A207260 Triangle read by rows: T(n,k) = k^2 + (1-(-1)^(n-k))/2.

0, 1, 1, 0, 2, 4, 1, 1, 5, 9, 0, 2, 4, 10, 16, 1, 1, 5, 9, 17, 25, 0, 2, 4, 10, 16, 26, 36, 1, 1, 5, 9, 17, 25, 37, 49, 0, 2, 4, 10, 16, 26, 36, 50, 64, 1, 1, 5, 9, 17, 25, 37, 49, 65, 81, 0, 2, 4, 10, 16, 26, 36, 50, 64, 82, 100, 1, 1, 5, 9, 17, 25, 37, 49, 65, 81, 101, 121

Offset

0,5

Comments

Row sums are A171218(n).

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Formula

T(n+k, n) = A002522(n) if k is odd.

T(n+k, n) = n^2 = A000290(n) if k is even.

T(2*n, n) = A137928(n), n>0.

T(2*n+1, n+1) = A080335(n).

T(n,0) = A000035(n).

T(n+1,1) = A000034(n).

T(n+2,2) = A010710(n).

T(n+3,3) = A010735(n).

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A007590(n), A000035(n), A171218(n)

for x = -1, 0, 1 respectively.

G.f.: x*(1 + y - x*y + x*(1 + 2*x)*y^2)/((1 - x^2)*(1 - x*y)^3). - Stefano Spezia, Nov 12 2024

Example

Triangle begins:
  0;
  1, 1;
  0, 2, 4;
  1, 1, 5,  9;
  0, 2, 4, 10, 16;
  1, 1, 5,  9, 17, 25;
  0, 2, 4, 10, 16, 26, 36;
  1, 1, 5,  9, 17, 25, 37, 49;
  0, 2, 4, 10, 16, 26, 36, 50, 64;
  1, 1, 5,  9, 17, 25, 37, 49, 65, 81;
  ...

Mathematica

Table[k^2 + (1-(-1)^(n-k))/2, {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Nov 13 2024 *)

Prog

(Magma) /* As triangle */ [[ k^2 + (1-(-1)^(n-k))/2: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 09 2024

Crossrefs

Cf. A000034, A000035, A000290, A002522, A007590, A010710, A010735, A080335, A171218, A137928.

Sequence in context: A201758 A096110 A383759 * A187913 A329458 A332010

Adjacent sequences: A207257 A207258 A207259 * A207261 A207262 A207263

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Feb 16 2012

Status

approved