A003984 -id:A003984 - cp's OEIS frontend

A004247 Multiplication table read by antidiagonals: T(i,j) = ij (i>=0, j>=0). Alternatively, multiplication triangle read by rows: P(i,j) = j(i-j) (i>=0, 0<=j<=i).

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 0, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30

Offset: 0

David W. Wilson

Table of x*y, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
Or, triangle read by rows, in which row n gives the numbers 0, n*1, (n-1)*2, (n-2)*3, ..., 2*(n-1), 1*n, 0.
Letting T(n,k) be the (k+1)st entry in the (n+1)st row (same numbering used for Pascal's triangle), T(n,k) is the dimension of the space of all k-dimensional subspaces of a (fixed) n-dimensional real vector space. - Paul Boddington, Oct 21 2003
From Dennis P. Walsh, Nov 10 2009: (Start)
Triangle P(n,k), 0<=k<=n, equals n^2 x the variance of a binary data set with k zeros and (n-k) ones. [For the case when n=0, let the variance of the empty set be defined as 0.]
P(n,k) is also the number of ways to form an opposite-sex dance couple from k women and (n-k) men. (End)
P(n,k) is the number of negative products of two numbers from a set of n real numbers, k of which are negative. - Logan Pipes, Jul 08 2021

			As the triangle P, sequence begins:
  0;
  0,0;
  0,1,0;
  0,2,2,0;
  0,3,4,3,0;
  0,4,6,6,4,0,;
  0,5,8,9,8,5,0;
  ...
From _Dennis P. Walsh_, Nov 10 2009: (Start)
P(5,2)=T(2,3)=6 since the variance of the data set <0,0,1,1,1> equals 6/25.
P(5,2)=6 since, with 2 women, say Alice and Betty, and with 3 men, say Charles, Dennis, and Ed, the dance couple is one of the following: {Alice, Charles}, {Alice, Dennis}, {Alice, Ed}, {Betty, Charles}, {Betty, Dennis} and {Betty, Ed}. (End)

T. D. Noe, Rows n = 0..50 of triangle, flattened
Dennis Walsh, Variance bounds on binary data sets

See A003991 for another version with many more comments.

Cf. A002262, A025581, A003056, A004197, A003984, A048720, A325820, A000292 (row sums of triangle), A002620.

seq(seq(k*(n-k),k=0..n),n=0..13); # Dennis P. Walsh, Nov 10 2009

Table[(x - y) y, {x, 0, 13}, {y, 0, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)

T(i,j)=i*j \\ Charles R Greathouse IV, Jun 23 2017

a(n) = A002262(n) * A025581(n). - Antti Karttunen
From Ridouane Oudra, Dec 14 2019: (Start)
a(n) = A004197(n)*A003984(n).
a(n) = (3/4 + n)*t^2 - (1/4)*t^4 - (1/2)*t - n^2 - n, where t = floor(sqrt(2*n+1)+1/2). (End)
P(n,k) = (P(n-1,k-1) + P(n-1,k) + n) / 2. - Robert FERREOL, Jan 16 2020
P(n,floor(n/2)) = A002620(n). - Logan Pipes, Jul 08 2021
From Stefano Spezia, Aug 19 2024: (Start)
G.f. as array: x*y/((1 - x)^2*(1 - y)^2).
E.g.f. as array: exp(x+y)*x*y. (End)

Edited by N. J. A. Sloane, Sep 30 2007

A001859 Triangular numbers plus quarter-squares: n*(n+1)/2 + floor((n+1)^2/4) (i.e., A000217(n) + A002620(n+1)).

Original entry on oeis.org

0, 2, 5, 10, 16, 24, 33, 44, 56, 70, 85, 102, 120, 140, 161, 184, 208, 234, 261, 290, 320, 352, 385, 420, 456, 494, 533, 574, 616, 660, 705, 752, 800, 850, 901, 954, 1008, 1064, 1121, 1180, 1240, 1302, 1365, 1430, 1496, 1564, 1633, 1704, 1776, 1850, 1925

Offset: 0

N. J. A. Sloane

Number of series-reduced planted trees with n+7 nodes and 3 internal nodes.
The trees enumerated with 3 internal nodes are of two types. Those with all internal nodes at different heights are enumerated by the triangular numbers. Those with two internal nodes at the same height are enumerated by the quarter squares. - Michael Somos, May 19 2000
Number of pairs (x,y) with x in {0,...,n}, y even in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012

			For n=1 we find 2 planted trees with 8 nodes, 3 of which are internal (i) and 5 are endpoints (e):
.e...e...e...e....e...e....
...i.......i........i...e..
.......i..............i...e
.......e................i..
........................e..
G.f. = 2*x + 5*x^2 + 10*x^3 + 16*x^4 + 24*x^5 + 33*x^6 + 44*x^7 + 56*x^8 + ...

John Riordan, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.3
S. V. Gervacio and H. Maehara, Partial order on a family of k-subsets of a linearly ordered set, Discr. Math., 306 (2006), 413-419.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. Riordan, Letter to N. J. A. Sloane, Oct. 1970
S. G. Wagner, An identity for the cycle indices of rooted tree automorphism groups, Elec. J. Combinat., 13 (2006), #R00.
Index entries for sequences related to rooted trees.
Index entries for sequences related to trees.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A002378, A002620, A004526, A006578, A045944, A049450.
First differences of A045947.
Antidiagonal sums of array A003984.
Cf. A107661, A077043.
Cf. A185212 (odd terms).

a001859 n = a000217 n + a002620 (n + 1)  -- Reinhard Zumkeller, Dec 20 2012

A001859:=(-1-z^2-2*z^3+z^4)/(z+1)/(z-1)^3; # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence with an additional leading 1
with (combinat):seq(count(Partition((3*n+2)), size=3), n=0..50); # Zerinvary Lajos, Mar 28 2008

With[{nn=60},Total/@Thread[{Accumulate[Range[0,nn]],Floor[Range[ nn+1]^2/4]}]] (* or *) LinearRecurrence[{2,0,-2,1},{0,2,5,10},60] (* Harvey P. Dale, Apr 01 2012 *)

PARI
```
{a(n) = n + (3*n^2 + 1) \ 4};
```

a(n) = A000217(n)+A002620(n+1).
a(n) = n + floor( (3n^2+1)/4 ).
G.f.: (2*x+x^2)/((1-x)^2*(1-x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + 3 = A002378(n) - A002620(n) = A006578(n-1) + A004526(n+1) - Henry Bottomley, Mar 08 2000
a(n) = A006578(-1-n) for all n in Z. - Michael Somos, May 10 2006
From Mitch Harris, Aug 22 2006: (Start)
a(n) = (6n^2 + 8n + 1 - (-1)^n)/8;
a(n) = Sum_{k=0..n} max(k, n-k). (End)
Starting (2, 5, 10, 16, 24, ...), = binomial transform of [2, 3, 2, -1, 2, -4, 8, -16, 32, ...]. - Gary W. Adamson, Nov 30 2007
a(0)=0, a(1)=2, a(2)=5, a(3)=10, a(n) = 2*a(n-1) + 0*a(n-2) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Apr 01 2012
a(n) = 3*n*(n+1)/2 - A006578(n). - Clark Kimberling, Jul 02 2012
a(2*n) = A045944(n), a(2*n - 1) = A049450(n) for all n in Z. - Michael Somos, Nov 03 2014
0 = -6 + a(n) - 2*a(n+2) + a(n+4) for all n in Z. - Michael Somos, Nov 03 2014
0 = a(n)*(+a(n+1) - a(n+2)) + a(n+1)*(-3 - a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Nov 03 2014
a(n) = Sum_{k=1..n} floor((n+k+2)/2). - Wesley Ivan Hurt, Mar 31 2017
Sum_{n>=1} 1/a(n) = 3/4 - Pi/(4*sqrt(3)) + 3*log(3)/4. - Amiram Eldar, May 28 2022
E.g.f.: (x*(7 + 3*x)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Aug 22 2023

Entry improved by Michael Somos

A051125 Table T(n,k) = max{n,k} read by antidiagonals (n >= 1, k >= 1).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Antidiagonal sums = A006578. - Reinhard Zumkeller, Nov 17 2011

			Table begins
  1, 2, 3, 4, 5, ...
  2, 2, 3, 4, 5, ...
  3, 3, 3, 4, 5, ...
  4, 4, 4, 4, 5, ...
  ...

Peter Kagey, Antidiagonals n = 1..126 of triangle, flattened

Cf. A003056, A003983, A003984, A004197.

Equals A003984(n) + 1.

Flat(List([1..15], n-> List([1..n], k-> Maximum(n-k+1,k) ))); # G. C. Greubel, Jul 23 2019

[Max(n-k+1,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 23 2019

seq(seq(max(r,d+1-r),r=1..d),d=1..15); # Robert Israel, Jul 22 2016

Flatten[Table[Max[n-k+1, k], {n, 13}, {k, n, 1, -1}]] (* Alonso del Arte, Nov 17 2011 *)

T(n,k) = max(n,k) \\ Charles R Greathouse IV, Feb 07 2017

[[max(n-k+1,k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 23 2019

From Robert Israel, Jul 22 2016: (Start)
G.f. as table: G(x,y) = x*y*(1-3*x*y+x*y^2+x^2*y)/((1-x*y)*(1-x)^2*(1-y)^2).
G.f. flattened: (1-x)^(-2)*(x^2 + Sum_{j >= 0} x^(2*j^2) *(x+x^2 -2*x^(j+2)-2*x^(-j+2)+2*x^(2*j+2))). (End)

More terms from Robert Lozyniak

A344838 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = max(n * 2^max(0, w(k)-w(n)), k * 2^max(0, w(n)-w(k))) (where w = A070939).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, May 29 2021

In other words, we right pad the binary expansion of the lesser of n and k with zeros (provided it is positive) so that both numbers have the same number of binary digits, and then take the greatest value.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+----------------------------------------------------------------
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|   1   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    2|   2   2   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    3|   3   3   3   3   6   6   6   7  12  12  12  12  12  13  14  15
    4|   4   4   4   6   4   5   6   7   8   9  10  11  12  13  14  15
    5|   5   5   5   6   5   5   6   7  10  10  10  11  12  13  14  15
    6|   6   6   6   6   6   6   6   7  12  12  12  12  12  13  14  15
    7|   7   7   7   7   7   7   7   7  14  14  14  14  14  14  14  15
    8|   8   8   8  12   8  10  12  14   8   9  10  11  12  13  14  15
    9|   9   9   9  12   9  10  12  14   9   9  10  11  12  13  14  15
   10|  10  10  10  12  10  10  12  14  10  10  10  11  12  13  14  15
   11|  11  11  11  12  11  11  12  14  11  11  11  11  12  13  14  15
   12|  12  12  12  12  12  12  12  14  12  12  12  12  12  13  14  15
   13|  13  13  13  13  13  13  13  14  13  13  13  13  13  13  14  15
   14|  14  14  14  14  14  14  14  14  14  14  14  14  14  14  14  15
   15|  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10

Cf. A003984, A070939.

Cf. A344834 (AND), A344835 (OR), A344836 (XOR), A344837 (min), A344839 (absolute difference).

T(n,k,op=max,w=m->#binary(m)) = { op(n*2^max(0, w(k)-w(n)), k*2^max(0, w(n)-w(k))) }

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = n.
T(n, 1) = max(1, n).

A377930 Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) = max(A007814(n), A007814(k)).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Nov 11 2024

Let K_0 = [0], and for any m > 0, K_m is obtained by arranging four copies of K_{m-1} around a "plus" shape made of m's as follows:
                                 +---------+---+---------+
                                 |         | m |         |
                                 |         |   |         |
                                 | K_{m-1} | . | K_{m-1} |
                                 |         | . |         |
                                 |         | . |         |
         +---+                   +---------+   +---------+
   K_0 = | 0 |, for m > 0, K_m = |m   ...    m    ...   m|
         +---+                   +---------+   +---------+
                                 |         | . |         |
                                 |         | . |         |
                                 | K_{m-1} | . | K_{m-1} |
                                 |         |   |         |
                                 |         | m |         |
                                 +---------+---+---------+
The square array A is the limit of K_m as m tends to infinity.

			Array A(n, k) begins:
    +---+---+---+---+---+---+---+
    | 0 | 1 | 0 | 2 | 0 | 1 | 0 |
    +---+   +---+   +---+   +---+
    | 1   1   1 | 2 | 1   1   1 |
    +---+   +---+   +---+   +---+
    | 0 | 1 | 0 | 2 | 0 | 1 | 0 |
    +---+---+---+   +---+---+---+
    | 2   2   2   2   2   2   2 |
    +---+---+---+   +---+---+---+
    | 0 | 1 | 0 | 2 | 0 | 1 | 0 |
    +---+   +---+   +---+   +---+
    | 1   1   1 | 2 | 1   1   1 |
    +---+   +---+   +---+   +---+
    | 0 | 1 | 0 | 2 | 0 | 1 | 0 |
    +---+---+---+---+---+---+---+

Cf. A003984, A007814, A053398.

A[n_,k_]:=Max[IntegerExponent[n,2],IntegerExponent[k,2]]; Table[A[n-k+1,k],{n,13},{k,n}]//Flatten (* Stefano Spezia, Nov 13 2024 *)

A(n, k) = max(valuation(n, 2), valuation(k, 2))

A(n, k) = A(k, n).

A(n, 0) = A(n, n) = A007814(n).

cp's OEIS Frontend

A004247 Multiplication table read by antidiagonals: T(i,j) = i*j (i>=0, j>=0). Alternatively, multiplication triangle read by rows: P(i,j) = j*(i-j) (i>=0, 0<=j<=i).

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A001859 Triangular numbers plus quarter-squares: n*(n+1)/2 + floor((n+1)^2/4) (i.e., A000217(n) + A002620(n+1)).

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A051125 Table T(n,k) = max{n,k} read by antidiagonals (n >= 1, k >= 1).

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A344838 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = max(n * 2^max(0, w(k)-w(n)), k * 2^max(0, w(n)-w(k))) (where w = A070939).

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A377930 Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) = max(A007814(n), A007814(k)).

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A004247 Multiplication table read by antidiagonals: T(i,j) = ij (i>=0, j>=0). Alternatively, multiplication triangle read by rows: P(i,j) = j(i-j) (i>=0, 0<=j<=i).