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

1,2

Links

Table of n, a(n) for n=1..68.

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.

Example

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

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 *)

Prog

(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

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.

Sequence in context: A298779 A131875 A095871 * A316671 A338157 A189235

Adjacent sequences: A284548 A284549 A284550 * A284552 A284553 A284554

Keyword

nonn,tabl,easy

Author

David Shane, Mar 29 2017

Status

approved