A281385 Triangular array T(n, k) = n^2 + n*k - k^2.

0, 1, 1, 4, 5, 4, 9, 11, 11, 9, 16, 19, 20, 19, 16, 25, 29, 31, 31, 29, 25, 36, 41, 44, 45, 44, 41, 36, 49, 55, 59, 61, 61, 59, 55, 49, 64, 71, 76, 79, 80, 79, 76, 71, 64, 81, 89, 95, 99, 101, 101, 99, 95, 89, 81, 100, 109, 116, 121, 124, 125, 124, 121, 116, 109, 100

Offset: 0

Author

Michel Marcus, Jan 23 2017

Keywords

• nonn
• tabl

Comments

Let {y0, y1, ...} a sequence satisfying y(m) = y(m-1) + y(m-2), then y(m)^2 - y(m-1)*y(m+1) = T(y0, y1)*(-1)^m. See the Fib. Quart. link.

Examples

			Triangle begins:
   0;
   1,  1;
   4,  5,  4;
   9, 11, 11,  9;
  16, 19, 20, 19, 16;
  25, 29, 31, 31, 29, 25;
  36, 41, 44, 45, 44, 41, 36;
  ...
A000032 begins {2, 1 ...} and satisfies y(m)^2-y(m-1)*y(m+1) = 5*(-1)^m.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
• F. D. Parker, The Very Existence: Problem H-248 and solution, The Fibonacci Quarterly, Vol 15, Nr 1, February 1977.

Crossrefs

Cf. A000032, A000045, A000290.

Programs

Maple

seq(seq(n^2+n*k-k^2, k=0..n),n=0..10); # Robert Israel, Jan 23 2017

Mathematica

Table[n^2+n*k-k^2,{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, May 25 2024 *)

PARI

T(n, k) = n^2 + n*k - k^2;
lista(nn) = for (n=0, nn, for (k=0,n, print1(T(n, k), ", ")); print());

Formula

From Robert Israel, Jan 23 2017: (Start)

G.f. as triangle: (1 + x + y - x*y - 4*x^2*y + x*y^2 - 4*x^2*y^2 + 5*x^3*y^2)*x/((1-x*y)^3*(1-x)^3).

G.f. as sequence: (1-4*x+x^2 + (3-4*x+x^2)*Sum_{k>=0} k*x^(k*(k+1)/2) + (-1+3*x-2*x^2)*Sum_{k>=0} x^(k*(k+1)/2))/(1-x)^3.

-(5*k-1)*T(n,k-1) + (5*k-2)*T(n,k) + (5*k-3)*T(n-1,k-1) - (5*k-4)*T(n-1,k) = 0 for 1 <= k <= n-1.

(End)