A028310 -id:A028310 - cp's OEIS frontend

A220073 Mirror of the triangle A130517.

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

Offset: 1

Reinhard Zumkeller, Dec 03 2012

T(n,k) = A130517(n,n-k+1), 1 <= k <= n;
T(n,n) = T(n,1) + 1.
From Boris Putievskiy, Jan 15 2013: (Start)
General case see A187760. Let m be natural number. Table T(n,k) n, k > 0, T(n,k)=n-k+1, if n>=k, T(n,k)=k-n+m-1, if n < k.  Table T(n,k) read by antidiagonals. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m}  shifts the sequence A000027. For m=1 the result is A220073, for m=2 the result is A143182. (End)
First inverse function (numbers of rows) for pairing function A209293. - Boris Putievskiy, Jan 28 2013

			From _Boris Putievskiy_, Jan 15 2013: (Start)
The start of the sequence as table:
1..1..2..3..4..5..6..7...
2..1..1..2..3..4..5..6...
3..2..1..1..2..3..4..5...
4..3..2..1..1..2..3..4...
5..4..3..2..1..1..2..3...
6..5..4..3..2..1..1..2...
7..6..5..4..3..2..1..1...
8..7..6..5..4..3..2..1...
. . .
The start of the sequence as triangle array read by rows:
1,
1, 2,
2, 1, 3,
3, 1, 2, 4,
4, 2, 1, 3, 5,
5, 3, 1, 2, 4, 6,
6, 4, 2, 1, 3, 5, 7,
7, 5, 3, 1, 2, 4, 6, 8,
. . .
Row number r contains r numbers: r-1, r-3,...,1,...r-2,r.
(End)

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A028310 (left edge), A000027 (right edge), A000012 (central terms), A000217 (row sums), A220075 (partial sums in rows), A002260, A000027, A143182, A187760, A209293.

a220073 n k = a220073_tabl !! (n-1) !! (k-1)
a220073_row n = a220073_tabl !! (n-1)
a220073_tabl = map reverse a130517_tabl

max = 13;
row[n_] := Join[Range[n, 1, -1], Range[max - n + 1]];
T = Array[row, max];
Table[T[[n - k + 1, k]], {n, 1, max}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 11 2017 *)

T(1,1)=1, for n>1: T(n,k)=T(n-1,n-k+1), 1<=k

From Boris Putievskiy, Jan 15 2013: (Start)

For the general case

a(n) = |(t+1)^2 - 2n| + m*floor((t^2+3t+2-2n)/(t+1)),

where t = floor((-1+sqrt(8*n-7))/2).

For m = 2

a(n) = |(t+1)^2 - 2n| + floor((t^2+3t+2-2n)/(t+1)),

where t=floor((-1+sqrt(8*n-7))/2). (End)

A253909 1 together with the positive squares.

Original entry on oeis.org

1, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500

Offset: 0

Omar E. Pol, Feb 12 2015

Also, right border of A246595 arranged as an irregular triangle.

a(n) are the Engel expansion of A070910. - Benedict W. J. Irwin, Dec 15 2016

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028310, A070910, A246595. Essentially the same as A000290 and A174902.

Join[{1}, Range[50]^2] (* Alonso del Arte, Feb 23 2015 *)
Range[0, 50]^2 /. 0 -> 1 (* Robert G. Wilson v, Dec 15 2016 *)

a(n)=max(n,1)^2 \\ Charles R Greathouse IV, Dec 16 2016

a(n) = A028310(n)^2.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>=4. - David Neil McGrath, May 23 2015
G.f.: (x^3-4*x^2+2*x-1)/(x-1)^3. - David Neil McGrath, May 25 2015
E.g.f.: 1 + exp(x)*x*(1 + x). - Stefano Spezia, Jan 30 2023

Keyword:mult added by Andrew Howroyd, Aug 06 2018

A069775 Permutation of natural numbers induced by the automorphism gma069775! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 17, 18, 16, 14, 15, 21, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 58, 59, 56, 51, 52, 57, 53, 54, 55, 63, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Antti Karttunen, Apr 16 2002

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Inverse of A069776. a(n) = A057163(A057509(A057163(n))) = A069773(A069770(n)). Cf. also A069787, A072797.

Number of cycles: A003239. Number of fixed points: A034731. Max. cycle size: A028310. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A069776 Permutation of natural numbers induced by the automorphism gma069776! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 17, 18, 16, 14, 15, 20, 21, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 54, 55, 57, 58, 59, 53, 56, 51, 52, 61, 62, 63, 60, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Antti Karttunen, Apr 16 2002

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Inverse of A069775. a(n) = A057163(A057510(A057163(n))) = A069770(A069774(n)). Cf. also A069787, A072797.

A199883 Number of distinct values taken by 6th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 48, 113, 262, 591, 1263, 2505, 4764, 8479, 14285, 22871, 35316, 52755, 76517, 107826, 148914, 202715, 270622

Offset: 1

Alois P. Heinz, Nov 11 2011

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 6th derivative at x=1: (x^(x^(x^x))) -> 2934; ((x^x)^(x^x)), ((x^(x^x))^x) -> 4908; (x^((x^x)^x)) -> 5034; (((x^x)^x)^x) -> 8322.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215836. Column k=6 of A216368.

f:= proc(n) option remember;
      `if`(n=1, {[0, 0, 0, 0, 0]},
                {seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
                 8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
                 10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
                 +15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4],
                 45*h[1]*g[1]^2+(120+60*h[2]+15*h[3]+60*g[2]+
                 270*h[1])*g[1]+54+15*h[3]+30*g[3]+6*g[4]+
                 60*h[1]*g[2]+15*h[1]*g[3]+30*h[1]+ 20*h[2]*g[2]+
                 100*h[2]+90*h[1]^2+g[5]+60*g[2]+6*h[4]],
                 h=f(n-j)), g=f(j)), j=1..n-1)})
    end:
a:= n-> nops(map(x-> x[5], f(n))):
seq(a(n), n=1..15);

a(22)-a(23) from Alois P. Heinz, Sep 26 2014

A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 18, 6, 3, 0, 192, 48, 12, 4, 0, 2500, 500, 100, 20, 5, 0, 38880, 6480, 1080, 180, 30, 6, 0, 705894, 100842, 14406, 2058, 294, 42, 7, 0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8, 0, 344373768, 38263752, 4251528, 472392, 52488, 5832, 648, 72, 9

Offset: 0

Alois P. Heinz, Aug 19 2013

			T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,2], [2,1].
T(2,2) = 2: [1,1], [2,2].
T(3,1) = 18: [1,1,2], [1,1,3], [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3], [3,3,1], [3,3,2].
T(3,2) = 6: [1,2,2], [1,3,3], [2,1,1], [2,3,3], [3,1,1], [3,2,2].
T(3,3) = 3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  1;
  0,        1;
  0,        2,       2;
  0,       18,       6,      3;
  0,      192,      48,     12,     4;
  0,     2500,     500,    100,    20,    5;
  0,    38880,    6480,   1080,   180,   30,   6;
  0,   705894,  100842,  14406,  2058,  294,  42,  7;
  0, 14680064, 1835008, 229376, 28672, 3584, 448, 56,  8;

Alois P. Heinz, Rows n = 0..140, flattened

Row sums give: A000312.
Columns k=0-4 give: A000007, A066274(n) = 2*A081131(n) for n>1, A053506(n) for n>2, A055865(n-1) = A085389(n-1) for n>3, A085390(n-1) for n>4.
Main diagonal gives: A028310.
Lower diagonals include (offsets may differ): A002378, A045991, A085537, A085538, A085539.
Cf. A228154, A228617.

T:= (n, k)-> `if`(n=0 and k=0, 1, `if`(k<1 or k>n, 0,
             `if`(k=n, n, (n-1)*n^(n-k)))):
seq(seq(T(n,k), k=0..n), n=0..12);

f[0,0]=1;
f[n_,k_]:=Which[1<=k<=n-1,n^(n-k)*(n-1),k<1,0,k==n,n,k>n,0];
Table[Table[f[n,k],{k,0,n}],{n,0,10}]//Grid (* Geoffrey Critzer, May 19 2014 *)

T(0,0) = 1, else T(n,k) = 0 for k<1 or k>n, else T(n,n) = n, else T(n,k) = (n-1)*n^(n-k).
Sum_{k=0..n}   T(n,k) = A000312(n).
Sum_{k=0..n} k*T(n,k) = A031972(n).

A347708 Number of distinct possible alternating products of odd-length factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 11 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
Note that it is sufficient to look at only length-1 and length-3 factorizations; cf. A347709.

			Representative factorizations for each of the a(180) = 7 alternating products:
  (2*2*3*3*5) -> 5
     (2*2*45) -> 45
     (2*3*30) -> 20
     (2*5*18) -> 36/5
     (2*9*10) -> 20/9
     (3*4*15) -> 45/4
        (180) -> 180

Antti Karttunen, Table of n, a(n) for n = 1..65537

The version for partitions is A028310, reverse A347707.
Positions of 1's appear to be A037143 \ {1}.
The even-length version for n > 1 is A072670, strict A211159.
Counting only integers appears to give A293234, with evens A046951.
This is the odd-length case of A347460, reverse A038548.
The any-length version for partitions is A347461, reverse A347462.
The length-3 case is A347709.
A001055 counts factorizations (strict A045778, ordered A074206).
A056239 adds up prime indices, row sums of A112798.
A276024 counts distinct positive subset-sums of partitions.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions.
A347050 = factorizations w/ an alternating permutation, complement A347706.
A347441 counts odd-length factorizations with integer alternating product.
Cf. A002033, A103919, A108917, A119620, A325770, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],OddQ[Length[#]]&]]],{n,100}]

altprod(facs) = prod(i=1,#facs,facs[i]^((-1)^(i-1)));
A347708aux(n, m=n, facs=List([])) = if(1==n, if((#facs)%2, altprod(facs), 0), my(newfacs, r, rats=List([])); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); r = A347708aux(n/d, d, newfacs); if(r, rats = concat(rats,r)))); (rats));
A347708(n) = if(1==n,0,#Set(A347708aux(n))); \\ Antti Karttunen, Jan 29 2025

Conjecture: For n > 1, a(n) = 1 + A347460(n) - A038548(n) + A072670(n).

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A101279 a(1) = 1; a(2k) = a(k), a(2k+1) = k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 3, 7, 1, 8, 4, 9, 2, 10, 5, 11, 1, 12, 6, 13, 3, 14, 7, 15, 1, 16, 8, 17, 4, 18, 9, 19, 2, 20, 10, 21, 5, 22, 11, 23, 1, 24, 12, 25, 6, 26, 13, 27, 3, 28, 14, 29, 7, 30, 15, 31, 1, 32, 16, 33, 8, 34, 17, 35, 4, 36, 18, 37, 9, 38, 19, 39, 2, 40, 20, 41, 10

Offset: 1

N. J. A. Sloane, May 22 2006; definition corrected May 23 2006

nonn

From Jeremy Gardiner, Mar 22 2015: (Start)
For n > 2 write n, n-1 in binary, then align bits from the left and take contiguous matching bits as a binary number.
For example:
n    = 19 10011
n-1  = 18 10010
a(n) =  9 1001
Also arrange the positive integers as a binary tree rooted at 1 as shown:
                                     1
                                     |
                 2................../ \..................3
                 |                                       |
       4......../ \........5                   6......../ \........7
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
   8       9          10      11          12       13         14       15
16 17   18 19      20  21  22  23      24  25   26  27     28  29   30  31
Each branch doubles the number above at the left fork or doubles and adds 1 at the right fork. Then for n > 2, a(n) is the greatest common ancestor of n and n-1, a(n) = gca(n,n-1).
(End)
From David James Sycamore, Mar 07 2023: (Start)
The following identical sequences, {b(n)} and {c(n)}, are the same as a(n+1) for n >= 1.
b(1) = 1, then reverse the conditions in Name: b(2k) = k, b(2k+1) = b(k).
c(1) = 1, then if c(n) is a first occurrence, c(n+1) = c(c(n)), else if c(n) has occurred previously, c(n+1) = n - c(n-1).
These are fractal sequences (b(2m+1) = c(2m+1), m >= 1, recovers the originals). Also {b(n)} and {c(n)} interleave A000027 with the present sequence.
(End)

			If n is a power of 2 then k=1.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A003602, A025480.

cf. A000027, A000079, A007283, A070939, A119387. - David James Sycamore, Mar 07 2023

a:=array(0..200); a[1]:=1; M:=200; for n from 2 to M do if n mod 2 = 1 then a[n]:=(n-1)/2; else a[n]:=a[n/2]; fi; od: [seq(a[n],n=1..M)];

a[1] = 1; a[n_] := a[n] = If[OddQ@n, (n - 1)/2, a[n/2]]; Array[a, 84] (* Robert G. Wilson v, May 23 2006 *)

a(n)=(n/2^valuation(n,2)-1)/2+if(n==2^valuation(n,2),1,0) /* Ralf Stephan, Aug 21 2013 */

a((n+1)/2) = A028310(n) if n is odd and a(n/2) = a(n) if n is even; thus this is a fractal sequence. - Robert G. Wilson v, May 23 2006; corrected by Clark Kimberling, Jul 07 2007
a(n) = A025480(n) + A036987(n) = (n/2^A007814(n) - 1)/2 + (n == 2^A007814(n)). - Ralf Stephan, Aug 21 2013
If n is a power of 2, A070939(a(n)) = 1, otherwise A070939(a(n)) = A119387(n-1).
Numbers m for which a(m) = 1 are A000079(m) and A007283(m), a(2^m + 1) = 2^(m-1); m >= 1. - David James Sycamore, Mar 07 2023

A163810 Expansion of (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^6) in powers of x.

Original entry on oeis.org

1, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1

Offset: 0

Michael Somos, Nov 07 2007

			G.f. = 1 - x - x^2 + x^4 + x^5 - x^7 - x^8 + x^10 + x^11 - x^13 - x^14 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-1).

A163806(n) = -a(n) unless n=0. A106510(n) = (-1)^n * a(n).

Convolution inverse of A028310. Series reversion of A109081.

Join[{1},LinearRecurrence[{1, -1},{-1, -1},104]] (* Ray Chandler, Sep 15 2015 *)

{a(n) = (n==0) + [0, -1, -1, 0, 1, 1][n%6 + 1]};

{a(n) = (n==0) + (-1)^n * kronecker(-3, n)};

Euler transform of length 6 sequence [ -1, -1, -1, 0, 0, 1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * u * (1 - u) * (2 - v) - (v - u^2).
a(3*n) = 0 unless n=0. a(6*n + 1) = a(6*n + 2) = -1, a(6*n + 4) = a(6*n + 5) = a(0) = 1.
a(-n) = -a(n) unless n=0. a(n+3) = -a(n) unless n=0 or n=-3.
G.f.: (1 - x)^2 / (1 - x + x^2).

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cp's OEIS Frontend

A199969 a(n) = the greatest non-divisor h of n (1 < h < n), or 0 if no such h exists.

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A220073 Mirror of the triangle A130517.

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A253909 1 together with the positive squares.

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A069775 Permutation of natural numbers induced by the automorphism gma069775! acting on the parenthesizations encoded by A014486.

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A069776 Permutation of natural numbers induced by the automorphism gma069776! acting on the parenthesizations encoded by A014486.

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A199883 Number of distinct values taken by 6th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

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A228273 T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A347708 Number of distinct possible alternating products of odd-length factorizations of n.

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