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
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
Programs
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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
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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
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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.