A054425 -id:A054425 - cp's OEIS frontend

A054431 Array read by antidiagonals: T(x, y) tells whether (x, y) are coprime (1) or not (0).

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1

Offset: 1

Antti Karttunen

Array is read along (x, y) = (1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1), ...

There are nontrivial infinite paths of 1's in this sequence, moving only 1 step down or to the right at each step. Starting at (1,1), move down to (2,1), then (3,1), ..., (13,1). Then move right to (13,2), (13,3), ..., (13,11). From this point, alternate moving down to the next prime row, and right to the next prime column. - Franklin T. Adams-Watters, May 27 2014

			Rows start:
  1, 1, 1, 1, 1, 1, ...;
  1, 0, 1, 0, 1, 0, ...;
  1, 1, 0, 1, 1, 0, ...;
  1, 0, 1, 0, 1, 0, ...;
  1, 1, 1, 1, 0, 1, ...;
  1, 0, 0, 0, 1, 0, ...;

Equal to A003989 with non-one values replaced with zeros.

Cf. A047999, A054432, A055088, A054521, A215200.

reduced_residue_set_0_1_array := n -> one_or_zero(igcd(((n-((trinv(n)*(trinv(n)-1))/2))+1), ((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1) ));
one_or_zero := n -> `if`((1 = n),(1),(0)); # trinv given at A054425
A054431_row := n -> seq(abs(numtheory[jacobi](n-k+1,k)),k=1..n);
for n from 1 to 14 do A054431_row(n) od; # Peter Luschny, Aug 05 2012

t[n_, k_] := Boole[CoprimeQ[n, k]]; Table[t[n-k+1, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 21 2012 *)

def A054431_row(n): return [abs(kronecker_symbol(n-k+1,k)) for k in (1..n)]
for n in (1..14): print(A054431_row(n)) # Peter Luschny, Aug 05 2012

T(n, k) = T(n, k-n) + T(n-k, k) starting with T(n, k)=0 if n or k are nonpositive and T(1, 1)=1. T(n, k) = A054521(n, k) if n>=k, = A054521(k, n) if n<=k. Antidiagonal sums are phi(n) = A000010(n). - Henry Bottomley, May 14 2002
As a triangular array for n>=1, 1<=k<=n, T(n,k) = |K(n-k+1|k)| where K(i|j) is the Kronecker symbol. - Peter Luschny, Aug 05 2012
Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} [gcd(n,k)=1]/n^s/k^c = zeta(s)*zeta(c)/zeta(s + c). - Mats Granvik, May 19 2021

A054424 Permutation of natural numbers: maps the canonical list of fractions (A020652/A020653) to whole Stern-Brocot (Farey) tree (top = 1/1 and both sides < 1 and > 1, but excluding the "fractions" 0/1 and 1/0).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 5, 6, 15, 16, 31, 32, 9, 11, 12, 14, 63, 64, 10, 13, 127, 128, 17, 23, 24, 30, 255, 256, 19, 28, 511, 512, 33, 18, 20, 47, 48, 27, 29, 62, 1023, 1024, 22, 25, 2047, 2048, 65, 35, 39, 21, 95, 96, 26, 56, 60, 126, 4095, 4096, 34, 40, 55, 61, 8191, 8192

Offset: 1

Antti Karttunen

nonn

			Whole Stern-Brocot tree: 1/1 1/2 2/1 1/3 2/3 3/2 3/1 1/4 2/5 3/5 3/4 4/3 5/3 5/2 4/1 1/5 2/7
Canonical fractions: 1/1 1/2 2/1 1/3 3/1 1/4 2/3 3/2 4/1 1/5 5/1 1/6 2/5 3/4 4/3 5/2 6/1

Cf. A047679, A007305, A007306, A054427, A057114. In table form: A054425. Inverse permutation: A054426.

cfrac2binexp := proc(c) local i,e,n; n := 0; for i from 1 to nops(c) do e := c[i]; if(i = nops(c)) then e := e-1; fi; n := ((2^e)*n) + ((i mod 2)*((2^e)-1)); od; RETURN(n); end;
frac2position_in_whole_SB_tree := proc(r) local k,msb; if(1 = r) then RETURN(1); else if(r > 1) then k := cfrac2binexp(convert(r,confrac)); else k := ReflectBinTreePermutation(cfrac2binexp(convert(1/r,confrac))); fi; msb := floor_log_2(k); if(r > 1) then RETURN(k + (2^(msb+1))); else RETURN(k + (2^(msb+1)) - (2^msb)); fi; fi; end;
canonical_fractions_to_whole_SternBrocot_permutation := proc(u) local a,n,i; a := []; for n from 2 to u do for i from 1 to n-1 do if (1 = igcd(n,i)) then a := [op(a),frac2position_in_whole_SB_tree(i/(n-i))]; fi; od; od; RETURN(a); end; # ReflectBinTreePermutation and floor_log_2 given in A054429

canonical_fractions_to_whole_SternBrocot_permutation(30);

A064645 Table where the entry (n,k) (n >= 0, k >= 0) gives number of Motzkin paths of the length n with the minimum peak width of k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 9, 2, 1, 1, 1, 21, 4, 1, 1, 1, 1, 51, 8, 2, 1, 1, 1, 1, 127, 17, 4, 1, 1, 1, 1, 1, 323, 37, 8, 2, 1, 1, 1, 1, 1, 835, 82, 16, 4, 1, 1, 1, 1, 1, 1, 2188, 185, 33, 8, 2, 1, 1, 1, 1, 1, 1, 5798, 423, 69, 16, 4, 1, 1, 1, 1, 1, 1, 1, 15511, 978, 146, 32, 8, 2, 1, 1, 1, 1, 1, 1, 1, 41835, 2283, 312, 65, 16, 4, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Oct 03 2001

			E.g., we have the following nine Motzkin paths of length 4, of which the last 4 have each peak at least of width 1 and the last 2 with each peak at least 2 dashes wide, so M(4,0) = 9, M(4,1) = 4 and M(4,2) = 2.
   /\                                 _       _     __
  /  \   /\/\   __/\   _/\_   /\__   / \_   _/ \   /  \   ____
The array starts:
      1    1   1   1   1   1   1   1   1   1   1
      1    1   1   1   1   1   1   1   1   1   1
      2    1   1   1   1   1   1   1   1   1   1
      4    2   1   1   1   1   1   1   1   1   1
      9    4   2   1   1   1   1   1   1   1   1
     21    8   4   2   1   1   1   1   1   1   1
     51   17   8   4   2   1   1   1   1   1   1
    127   37  16   8   4   2   1   1   1   1   1
    323   82  33  16   8   4   2   1   1   1   1
    835  185  69  32  16   8   4   2   1   1   1
   2188  423 146  65  32  16   8   4   2   1   1
   5798  978 312 133  64  32  16   8   4   2   1
  15511 2283 673 274 129  64  32  16   8   4   2

Seiichi Manyama, Table of n, a(n) for n = 0..9869 (rows n=0..139 of triangle, flattened)

Column k=0: Motzkin numbers (A001006), column k=1: A004148, column k=2: A004149, column k=3: A023421, column k=4: A023422, column k=5: A023423. Uses the table A001263(n, k).

# trinv() given in A054425
[seq(A064645(j),j=0..104)]; A064645 := (n) -> Mpw((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), (n-((trinv(n)*(trinv(n)-1))/2)));
C := (n,k) -> `if`((n <= 0),0,binomial(n,k));
Mpw := proc(n,m) local i,k; 1+add(add(A001263(i,k)*C(n-(m*k),2*i),k=1..i),i=0..floor(n/2)); end;

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
CC[n_, k_] := If[n <= 0, 0, Binomial[n, k]];
a[n_] := Mpw[(((trinv[n] - 1)*(((1/2) trinv[n]) + 1)) - n), (n - ((trinv[n] (trinv[n] - 1))/2))];
Mpw[n_, m_] := 1 + Sum[Sum[If[k == 0, 0, Binomial[i - 1, k - 1] Binomial[i, k - 1]/k] CC[n - m*k, 2i], {k, 1, i}], {i, 0, n/2}];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)

A061857 Triangle in which the k-th item in the n-th row (both starting from 1) is the number of ways in which we can add 2 distinct integers from 1 to n in such a way that the sum is divisible by k.

Original entry on oeis.org

0, 1, 0, 3, 1, 1, 6, 2, 2, 1, 10, 4, 4, 2, 2, 15, 6, 5, 3, 3, 2, 21, 9, 7, 5, 4, 3, 3, 28, 12, 10, 6, 6, 4, 4, 3, 36, 16, 12, 8, 8, 5, 5, 4, 4, 45, 20, 15, 10, 9, 7, 6, 5, 5, 4, 55, 25, 19, 13, 11, 9, 8, 6, 6, 5, 5, 66, 30, 22, 15, 13, 10, 10, 7, 7, 6, 6, 5, 78, 36, 26, 18, 16, 12, 12, 9, 8, 7, 7

Offset: 1

Antti Karttunen, May 11 2001

Since the sum of two distinct integers from 1 to n can be as much as 2n-1, this triangular table cannot show all the possible cases. For larger triangles showing all solutions, see A220691 and A220693. - Antti Karttunen, Feb 18 2013 [based on Robert Israel's mail, May 07 2012]

			The second term on the sixth row is 6 because we have 6 solutions: {1+3, 1+5, 2+4, 2+6, 3+5, 4+6} and the third term on the same row is 5 because we have solutions {1+2,1+5,2+4,3+6,4+5}.
Triangle begins:
   0;
   1,  0;
   3,  1,  1;
   6,  2,  2,  1;
  10,  4,  4,  2,  2;
  15,  6,  5,  3,  3,  2;
  21,  9,  7,  5,  4,  3,  3;
  28, 12, 10,  6,  6,  4,  4,  3;
  36, 16, 12,  8,  8,  5,  5,  4,  4;
  45, 20, 15, 10,  9,  7,  6,  5,  5,  4;

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Stackexchange, Question 142323
Index entries for sequences related to subset sums modulo m

This is the lower triangular region of square array A220691. See A220693 for all nonzero solutions.

The left edge (first diagonal) of the triangle: A000217, the second diagonal is given by C(((n+(n mod 2))/2), 2)+C(((n-(n mod 2))/2), 2) = A002620, the third diagonal by A058212, the fourth by A001971, the central column by A042963? trinv is given at A054425. Cf. A061865.

a061857 n k = length [()| i <- [2..n], j <- [1..i-1], mod (i + j) k == 0]
a061857_row n = map (a061857 n) [1..n]
a061857_tabl = map a061857_row [1..]
-- Reinhard Zumkeller, May 08 2012

[seq(DivSumChoose2Triangle(j),j=1..120)]; DivSumChoose2Triangle := (n) -> nops(DivSumChoose2(trinv(n-1),(n-((trinv(n-1)*(trinv(n-1)-1))/2))));
DivSumChoose2 := proc(n,k) local a,i,j; a := []; for i from 1 to (n-1) do for j from (i+1) to n do if(0 = ((i+j) mod k)) then a := [op(a),[i,j]]; fi; od; od; RETURN(a); end;

a[n_, 1] := n*(n-1)/2; a[n_, k_] := Module[{r}, r = Reduce[1 <= i < j <= n && Mod[i + j, k] == 0, {i, j}, Integers]; Which[Head[r] === Or, Length[r], Head[r] === And, 1, r === False, 0, True, Print[r, " not parsed"]]]; Table[a[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 04 2014 *)

(define (A061857 n) (A220691bi (A002024 n) (A002260 n))) ;; Antti Karttunen, Feb 18 2013. Needs A220691bi from A220691.

From Robert Israel, May 08 2012: (Start)
Let n+1 = b mod k with 0 <= b < k, q = (n+1-b)/k.  Let k = c mod 2, c = 0 or 1.
If b = 0 or 1 then a(n,k) = q^2*k/2 + q*b - 2*q - b + 1 + c*q/2.
If b >= (k+3)/2 then a(n,k) = q^2*k/2 + q*b - 2*q + b - 1 - k/2 + c*(q+1)/2.
Otherwise a(n,k) = q^2*k/2 + q*b - 2*q + c*q/2. (End)

Offset corrected by Reinhard Zumkeller, May 08 2012

A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 1, 6, 6, 5, 6, 2, 8, 8, 6, 3, 8, 4, 10, 10, 7, 8, 1, 10, 6, 12, 12, 8, 5, 10, 2, 12, 8, 14, 14, 9, 10, 3, 12, 4, 14, 10, 16, 16, 10, 7, 12, 1, 14, 6, 16, 12, 18, 18, 11, 12, 5, 14, 2, 16, 8, 18, 14, 20, 20, 12, 9, 14, 3, 16, 4, 18, 10, 20, 16, 22, 22, 13, 14, 7, 16, 1

Offset: 0

Antti Karttunen, Oct 19 2001

Simple periodic site swap permutations of natural numbers.

Row n of the table (starting from n=0) gives a permutation of natural numbers corresponding to the simple, infinite, periodic site swap pattern ...nnnnn...

			Table begins:
1 2 3 4 5 6 7 ...
2 4 1 6 3 8 5 ...
4 6 2 8 1 10 3 ...
6 8 4 10 2 12 1 ...

Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.
Juggling Information Service, Site Swap FAQs

Successive rows and associated site swap sequences, starting from the zeroth row: (A000027, A000004), (A065164, A000012), (A065165, A007395), (A065166, A010701). Cf. also A065171, A065174, A065177. trinv given at A054425.

PerSS_table := (n) -> PerSS((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1, (n-((trinv(n)*(trinv(n)-1))/2))); PerSS := (n,c) -> Z2N(N2Z(n)+c);
N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
[seq(PerSS_table(j),j=0..119)];

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then the n-th term of the k-th row is f(g(n)+k).

A065177 Table M(n,b) (columns: n >= 1, rows: b >= 0) gives the number of site swap juggling patterns with exact period n, using exactly b balls, where cyclic shifts are not counted as distinct.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 6, 3, 1, 0, 6, 15, 12, 4, 1, 0, 9, 42, 42, 20, 5, 1, 0, 18, 107, 156, 90, 30, 6, 1, 0, 30, 294, 554, 420, 165, 42, 7, 1, 0, 56, 780, 2028, 1910, 930, 273, 56, 8, 1, 0, 99, 2128, 7350, 8820, 5155, 1806, 420, 72, 9, 1, 0, 186, 5781, 26936

Offset: 0

Antti Karttunen, Oct 19 2001

			Upper left corner starts as:
  1, 0,  0,   0,    0,     0,     0, ...
  1, 1,  2,   3,    6,     9,    18, ...
  1, 2,  6,  15,   42,   107,   294, ...
  1, 3, 12,  42,  156,   554,  2028, ...
  1, 4, 20,  90,  420,  1910,  8820, ...
  1, 5, 30, 165,  930,  5155, 28830, ...
  1, 6, 42, 273, 1806, 11809, 77658, ...
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.
Juggling Information Service, Site Swap FAQs

Row 1: A059966, row 2: A065178, row 3: A065179, row 4: A065180.
Column 1: A002378, column 2: A059270.
Main diagonal gives A306173.
Cf. also A065167. trinv given at A054425.

[seq(DistSS_table(j),j=0..119)]; DistSS_table := (n) -> DistSS((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1, (n-((trinv(n)*(trinv(n)-1))/2)));
with(numtheory); DistSS := proc(n,b) local d,s; s := 0; for d in divisors(n) do s := s+mobius(n/d)*((b+1)^d - b^d); od; RETURN(s/n); end;

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
DistSS[n_, b_] := DivisorSum[n, MoebiusMu[n/#]*((b + 1)^# - b^#)&] /n;
a[n_] := DistSS[(((trinv[n] - 1)*(((1/2)*trinv[n]) + 1)) - n) + 1, (n - ((trinv[n]*(trinv[n] - 1))/2))];
Table[a[n], {n, 0, 119}] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)

Row n is the inverse Euler transform of j-> n^(j-1). - Alois P. Heinz, Jun 23 2018

A062103 Number of paths by which an unpromoted knight (keima) of Shogi can move to various squares on infinite board, if it starts from its origin square, the second leftmost square of the back rank.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 14

Offset: 1

Antti Karttunen, May 30 2001

Table formatted as a square array shows the top-left corner of the infinite board. This is an aerated and sligthly skewed variant of Catalan's triangle A009766.

Hans L. Bodlaender, The Chess Variant Pages
Fairbairn, Leggett et al., Information about Shogi (Japanese chess)

A009766, A049604, A062104, trinv given at A054425.

[seq(ShoogiKnightSeq(j),j=1..120)]; ShoogiKnightSeq := n -> ShoogiKnightTriangle(trinv(n-1)-1,(n-((trinv(n-1)*(trinv(n-1)-1))/2))-1);
ShoogiKnightTriangle := proc(r,m) option remember; if(m < 0) then RETURN(0); fi; if(r < 0) then RETURN(0); fi; if(m > r) then RETURN(0); fi; if((1 = r) and (0 = m)) then RETURN(1); fi; RETURN(ShoogiKnightTriangle(r-3,m-2) + ShoogiKnightTriangle(r-1,m-2)); end;

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
ShoogiKnightSeq[n_] := ShoogiKnightTriangle[trinv[n - 1] - 1, (n - ((trinv[n - 1]*(trinv[n - 1] - 1))/2)) - 1];
ShoogiKnightTriangle[r_, m_] := ShoogiKnightTriangle[r, m] = Which[m < 0, 0, r < 0, 0, m > r, 0, r == 1 && m == 0, 1, True, ShoogiKnightTriangle[r - 3, m - 2] + ShoogiKnightTriangle[r - 1, m - 2]];
Array[ShoogiKnightSeq, 120] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)

A062104 Square array read by antidiagonals: number of ways a black pawn (starting at any square on the second rank) can (theoretically) end at various squares on an infinite chessboard.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 6, 0, 1, 3, 9, 15, 0, 1, 3, 10, 25, 40, 0, 1, 3, 10, 29, 69, 109, 0, 1, 3, 10, 30, 84, 193, 302, 0, 1, 3, 10, 30, 89, 242, 544, 846, 0, 1, 3, 10, 30, 90, 263, 698, 1544, 2390, 0, 1, 3, 10, 30, 90, 269, 774, 2016, 4406, 6796, 0, 1, 3, 10, 30, 90, 270

Offset: 0

Antti Karttunen, May 30 2001

Table formatted as a square array shows the top-left corner of the infinite board.

			Array begins:
0       0       0       0       0       0       0       0       0       0       0       0 ...
1       1       1       1       1       1       1       1       1       1       1 ...
2       3       3       3       3       3       3       3       3       3 ...
6       9       10      10      10      10      10      10      10 ...
15      25      29      30      30      30      30      30 ...
40      69      84      89      90      90      90 ...
109     193     242     263     269     270 ...
302     544     698     774 ...
846     1544    2016 ...
2390    4406 ...
6796 ...

A062106 gives the left column and A062107 the diagonal of the table. A062105 is a more regular variant. Cf. also A062103. Trinv given at A054425.

[seq(CPTSeq(j),j=0..91)]; CPTSeq := n -> ChessPawnTriangle( (1+(n-((trinv(n)*(trinv(n)-1))/2))), ((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1) );
ChessPawnTriangle := proc(r,c) option remember; if(r < 2) then RETURN(0); fi; if(c < 1) then RETURN(0); fi; if(2 = r) then RETURN(1); fi; if(4 = r) then RETURN(1+ChessPawnTriangle(r-1,c-1)+ChessPawnTriangle(r-1,c)+ChessPawnTriangle(r-1,c+1));
else RETURN(ChessPawnTriangle(r-1,c-1)+ChessPawnTriangle(r-1,c)+ChessPawnTriangle(r-1,c+1)); fi; end;

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
CPTSeq[n_] := ChessPawnTriangle[(1 + (n - ((trinv[n]*(trinv[n] - 1))/2))), ((((trinv[n] - 1)*(((1/2)*trinv[n]) + 1)) - n) + 1)];
ChessPawnTriangle[r_, c_] := ChessPawnTriangle[r, c] = Which[r < 2, 0, c < 1, 0, 2 == r, 1, 4 == r, 1 + ChessPawnTriangle[r - 1, c - 1] + ChessPawnTriangle[r - 1, c] + ChessPawnTriangle[r - 1, c + 1], True, ChessPawnTriangle[r - 1, c - 1] + ChessPawnTriangle[r - 1, c] + ChessPawnTriangle[r - 1, c + 1]];
Table[CPTSeq[j], {j, 0, 91}] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)

Edited by N. J. A. Sloane, May 22 2014

A338579 Triangle T(D,N) read by rows, 1 <= N < D >= 2, where T(D,N) is the position of the fraction N/D in the Farey tree (or Stern-Brocot subtree) A007305/A007306.

Original entry on oeis.org

2, 3, 4, 5, 2, 8, 9, 6, 7, 16, 17, 3, 2, 4, 32, 33, 10, 12, 13, 15, 64, 65, 5, 11, 2, 14, 8, 128, 129, 18, 3, 24, 25, 4, 31, 256, 257, 9, 20, 6, 2, 7, 29, 16, 512, 513, 34, 19, 21, 48, 49, 28, 30, 63, 1024, 1025, 17, 5, 3, 23, 2, 26, 4, 8, 32, 2048

Offset: 2

Hugo Pfoertner, Nov 10 2020

Fractions are reduced to lowest terms.

			The triangle begins
     N     1   2  3  4  5  6   7   8   9   10   11   12   13    14    15
   D \------------------------------------------------------------------
   2 |     2   .  .  .  .  .   .   .   .    .    .    .    .     .     .
   3 |     3   4  .  .  .  .   .   .   .    .    .    .    .     .     .
   4 |     5   2  8  .  .  .   .   .   .    .    .    .    .     .     .
   5 |     9   6  7 16  .  .   .   .   .    .    .    .    .     .     .
   6 |    17   3  2  4 32  .   .   .   .    .    .    .    .     .     .
   7 |    33  10 12 13 15 64   .   .   .    .    .    .    .     .     .
   8 |    65   5 11  2 14  8 128   .   .    .    .    .    .     .     .
   9 |   129  18  3 24 25  4  31 256   .    .    .    .    .     .     .
  10 |   257   9 20  6  2  7  29  16 512    .    .    .    .     .     .
  11 |   513  34 19 21 48 49  28  30  63 1024    .    .    .     .     .
  12 |  1025  17  5  3 23  2  26   4   8   32 2048    .    .     .     .
  13 |  2049  66 36 40 22 96  97  27  57   61  127 4096    .     .     .
  14 |  4097  33 35 10 41 12   2  13  56   15   62   64 8192     .     .
  15 |  8193 130  9 37  3  6 192 193   7    4   60   16  255 16384     .
  16 | 16385  65 68  5 80 11  47   2  50   14  113    8  125   128 32768
.
T(7,2) = 10 because A007306(10) = 7 and A007305(10) = 2 is the required double match, i.e., the position of the fraction 2/7 in the Farey tree is 10.
T(14,4) = T(7,2) = 10, because the fraction 4/14 reduced to lowest terms is 2/7.
T(16,12) = 8, because the fraction 12/16 reduced to lowest terms is 3/4, with the double match A007306(8)=4 and A007305(8)=3.

Hugo Pfoertner, Table of n, a(n) for n = 2..1226, rows 2..50, flattened.

Cf. A007305, A007306, A054424, A054425.

\\ using Yosu Yurramendi's formulas
a338579(lim)={
my(a7305=vectorsmall(2+2^(lim+2)),a7306=vectorsmall(2+2^(lim+2)));
  a7305[1]=1;
  for(m=1,lim,
     for(k=0,2^(m-1)-1,
      a7305[2^m+k]=a7305[2^(m-1)+k];
      a7305[2^m+2^(m-1)+k]=a7305[2^(m-1)+k]+a7305[2^m-k-1]
     )
  );
  a7306[1]=1;a7306[2]=2;
  for(m=0,lim,
     for(k=1,2^m,
      a7306[2^(m+1)+k]=a7306[2^m+k] + a7306[k];
      a7306[2^(m+1)-k+1]=a7306[2^m+k]
     )
  );
   my(findinFS(x)=for(k=2,#a7306,
      if(!(a7305[k-1]/a7306[k]-x),return(k)));0);
  for(de=2,lim+2,for(nu=1,de-1,my(q=nu/de);print1(findinFS(q),", ")))
};
a338579(10);

T(d,n) = my(ret=1); d-=n; while(n!=d, ret<<=1; if(n>d, n-=d;ret++, d-=n)); ret+1; \\ Kevin Ryde, Nov 11 2020

A061859 Differences between the ordinary multiplication table A004247 and Xmult table A048720, computed for {3..n} * {3..n}.

Original entry on oeis.org

4, 0, 0, 0, 0, 0, 8, 0, 0, 8, 12, 0, 8, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 8, 16, 8, 0, 0, 0, 0, 0, 24, 24, 0, 0, 0, 4, 0, 0, 0, 28, 0, 0, 0, 4, 16, 0, 16, 0, 0, 0, 0, 16, 0, 16, 16, 0, 16, 0, 0, 0, 0, 0, 16, 0, 16, 24, 0, 0, 8, 16, 0, 0, 16, 8, 0, 0, 24, 28, 0, 8, 32, 28, 0, 16, 0, 28, 32, 8, 0, 28

Offset: 0

Antti Karttunen, May 11 2001

Cf. "Zoomed out" variant: A061858, trinv given at A054425. The first, third and fifth diagonals are given by A048728-A048730.

[seq(diff_mult_Xmult_table3(j),j=0..119)]; diff_mult_Xmult_table3 := (n) -> (mult_table3(n) - Xmult_table3(n));
mult_table3 := (n) -> floor(evalf(((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+3) * (3+(n-((trinv(n)*(trinv(n)-1))/2))) ));
Xmult_table3 := (n) -> Xmult( ((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+3),(3+(n-((trinv(n)*(trinv(n)-1))/2))) );

cp's OEIS Frontend

A054431 Array read by antidiagonals: T(x, y) tells whether (x, y) are coprime (1) or not (0).

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A054424 Permutation of natural numbers: maps the canonical list of fractions (A020652/A020653) to whole Stern-Brocot (Farey) tree (top = 1/1 and both sides < 1 and > 1, but excluding the "fractions" 0/1 and 1/0).

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A064645 Table where the entry (n,k) (n >= 0, k >= 0) gives number of Motzkin paths of the length n with the minimum peak width of k.

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A061857 Triangle in which the k-th item in the n-th row (both starting from 1) is the number of ways in which we can add 2 distinct integers from 1 to n in such a way that the sum is divisible by k.

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A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).

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A065177 Table M(n,b) (columns: n >= 1, rows: b >= 0) gives the number of site swap juggling patterns with exact period n, using exactly b balls, where cyclic shifts are not counted as distinct.

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A062103 Number of paths by which an unpromoted knight (keima) of Shogi can move to various squares on infinite board, if it starts from its origin square, the second leftmost square of the back rank.

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