A007590 -id:A007590 - cp's OEIS frontend

A274228 Triangle read by rows: T(n,k) (n>=3, 0<=k<=n-3) = number of n-sequences of 0's and 1's with exactly one pair of adjacent 0's and exactly k pairs of adjacent 1's.

2, 3, 2, 4, 4, 2, 5, 8, 5, 2, 6, 12, 12, 6, 2, 7, 18, 21, 16, 7, 2, 8, 24, 36, 32, 20, 8, 2, 9, 32, 54, 60, 45, 24, 9, 2, 10, 40, 80, 100, 90, 60, 28, 10, 2, 11, 50, 110, 160, 165, 126, 77, 32, 11, 2, 12, 60, 150, 240, 280, 252, 168, 96, 36, 12, 2, 13, 72, 195, 350, 455, 448, 364, 216, 117, 40, 13, 2

Offset: 3

Jeremy Dover, Jun 14 2016

			n=3 => 100, 001 -> T(3,0) = 2.
n=4 => 0010, 0100, 1001 -> T(4,0) = 3; 0011, 1100 -> T(4,1) = 2.
Triangle starts:
2,
3, 2,
4, 4, 2,
5, 8, 5, 2,
6, 12, 12, 6, 2,
7, 18, 21, 16, 7, 2,
8, 24, 36, 32, 20, 8, 2,
9, 32, 54, 60, 45, 24, 9, 2,
10, 40, 80, 100, 90, 60, 28, 10, 2,
11, 50, 110, 160, 165, 126, 77, 32, 11, 2,
12, 60, 150, 240, 280, 252, 168, 96, 36, 12, 2,
13, 72, 195, 350, 455, 448, 364, 216, 117, 40, 13, 2,
...

Row sums give A001629.
Cf. A073044.
Columns of table:
  T(n,0)=A000027(n-1)
  T(n,1)=A007590(n-1)
  T(n,2)=A080838(n-1)
  T(n,3)=A032091(n)

Table[(k + 1) (Binomial[Floor[(n + k - 2)/2], k + 1] + Binomial[Floor[(n + k - 3)/2], k + 1]) + 2 Binomial[Floor[(n + k - 3)/2], k], {n, 3, 14}, {k, 0, n - 3}] // Flatten (* Michael De Vlieger, Jun 16 2016 *)

T(n,k) = (k+1)*(binomial((n+k-2)\2,k+1)+binomial((n+k-3)\2,k+1))+2*binomial((n+k-3)\2,k); \\ Michel Marcus, Jun 17 2016

T(n,k) = (k+1)*(binomial(floor((n+k-2)/2),k+1)+binomial(floor((n+k-3)/2),k+1))+2*binomial(floor((n+k-3)/2),k).
T(n,k) = (k+1)*A073044(n-2,k+1) + 2*A046854(n-3,k).
T(n,k) = A274742(n,k)+A274742(n-1,k)+A046854(n-3,k).

A307707 Lexicographically earliest sequence of nonnegative integers in which, for all k >= 0, there are exactly k pairs of consecutive terms whose sum is k.

Original entry on oeis.org

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

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Apr 23 2019

The old definition was "Lexicographically earliest sequence starting with a(1) = 0 such that a(n) is the number of pairs of contiguous terms whose sum is a(n)".
From Paul Curtz, Apr 27 2019: This can be written as a triangle:
                    0
                  1   1
                1   2   1
              2   2   2   2
            2   3   2   3   2
          3   3   3   3   3   3
        3   4   3   4   3   4   3
...

Jean-Marc Falcoz, Table of n, a(n) for n = 1..11326

Cf. A002024.
Cf. also A007590, A057353, A106466 and A238410.
For other versions see A307720 and A378117.

m = 107; a[1]=0;
a24[n_] := Ceiling[(Sqrt[8n+1]-1)/2];
Array[a, m] /. Solve[Table[a[n] + a[n+1] == a24[n], {n, 1, m-1}]][[1]] (* Jean-François Alcover, Jun 02 2019, after Rémy Sigrist's formula *)

v=0; rem=wanted=1; for (n=1, 107, print1 (v", "); v=wanted-v; if (rem--==0, rem=wanted++)) \\ Rémy Sigrist, Apr 23 2019

a(n) + a(n+1) = A002024(n). - Rémy Sigrist, Apr 24 2019

Let t_m = m*(m+1)/2. Write n = t_m - i with m >= 1 and 0 <= i < m. Then a(n) = m/2 if m is even, or if m is odd, a(n) = (m-1)/2 + (i-1 mod 2). - N. J. A. Sloane, Nov 16 2024

Definition clarified by Rémy Sigrist and N. J. A. Sloane, Nov 17 2024

A355509 Peaceable coexisting armies of knights: a(n) is the maximum number m such that m white knights and m black knights can coexist on an n X n chessboard without attacking each other.

Original entry on oeis.org

0, 2, 3, 6, 10, 14, 18, 24, 32, 40, 50, 60, 72, 84, 98, 112, 128, 144, 162, 180, 200, 220, 242, 264, 288, 312, 338, 364, 392, 420, 450, 480, 512, 544, 578, 612, 648, 684, 722, 760, 800, 840, 882, 924, 968, 1012, 1058, 1104, 1152, 1200, 1250, 1300, 1352, 1404

Offset: 1

Aaron Khan, Jul 04 2022

After the first 7 terms, the first differences are terms of A052928: for n >= 8, a(n) - a(n-1) = A052928(n-1).

The increase in differences going from an even n to an odd n, but not from an odd n to an even n, is due to the differing optimal layouts for odd vs. even n values. See example section for a(7) and a(8).

			Examples for n=2 to n=6 have been included as they do not follow the general formula.
.
A solution illustrating a(2)=2:
  +-----+
  | B B |
  | W W |
  +-----+
.
A solution illustrating a(3)=3:
  +-------+
  | . . . |
  | B B W |
  | W W B |
  +-------+
.
A solution illustrating a(4)=6:
  +---------+
  | B B . W |
  | W W . B |
  | B B . W |
  | W W . B |
  +---------+
.
A solution illustrating a(5)=10:
  +-----------+
  | W B W B W |
  | W B W B W |
  | . . . . . |
  | B W B W B |
  | B W B W B |
  +-----------+
.
A solution illustrating a(6)=14:
  +-------------+
  | B B W W B B |
  | W W B B W W |
  | B . . . . B |
  | W . . . . W |
  | B B W W B B |
  | W W B B W W |
  +-------------+
.
Examples for n=7 and n=8 are provided, as while both follow the same formula, the layout for even values of n differs from the layout for odd values of n (related to the fact that, for even values of n, the floor function rounds down a non-integer value).
.
A solution illustrating a(7)=18:
  +---------------+
  | B B B B B B B |
  | B B B B B B B |
  | B . B . B . B |
  | . . . . . . . |
  | W . W . W . W |
  | W W W W W W W |
  | W W W W W W W |
  +---------------+
.
A solution illustrating a(8)=24:
  +-----------------+
  | B B B B B B B B |
  | B B B B B B B B |
  | B B B B B B B B |
  | . . . . . . . . |
  | . . . . . . . . |
  | W W W W W W W W |
  | W W W W W W W W |
  | W W W W W W W W |
  +-----------------+

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A007590, A052928, A176222 (peaceable kings), A250000 (peaceable queens), A002620 (peaceable rooks).

For n > 6, a(n) = floor(((n-1)^2)/2).

G.f.: x^2*(2 - x + 2*x^3 - 2*x^4 - x^5 + 2*x^6 + 2*x^7 - 2*x^8)/((1 - x)^3*(1 + x)). - Stefano Spezia, Jul 05 2022

A101037 Triangle read by rows: T(n,1) = T(n,n) = n and for 1

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 27 2004

For n>1: sum of n-th row = A007590(n+1).

			Triangle begins:
  1;
  2, 2;
  3, 2, 3;
  4, 2, 2, 4;
  5, 3, 2, 3, 5;
  6, 4, 2, 2, 4, 6;
  7, 5, 3, 2, 3, 5, 7;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

T:= proc(n,k) if k < (n+1)/2 then n-2*k+2 elif k=(n+1)/2 then 2 else 2*k-n fi end proc:
T(1,1):= 1:
seq(seq(T(n,k),k=1..n),n=1..20); # Robert Israel, Jan 30 2018

T[n_, 1] := n; T[n_, n_] := n; T[n_, k_] := T[n, k] = Which[k < (n + 1)/2, n - 2*k + 2, k == (n + 1)/2, 2, True, 2*k - n];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 04 2019 *)

From Robert Israel, Jan 30 2018: (Start)
T(n,k) = n - 2*k + 2 if k < (n+1)/2.
T(n,(n+1)/2) = 2 if n>1 is odd.
T(n,k) = 2*k - n if k > (n+1)/2.
G.f. as triangle: x*y*(x^6*y^3-2*x^5*y^3-2*x^5*y^2+x^4*y^3+3*x^4*y^2+x^4*y-3*x^2*y+1)/((1-x^2*y)*(1-x)^2*(1-x*y)^2).
(End)

A131478 a(n) = ceiling(n^4/4).

Original entry on oeis.org

0, 1, 4, 21, 64, 157, 324, 601, 1024, 1641, 2500, 3661, 5184, 7141, 9604, 12657, 16384, 20881, 26244, 32581, 40000, 48621, 58564, 69961, 82944, 97657, 114244, 132861, 153664, 176821, 202500, 230881, 262144, 296481, 334084, 375157, 419904, 468541, 521284

Offset: 0

Mohammad K. Azarian, Jul 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A000982, A004526, A007590, A008619, A036487, A058919.

Cf. A131479.

[Ceiling(n^4/4) : n in [0..50]]; // Vincenzo Librandi, Oct 01 2011

Ceiling[Range[0,40]^4/4] (* Harvey P. Dale, May 17 2019 *)
CoefficientList[Series[(x(x^3 + 6x^2 + 7x + 1)Cosh[x]+ (x^4 + 6x^3 + 7x^2 + x + 3)Sinh[x])/4,{x,0,35}],x]Table[n!,{n,0,35}] (* Stefano Spezia, Feb 19 2023 *)

vector(50, n, n--;ceil(n^4/4)) \\ Michel Marcus, Jun 16 2015

def A131478(n): return n**4+3>>2 # Chai Wah Wu, Jan 30 2023

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: x*(1 + 10*x^2 + x^4)/((1 - x)^5*(1 + x)).
a(n) + a(n+1) = A058919(n+1). (End)
a(n) = floor(n^4/4 + 3/4). - Bruno Berselli, Dec 21 2017
E.g.f.: (x*(x^3 + 6*x^2 + 7*x + 1)*cosh(x) + (x^4 + 6*x^3 + 7*x^2 + x + 3)*sinh(x))/4. - Stefano Spezia, Feb 18 2023

A143183 Triangle T(n,k) = 1 + (2+n)abs(n-2k), read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 4, 4, 9, 1, 9, 16, 6, 6, 16, 25, 13, 1, 13, 25, 36, 22, 8, 8, 22, 36, 49, 33, 17, 1, 17, 33, 49, 64, 46, 28, 10, 10, 28, 46, 64, 81, 61, 41, 21, 1, 21, 41, 61, 81, 100, 78, 56, 34, 12, 12, 34, 56, 78, 100, 121, 97, 73, 49, 25, 1, 25, 49, 73, 97, 121

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 17 2008

			Triangle begins as:
    1;
    4,  4;
    9,  1,  9;
   16,  6,  6, 16;
   25, 13,  1, 13, 25;
   36, 22,  8,  8, 22, 36;
   49, 33, 17,  1, 17, 33, 49;
   64, 46, 28, 10, 10, 28, 46, 64;
   81, 61, 41, 21,  1, 21, 41, 61, 81;
  100, 78, 56, 34, 12, 12, 34, 56, 78, 100;
  121, 97, 73, 49, 25,  1, 25, 49, 73,  97, 121;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000290, A005843, A007590, A143182.

[1+(n+2)*Abs(n-2*k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 23 2024

A143183 := proc(n,k)
        1+(2+n)*abs(n-2*m) ;
end proc: # R. J. Mathar, Jul 12 2012

T[n_, m_]:= 1 + Abs[(n-m+1)^2 - (m+1)^2];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

flatten([[1+(n+2)*abs(n-2*k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 23 2024

T(n, k) = 1 + (2+n)*abs(n-2*k), for 0 <= k <= n.
T(n, k) = T(n, n-k).
Sum_{k=0..n} T(n, k) = (n+2)*A007590(n+1) + n + 1 (row sums).
From G. C. Greubel, Apr 23 2024: (Start)
T(n, 0) = A000290(n+1).
T(2*n-1, n) = A005843(n+1), n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1 + (-1)^n)*((n^2 + 3*n + 3) - (-1)^(n/2)*(n + 2)). (End)

Row sums corrected by R. J. Mathar, Jul 12 2012

A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2k, 1 + 2(n-k), n).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 4, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 6, 5, 3, 1, 1, 3, 5, 7, 7, 5, 3, 1, 1, 3, 5, 7, 8, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 10, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 12, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 13, 11, 9, 7, 5, 3, 1

Offset: 0

Roger L. Bagula, Mar 01 2009

The general form of this, and related triangular sequences, takes the form A(n, k, m) = (m*(n-k) + 1)*A(n-1, k-1, m) + (m*k + 1)*A(n-1, k, m) + m*f(n, k)* A(n-2, k-1, m), where f(n,k) is a polynomial in n and k.

Row sums are: 0, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, ... = A007590(n+1). - N. J. A. Sloane, Aug 27 2009

			Triangle begins as:
  0;
  1, 1;
  1, 2, 1;
  1, 3, 3, 1;
  1, 3, 4, 3, 1;
  1, 3, 5, 5, 3, 1;
  1, 3, 5, 6, 5, 3, 1;
  1, 3, 5, 7, 7, 5, 3, 1;
  1, 3, 5, 7, 8, 7, 5, 3, 1;
  1, 3, 5, 7, 9, 9, 7, 5, 3, 1;
  1, 3, 5, 7, 9, 10, 9, 7, 5, 3, 1;

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A003983, A157457.

T := proc(m,n) return min(1+2*m, 1+2*(n-m), n): end: seq(seq(T(m,n),m=0..n),n=0..14); # Nathaniel Johnston, Apr 29 2011

T[n_, k_]:= Min[1+2*k, 1+2*(n-k), n]; Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten

T(n, k) = min(1 + 2*k, 1 + 2*(n - k), n).
From Yu-Sheng Chang, May 19 2020: (Start)
O.g.f.: F(z,v) = (1+v)*z/((1-v*z-1)*(1-z)*(1-v*z^2)).
T(n,k) = [v^k] (1+v)*(2*v^(n+1)+2-((sqrt(v)-1)^2 * (-1)^n + (sqrt(v)+1)^2) * v^((1/2)*n))/(2*(v-1)^2). (End)

Edited by N. J. A. Sloane, Aug 27 2009

More terms from and partially edited by G. C. Greubel, May 21 2020

A184532 Array, read by rows: T(n,h)=floor[1/{(h+n^3)^(1/3)}], where h=1,2,...,3n^2+3n and {}=fractional part.

Original entry on oeis.org

3, 2, 1, 1, 1, 1, 12, 6, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 13, 9, 7, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 48, 24, 16, 12, 9, 8, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 75, 37, 25, 18, 15, 12, 10, 9, 8, 7, 7, 6, 5, 5, 5, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1

Offset: 1

Clark Kimberling, Jan 16 2011

			First 2 rows:
  3, 2, 1, 1, 1, 1
  12, 6, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Cf. A013942 (analogous array for sqrt(h+n^2)), A184533.

Columns 1 to 6: A033428 (3n^2), A184532=A000290+A007590, A000290 (n^2), A184534, A184535, A080476.

f[n_,h_]:=FractionalPart[(n^3+h)^(1/3)];
g[n_,h_]:=Floor[1/f[n,h]];
Table[Flatten[Table[g[n,h],{n,1,5},{h,1,3n^2+3n}]]]
TableForm[Table[g[n,h],{n,1,5},{h,1,3n^2+3n}]]

T(n,h)=floor[1/{(h+n^3)^(1/3)}], where h=1,2,...,3n^2+3n and {}=fractional part.

A190650 Product of iterated integral part of square root.

Original entry on oeis.org

1, 2, 3, 8, 10, 12, 14, 16, 27, 30, 33, 36, 39, 42, 45, 128, 136, 144, 152, 160, 168, 176, 184, 192, 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, 350, 432, 444, 456, 468, 480, 492, 504, 516, 528, 540, 552, 564, 576, 686, 700, 714, 728, 742, 756, 770, 784, 798, 812, 826, 840, 854, 868, 882, 1024, 1040, 1056, 1072, 1088, 1104, 1120, 1136, 1152, 1168, 1184, 1200, 1216, 1232, 1248, 1264, 1280

Offset: 1

Franklin T. Adams-Watters, May 16 2011

a(n) = n * f(n) * f(f(n)) * ..., where f(n) = floor(sqrt(n)). Although this is written as an infinite product, all but finitely many terms are 1.

			a(1) = 1, a(2) = 2*1, a(3) = 3*1, a(4) = 4*2*1, a(5) = 5*2*1, ....

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000196, A007590, A001146, A190668.

a(n)=local(r);r=n;while((n=sqrtint(n))>1,r*=n);r

a(1) = 1; for n>1, a(n) = n*a(floor(sqrt(n))).

a(n) <= n^2/2 for n > 1. Equality holds for n = 2^2^k.

A192396 Square array T(n, k) = floor(((k+1)^n - (1+(-1)^k)/2)/2) read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 4, 4, 2, 0, 0, 8, 13, 8, 2, 0, 0, 16, 40, 32, 12, 3, 0, 0, 32, 121, 128, 62, 18, 3, 0, 0, 64, 364, 512, 312, 108, 24, 4, 0, 0, 128, 1093, 2048, 1562, 648, 171, 32, 4, 0, 0, 256, 3280, 8192, 7812, 3888, 1200, 256, 40, 5, 0

Offset: 0

Adi Dani, Jun 29 2011

T(n,k) is the number of compositions of odd natural numbers into n parts <=k.

			T(2,4)=12: there are 12 compositions of odd natural numbers into 2 parts <=4
  1: (0,1), (1,0);
  3: (1,2), (2,1), (0,3), (3,0);
  5: (1,4), (4,1), (2,3), (3,2);
  7: (3,4), (4,3).
The table starts
    0,  0,   0,   0,    0,    0, ... A000004;
    0,  1,   1,   2,    2,    3, ... A004526;
    0,  2,   4,   8,   12,   18, ... A007590;
    0,  4,  13,  32,   62,  108, ... A036487;
    0,  8,  40, 128,  312,  648, ... A191903;
    0, 16, 121, 512, 1562, 3888, ... A191902;
    .        .      .       .    ...
with columns: A000004, A000079, A003462, A004171, A128531, A081341, ... .
Antidiagonal triangle begins:
  0;
  0,  0;
  0,  1,   0;
  0,  2,   1,   0;
  0,  4,   4,   2,   0;
  0,  8,  13,   8,   2,   0;
  0, 16,  40,  32,  12,   3,  0;
  0, 32, 121, 128,  62,  18,  3,  0;
  0, 64, 364, 512, 312, 108, 24,  4,  0;

G. C. Greubel, Antidiagonals n = 0..50, flattened
Adi Dani, Restricted compositions of natural numbers

Rows: A000004, A004526, A007590, A036487, A191902, A191902.

Columns: A000004, A000079, A003462, A004171, A081341, A128531.

A192396:= func< n,k | Floor(((k+1)^n - (1+(-1)^k)/2)/2) >;
[A192396(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 11 2023

A192396 := proc(n,k) (k+1)^n-(1+(-1)^k)/2 ; floor(%/2) ; end proc:
seq(seq( A192396(d-k,k),k=0..d),d=0..10) ; # R. J. Mathar, Jun 30 2011

T[n_, k_]:= Floor[((k+1)^n - (1+(-1)^k)/2)/2];
Table[T[n-k,k], {n,0,12}, {k,0,n}]//Flatten

def A192396(n,k): return ((k+1)^n - ((k+1)%2))//2
flatten([[A192396(n-k,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 11 2023

cp's OEIS Frontend

A274228 Triangle read by rows: T(n,k) (n>=3, 0<=k<=n-3) = number of n-sequences of 0's and 1's with exactly one pair of adjacent 0's and exactly k pairs of adjacent 1's.

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A307707 Lexicographically earliest sequence of nonnegative integers in which, for all k >= 0, there are exactly k pairs of consecutive terms whose sum is k.

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A355509 Peaceable coexisting armies of knights: a(n) is the maximum number m such that m white knights and m black knights can coexist on an n X n chessboard without attacking each other.

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A101037 Triangle read by rows: T(n,1) = T(n,n) = n and for 1

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A131478 a(n) = ceiling(n^4/4).

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A143183 Triangle T(n,k) = 1 + (2+n)*abs(n-2*k), read by rows, for 0 <= k <= n.

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A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2*k, 1 + 2*(n-k), n).

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A143183 Triangle T(n,k) = 1 + (2+n)abs(n-2k), read by rows, for 0 <= k <= n.

A157458 Triangle, read by rows, double tent function: T(n,k) = min(1 + 2k, 1 + 2(n-k), n).