OFFSET
0,4
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.
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.
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)
EXAMPLE
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.
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 *)
PROG
(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());
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Michel Marcus, Jan 23 2017
STATUS
approved