A284551 Triangular array read by rows, demonstrating that the difference between a pentagonal number (left edge of triangle) and a square (right edge) is a triangular number.

1, 5, 4, 12, 11, 9, 22, 21, 19, 16, 35, 34, 32, 29, 25, 51, 50, 48, 45, 41, 36, 70, 69, 67, 64, 60, 55, 49, 92, 91, 89, 86, 82, 77, 71, 64, 117, 116, 114, 111, 107, 102, 96, 89, 81, 145, 144, 142, 139, 135, 130, 124, 117, 109, 100, 176, 175, 173, 170, 166, 161, 155, 148, 140, 131, 121, 210, 209

Offset: 1

Author

David Shane, Mar 29 2017

Keywords

• nonn
• tabl
• easy

Examples

			Rows: {1}; {5,4}; {12,11,9}; ...
Triangle begins:
               1
            5     4
        12    11     9
     22    21    19    16
  35    34    32    29    25
		

Crossrefs

Cf. A049777, A049780, which have a similar layout based on subtracting triangular numbers of increasing value from the leftmost element of the row.

A051662 gives row sums.

Programs

Maple

A284551 := proc(n,m)
    n*(3*n-1)-m*(m-1) ;
    %/2 ;
end proc:
seq(seq(A284551(n,m),m=1..n),n=1..15) ; # R. J. Mathar, Mar 30 2017

Mathematica

T[n_,m_]:= Floor[n(3n - 1) - m(m - 1)]/2; Table[T[n, k], {n, 12}, {k, n}] // Flatten (* Indranil Ghosh, Mar 30 2017 *)

PARI

T(n,m) = floor(n*(3*n - 1) - m*(m - 1))/2;
for(n=1, 12, for(k=1, n, print1(T(n,k),", ");); print();); \\ Indranil Ghosh, Mar 30 2017

Python

def T(n, m): return (n*(3*n - 1) - m*(m - 1))//2
for n in range(1, 13):
    print([T(n,k) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 30 2017

Formula

P(m,n) = (m(3m-1) - n(n-1))/2. Alternatively, P(n) - T(n-1) = S(n) where P(n) is a pentagonal number, T(n-1) is a triangular number, and S(n) is a square number.