) - [see the attached notes.].

A073430 gives the upper triangular region of this array. Used to compute A073431. Entries on row k are all divisible by 2^k, thus dividing them out yields the array/triangle A074079/A074080.

A073346 := n -> A073346bi(A025581(n), A002262(n));
A073346bi := proc(n,k) option remember; local i,j; if(0 = k) then RETURN(A036987(n)); fi; if(0 = n) then RETURN(0); fi; 2 * add(A073346bi(n-i-1,k-1) * add(A073346bi(i,j),j=0..(k-1)),i=0..floor((n-1)/2)) + 2 * add(A073346bi(n-i-1,k-1) * add(A073346bi(i,j),j=0..(k-2)),i=(floor((n-1)/2)+1)..(n-1)) - (`mod`(n,2))*(A073346bi(floor((n-1)/2),k-1)^2) - (`if`((1=k),1,0))*A036987(n); end;
A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);

(See the Maple code below. Note that here we use the same convolution recurrence as with A073345, but only the initial conditions for the first two rows (k=0 and k=1) are different. Is there a nicer formula?)

Sequence number in comments corrected

A114327 Table T(n,m) = n - m read by upwards antidiagonals.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Feb 06 2006

From Clark Kimberling, May 31 2011: (Start)
If we arrange A000027 as an array with northwest corner
   1    2    4    7    17 ...
   3    5    8   12    18 ...
   6    9   13   18    24 ...
  10   14   19   25    32 ...
diagonals can be numbered as follows depending on their distance to the main diagonal:
  Diag 0:  1, 5, 13, 25, ...
  Diag 1:  2, 8, 18, 32, ...
  Diag -1: 3, 9, 19, 33, ...,
then a(n) in the flattened array is the number of the diagonal that contains n+1. (End)
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in Jordan-Schwinger form (cf. Harter, Klee, Schwinger). Triangle terms T(n,k) = T(2j,j-m) satisfy: (1/2) T(2j,j-m) =  = m. Matrix J_3 is diagonal, so this equality determines the only nonzero entries. - Bradley Klee, Jan 29 2016
For the characteristic polynomial of the n X n matrix M_n (Det(x*1_n - M_n)) with elements M_n(i, j) = i-j see the Michael Somos, Nov 14 2002, comment on A002415. - Wolfdieter Lang, Feb 05 2018
The entries of the n-th antidiagonal, T(n,1), T(n-1,2), ... , T(1,n), are the eigenvalues of the Hamming graph H(2,n-1) (or hypercube Q(n-1)). - Miquel A. Fiol, May 21 2024

			From _Wolfdieter Lang_, Feb 05 2018: (Start)
The table T(n, m) begins:
  n\m 0  1  2  3  4  5 ...
  0:  0 -1 -2 -3 -4 -5 ...
  1:  1  0 -1 -2 -3 -4 ...
  2:  2  1  0 -1 -2 -3 ...
  3:  3  2  1  0 -1 -2 ...
  4:  4  3  2  1  0 -1 ...
  5:  5  4  3  2  1  0 ...
  ...
The triangle t(n, k) begins:
  n\k  0  1  2  3  4  5  6  7  8  9  10 ...
  0:   0
  1:   1 -1
  2:   2  0 -2
  3:   3  1 -1 -3
  4:   4  2  0 -2 -4
  5:   5  3  1 -1 -3 -5
  6:   6  4  2  0 -2 -4 -6
  7:   7  5  3  1 -1 -3 -5 -7
  8:   8  6  4  2  0 -2 -4 -6 -8
  9:   9  7  5  3  1 -1 -3 -5 -7 -9
  10: 10  8  6  4  2  0 -2 -4 -6 -8 -10
  ... Reformatted and corrected. (End)

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum, Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

Apart from signs, same as A049581. Cf. A003056, A025581, A002262, A002260, A004736. J_1,J_2: A094053; J_1^2,J_2^2: A141387, A268759. A002415.

a114327 n k = a114327_tabl !! n !! k
a114327_row n = a114327_tabl !! n
a114327_tabl = zipWith (zipWith (-)) a025581_tabl a002262_tabl
-- Reinhard Zumkeller, Aug 09 2014

seq(seq(i-2*j,j=0..i),i=0..30); # Robert Israel, Jan 29 2016

max = 12; a025581 = NestList[Prepend[#, First[#]+1]&, {0}, max]; a002262 = Table[Range[0, n], {n, 0, max}]; a114327 = a025581 - a002262 // Flatten (* Jean-François Alcover, Jan 04 2016 *)
Flatten[Table[-2 m, {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

T(n,m) = n-m \\ Charles R Greathouse IV, Feb 07 2017

from math import isqrt
def A114327(n): return ((m:=isqrt(k:=n+1<<1))+(k>m*(m+1)))**2+1-k # Chai Wah Wu, Nov 09 2024

G.f. for the table: Sum_{n, m>=0} T(n,m)*x^n*y^n = (x-y)/((1-x)^2*(1-y)^2).
E.g.f. for the table: Sum_{n, m>=0} T(n,m)x^n/n!*y^m/m! = (x-y)*e^{x+y}.
T(n,k) = A025581(n,k) - A002262(n,k).
a(n+1) = A004736(n) - A002260(n) or a(n+1) = ((t*t+3*t+4)/2-n) - (n-t*(t+1)/2), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012
G.f. as sequence: -(1+x)/(1-x)^2 + Sum_{j>=0} (2*j+1)*x^(j*(j+1)/2) / (1-x). The sum is related to Jacobi theta functions. - Robert Israel, Jan 29 2016
Triangle t(n, k) = n - 2*k, for n >= 0, k = 0..n. (see the Maple program). - Wolfdieter Lang, Feb 05 2018

Formula improved by Reinhard Zumkeller, Aug 09 2014

A163358 Inverse permutation to A163357.

Original entry on oeis.org

0, 1, 4, 2, 5, 9, 13, 8, 12, 18, 24, 17, 11, 7, 3, 6, 10, 16, 22, 15, 21, 28, 37, 29, 38, 47, 58, 48, 39, 30, 23, 31, 40, 50, 60, 49, 59, 70, 83, 71, 84, 97, 112, 98, 85, 72, 61, 73, 62, 52, 42, 51, 41, 32, 25, 33, 26, 19, 14, 20, 27, 34, 43, 35, 44, 54, 64, 53, 63, 74, 87

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

abs(A025581(a(n+1)) - A025581(a(n))) + abs(A002262(a(n+1)) - A002262(a(n))) = 1 for all n.

Inverse: A163357. a(n) = A054239(A163356(n)). One-based version: A163362. See also A163334 and A163336.

A194847 Write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; sequence gives i values.

Original entry on oeis.org

2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 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, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10

Offset: 0

N. J. A. Sloane, Sep 03 2011

nonn

Each n >= 0 has a unique representation as n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0. This is the combinatorial number system of degree t = 3, where we get [A194847, A194848, A056558]. For degree t = 2 we get [A002024, A002262] and A138036.

			The i,j,k coordinates for n equal to 0 through 10 are:
0, [2, 1, 0]
1, [3, 1, 0]
2, [3, 2, 0]
3, [3, 2, 1]
4, [4, 1, 0]
5, [4, 2, 0]
6, [4, 2, 1]
7, [4, 3, 0]
8, [4, 3, 1]
9, [4, 3, 2]
10, [5, 1, 0]

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

The [i,j,k] values are [A194847, A194848, A056558], or equivalently [A056556+2, A056557+1, A056558]. See A194849 for the union list of triples.

Cf. also A002024, A002262, A138036.

# Given x and a list a, returns smallest i such that x >= a[i].
whereinlist:=proc(x,a)  local i:
if whattype(a) <> list then ERROR(`a not a list`); fi:
for i from 1 to nops(a) do if x < a[i] then break; fi; od:
RETURN(i-1); end:
t3:=[seq(binomial(n,3),n=0..50)];
t2:=[seq(binomial(n,2),n=0..50)];
t1:=[seq(binomial(n,1),n=0..50)];
for n from 0 to 200 do
i3:=whereinlist(n,t3);
i2:=whereinlist(n-t3[i3],t2);
i1:=whereinlist(n-t3[i3]-t2[i2],t1);
L[n]:=[i3-1,i2-1,i1-1];
od:
[seq(L[n][1],n=0..200)];

from math import comb
from sympy import integer_nthroot
def A194847(n): return (m:=integer_nthroot(6*(n+1),3)[0])+(n>=comb(m+2,3))+1 # Chai Wah Wu, Nov 05 2024

Equals A056556(n) + 2.

A242422 Numbers in whose prime factorization the indices of primes sum to a triangular number.

Original entry on oeis.org

1, 2, 5, 6, 8, 13, 21, 22, 25, 27, 28, 29, 30, 36, 40, 46, 47, 48, 57, 64, 73, 76, 85, 86, 91, 102, 107, 117, 121, 123, 130, 136, 142, 147, 151, 154, 156, 164, 165, 175, 185, 189, 196, 197, 198, 201, 206, 208, 210, 217, 220, 222, 225, 243, 250, 252, 257, 264, 268, 270, 279, 280, 296, 298, 299, 300

Offset: 1

Antti Karttunen, May 16 2014

nonn

Numbers k such that A010054(A056239(k)) is one, or equally, that A002262(A056239(k)) is zero.
In "Bulgarian solitaire" a deck of cards or another finite set of objects is divided into one or more piles, and the "Bulgarian operation" is performed by taking one card from each pile, and making a new pile of them. The question originally posed was: on what condition the resulting partitions will eventually reach a fixed point, that is, a collection of piles that will be unchanged by the operation. See Martin Gardner reference and the Wikipedia-page.
This sequence answers the question when we implement the operation on the partition list A112798: These are all such numbers that starting iterating A242424 from them leads eventually to a fixed point, which will be one of the primorial numbers, A002110.
Contains the same terms as rows of A215366 indexed with triangular numbers (A000217: 0, 1, 3, 6, ...), although not in the same order. {1}, {2}, {5, 6, 8}, {13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64}, etc.
Heinz numbers of integer partitions of triangular numbers. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). - Gus Wiseman, Nov 13 2018

			1 is present as it has an empty factorization, for which the sum of prime indices is zero, and zero is also a triangular number.
2 = p_1 is present as 1 is a triangular number.
6 = p_1 * p_2 is present, as 1+2 = 3 is a triangular number.
300 = 2*2*3*5*5 = p_1 * p_1 * p_2 * p_3 * p_3 is present, as 1+1+2+3+3 = 10 is a triangular number.
Any primorial number p_1 * p_2 * p_3 * ... * p_n is present, as 1+2+3+...+n is by definition a triangular number.
The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (), (1), (3), (2,1), (1,1,1), (6), (4,2), (5,1), (3,3), (2,2,2), (4,1,1), (10), (3,2,1), (2,2,1,1), (3,1,1,1), (9,1), (15), (2,1,1,1,1), (8,2), (1,1,1,1,1,1), (21), (8,1,1), (7,3), (14,1), (6,4). - _Gus Wiseman_, Nov 13 2018

Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.

Antti Karttunen, Table of n, a(n) for n = 1..2455
Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).
Wikipedia, Bulgarian solitaire

Complement: A242423.
A002110 (primorial numbers) is a subsequence.
Cf. A112798, A215366, A242424, A000217, A002262, A010054, A056239, A226062, A037481.
Cf. A066723, A321470, A321471, A321472.

triQ[n_]:=Module[{k,i},For[k=n;i=1,k>0,i++,k-=i];k==0];
Select[Range[100],triQ[Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]]&] (* Gus Wiseman, Nov 13 2018 *)

A268715 Square array A(i,j) = A003188(A006068(i) + A006068(j)), read by antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 3, 6, 6, 3, 4, 2, 5, 2, 4, 5, 12, 7, 7, 12, 5, 6, 4, 15, 6, 15, 4, 6, 7, 7, 13, 13, 13, 13, 7, 7, 8, 5, 4, 12, 9, 12, 4, 5, 8, 9, 24, 12, 5, 11, 11, 5, 12, 24, 9, 10, 8, 27, 4, 14, 10, 14, 4, 27, 8, 10, 11, 11, 25, 25, 10, 15, 15, 10, 25, 25, 11, 11, 12, 9, 8, 24, 29, 14, 12, 14, 29, 24, 8, 9, 12, 13, 13, 24, 9, 31, 31, 13, 13, 31, 31, 9, 24, 13, 13

Offset: 0

Antti Karttunen, Feb 12 2016

Each row n is row A006068(n) of array A268820 without its A006068(n) initial terms.

			The top left [0 .. 15] x [0 .. 15] section of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15
   1,  3,  6,  2, 12,  4,  7,  5, 24,  8, 11,  9, 13, 15, 10, 14
   2,  6,  5,  7, 15, 13,  4, 12, 27, 25,  8, 24, 14, 10,  9, 11
   3,  2,  7,  6, 13, 12,  5,  4, 25, 24,  9,  8, 15, 14, 11, 10
   4, 12, 15, 13,  9, 11, 14, 10, 29, 31, 26, 30,  8, 24, 27, 25
   5,  4, 13, 12, 11, 10, 15, 14, 31, 30, 27, 26,  9,  8, 25, 24
   6,  7,  4,  5, 14, 15, 12, 13, 26, 27, 24, 25, 10, 11,  8,  9
   7,  5, 12,  4, 10, 14, 13, 15, 30, 26, 25, 27, 11,  9, 24,  8
   8, 24, 27, 25, 29, 31, 26, 30, 17, 19, 22, 18, 28, 20, 23, 21
   9,  8, 25, 24, 31, 30, 27, 26, 19, 18, 23, 22, 29, 28, 21, 20
  10, 11,  8,  9, 26, 27, 24, 25, 22, 23, 20, 21, 30, 31, 28, 29
  11,  9, 24,  8, 30, 26, 25, 27, 18, 22, 21, 23, 31, 29, 20, 28
  12, 13, 14, 15,  8,  9, 10, 11, 28, 29, 30, 31, 24, 25, 26, 27
  13, 15, 10, 14, 24,  8, 11,  9, 20, 28, 31, 29, 25, 27, 30, 26
  14, 10,  9, 11, 27, 25,  8, 24, 23, 21, 28, 20, 26, 30, 29, 31
  15, 14, 11, 10, 25, 24,  9,  8, 21, 20, 29, 28, 27, 26, 31, 30

Antti Karttunen, Table of n, a(n) for n = 0..15050; the first 173 antidiagonals of the array

Cf. A003188, A006068, A268714, A268820.
Main diagonal: A001969.
Row 0, column 0: A001477.
Row 1, column 1: A268717.
Antidiagonal sums: A268837.
Cf. A268719 (the lower triangular section).
Cf. also A268725.

A003188[n_] := BitXor[n, Floor[n/2]]; A006068[n_] := BitXor @@ Table[Floor[ n/2^m], {m, 0, Log[2, n]}]; A006068[0]=0; A[i_, j_] := A003188[A006068[i] + A006068[j]]; Table[A[i-j, j], {i, 0, 13}, {j, 0, i}] // Flatten (* Jean-François Alcover, Feb 17 2016 *)

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def T(n, k): return a003188(a006068(n) + a006068(k))
for n in range(21): print([T(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268715 n) (A268715bi (A002262 n) (A025581 n)))
(define (A268715bi row col) (A003188 (+ (A006068 row) (A006068 col))))
;; Alternatively, extracting data from array A268820:
(define (A268715bi row col) (A268820bi (A006068 row) (+ (A006068 row) col)))

A(i,j) = A003188(A006068(i) + A006068(j)) = A003188(A268714(i,j)).

A(row,col) = A268820(A006068(row), (A006068(row)+col)).

A053616 Pyramidal sequence: distance to nearest triangular number.

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 20 2000

From Wolfdieter Lang, Oct 24 2020: (Start)
If this sequence is written with offset 1 as a number triangle T(n, k), with n the length of row n, for n >= 1, then row n gives the primitive period of the periodic sequence {k (mod* n)}_{k>=0}, where k (mod* n) = k (mod n) if k <= floor(n/2) and otherwise it is -k (mod n). Such a modified modular relation mod* n has been used by Brändli and Beyne, but for integers relative prime to n.
These periodic sequences are given in A000007, A000035, A011655, A007877, |A117444|, A260686, A279316, for n = 1, 2, ..., 7. For n = 10 A271751, n = 12 A271832, n = 14 A279313. (End)

			a(12) = |12 - 10| = 2 since 10 is the nearest triangular number to 12.
From _M. F. Hasler_, Dec 06 2019: (Start)
Ignoring a(0) = 0, the sequence can be written as triangle indexed by m >= k >= 1, in which case the terms are (m - |k - |m-k||)/2, as follows:
   0,      (Row 0: ignore)
   0,      (Row m=1, k=1: For k=m, m - |k - |m-k|| = m - |m - 0| = 0.)
   1, 0,        (Row m=2: for k=1, |m-k| = 1, k-|m-k| = 0, m-0 = 2, (...)/2 = 1.)
   1, 1, 0,
   1, 2, 1, 0,    (Row m=4: for k=2, we have twice the value of (m=2, k=1) => 2.)
   1, 2, 2, 1, 0,
   (...)
This is related to the non-associative operation A049581(x,y) = |x - y| =: x @ y. Specifically, @ is commutative and any x is its own inverse, so non-associativity of @ can be measured through the commutator ((x @ y) @ y) @ x which equals twice the element indexed {m,k} = {x,y} in the above triangle.
(End)

T. D. Noe, Table of n, a(n) for n = 0..1000
Gerold Brändli and Tim Beyne, Modified Congruence Modulo n with Half the Amount of Residues, arXiv:1504.02757 [math.NT], 2015-2016.
Index entries for sequences related to distance to nearest element of some set

Cf. A000217, A002262, A053188, A172471.

a(n) = abs(A305258(n)).

a[n_] := (k =.; k = Reduce[k > 0 && k*(k+1)/2 == n, Reals][[2]] // Floor; Min[(k+1)*(k+2)/2 - n, n - k*(k+1)/2]); Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Jan 08 2013 *)
Module[{trms=120,t},t=Accumulate[Range[Ceiling[(Sqrt[8*trms+1]-1)/2]]]; Join[{0},Flatten[Table[Abs[Nearest[t,n][[1]]-n],{n,trms}]]]] (* Harvey P. Dale, Nov 08 2013 *)

print1(x=0, ", ");for(stride=1,13,x+=stride;y=x+stride+1;for(k=x,y-1,print1(min(k-x,y-k), ", "))) \\ Hugo Pfoertner, Jun 02 2018

apply( {a(n)=if(n,-abs(n*2-(n=sqrtint(8*n-7)\/2)^2)+n)\2}, [0..40]) \\ same as (i - |j - |i-j||)/2 with i=sqrtint(8*n-7)\/2, j=n-i(i-1)/2. - M. F. Hasler, Dec 06 2019

from math import isqrt
def A053616(n): return abs((m:=isqrt(k:=n<<1))*(m+1)-k)>>1 # Chai Wah Wu, Jul 15 2022

a(n) = (x - |y - |x-y||)/2, when (x,y) is the n-th element in the triangle x >= y >= 1. - M. F. Hasler, Dec 06 2019
a(n) = (1/2)*abs(t^2 + t - 2*n), where t = floor(sqrt(2*n)) = A172471. - Ridouane Oudra, Dec 15 2021
From Ctibor O. Zizka, Nov 12 2024: (Start)
For s >= 1, t from [0, s] :
a(2*s^2 + t) = s - t.
a(2*s^2 - t) = s - t.
a(2*s^2 + 2*s - t) = s - t.
a(2*s^2 + 2*s + 1 + t) = s - t. (End)

cp's OEIS Frontend

A268820 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = A003188(1+A006068(A(r-1,c-1))) = A268717(1+A(r-1,c-1)), read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

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A071673 Sequence a(n) obtained by setting a(0) = 0; then reading the table T(x,y)=a(x)+a(y)+1 in antidiagonal fashion.

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A073345 Table T(n,k), read by ascending antidiagonals, giving the number of rooted plane binary trees of size n and height k.

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A073346 Table T(n,k) (listed antidiagonalwise in order T(0,0), T(1,0), T(0,1), T(2,0), T(1,1), ...) giving the number of rooted plane binary trees of size n and "contracted height" k.

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A114327 Table T(n,m) = n - m read by upwards antidiagonals.

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A163358 Inverse permutation to A163357.

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A194847 Write n = C(i,3)+C(j,2)+C(k,1) with i>j>k>=0; sequence gives i values.

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