A049777 -id:A049777 - cp's OEIS frontend

A160651 a(n) is the number of triangular nonnegative integers that are each equal to n(n+1)/2 - m(m+1)/2, for some m's where 0 <= m <= n.

1, 2, 2, 3, 2, 2, 4, 2, 4, 2, 4, 4, 2, 4, 2, 4, 4, 2, 4, 2, 3, 6, 2, 8, 2, 2, 4, 4, 8, 2, 2, 4, 2, 4, 2, 2, 8, 4, 4, 2, 4, 8, 2, 4, 4, 4, 6, 2, 4, 6, 2, 4, 4, 6, 4, 4, 4, 4, 6, 4, 2, 8, 4, 4, 4, 2, 8, 4, 4, 2, 2, 6, 2, 4, 4, 4, 4, 4, 12, 2, 4, 4, 2, 4, 2, 2, 8, 2, 8, 4, 2, 8, 4, 8, 4, 8, 8, 2, 4, 2, 2, 8, 2, 6, 2

Offset: 0

Leroy Quet, May 21 2009

nonn

			For n = 6, the values of n(n+1)/2 - m(m+1)/2, 0 <= m <= n, are 21, 20, 18, 15, 11, 6, and 0. Of these, 21, 15, 6, and 0 are triangular numbers, so a(6) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000217, A001652, A049777, A049780.

a:= n-> add(`if`(issqr(4*(n+m+1)*(n-m)+1), 1, 0), m=0..n):
seq(a(n), n=0..100);  # Alois P. Heinz, May 27 2018

a(n) = sum(m=0, n, ispolygonal(n*(n+1)/2 - m*(m+1)/2, 3)); \\ Michel Marcus, May 27 2018

a(n) == 1 (mod 2) <=> n in { A001652 }. - Alois P. Heinz, May 27 2018

Extended by Ray Chandler, Jun 16 2009

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.

Original entry on oeis.org

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

David Shane, Mar 29 2017

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

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.

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

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 *)

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

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

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.

A120476 Triangle read by rows: a(n,m)=(2n-1)(n-m)*(n+m+1)/2, where n is the column and m the row index.

Original entry on oeis.org

1, 3, -2, 6, -5, -9, 10, -9, -21, -20, 15, -14, -36, -45, -35, 21, -20, -54, -75, -77, -54, 28, -27, -75, -110, -126, -117, -77, 36, -35, -99, -150, -182, -189, -165, -104, 45, -44, -126, -195, -245, -270, -264, -221, -135, 55, -54, -156, -245, -315, -360, -374, -351, -285, -170

Offset: 0

Roger L. Bagula, Jul 19 2006

Triangular array based on recurrence in Laplace function in J. W. S. Rayleigh.

			1,
3, -2,
6, -5, -9,
10, -9, -21,-20,
15, -14,-36,-45, -35

J. W. S. Rayleigh, The Theory of Sound, volume 2, page 237,Dover, New York,1945

Cf. A006472.

a = Table[Table[(m + 1)*(2*n - 1)*(n - m)*(n + m + 1)/(2*(m + 1)), {n, 0, m - 1}], {m, 1, 10}] Flatten[a]

a(n,m) = (2n-1)*[A000217(n)-A000217(m)] = (1-2n)*A049777(n,m) . - R. J. Mathar, Dec 05 2007

Row sums: sum_{n=0..m-1} a(n,m) = -m(m+1)(3m^2-5m-4)/12. [From R. J. Mathar, Jan 15 2009]

Edited by N. J. A. Sloane, Oct 01 2006

A132169 Irregular triangle read by rows. A141616(n)/4.

Original entry on oeis.org

2, 3, 6, 4, 8, 5, 12, 10, 6, 15, 12, 7, 20, 18, 14, 8, 24, 21, 16, 9, 30, 28, 24, 18, 10, 35, 32, 27, 20, 11, 42, 40, 36, 30, 22, 12, 48, 45, 40, 33, 24, 13, 56, 54, 50, 44, 36, 26, 14, 63, 60, 55, 48, 39, 28, 15, 72, 70, 66, 60, 52, 42, 30, 16

Offset: 0

Paul Curtz, Aug 26 2008

From Paul Curtz, Apr 14 2016: (Start)
Row sums: A023856.
Even rows: A120070.
Odd rows:
2,
6,   4,
12, 10, 6,
etc.
Divided by 2:
1,
3,   2,
6,   5,  3,
10,  9,  7, 4,
15, 14, 12, 9, 5,
etc.
This is A049777. Or positive A049780.
Also A271668 without the first column and bordered by the natural numbers as main diagonal.
(End)

			Irregular triangle:
2,
3,
6,   4,
8,   5,
12, 10, 6,
15, 12, 7,
20, 18, 14,  8,
24, 21, 16,  9,
30, 28, 24, 18, 10,
35, 32, 27, 20, 11,
etc.

Cf. A023856, A049777, A049780, A052938, A120070, A131723, A141616, A168230, A198442, A271668.

(Table[n^2 - k^2, {n, 3, 18}, {k, n}] /. m_ /; Or[OddQ@ m, m == 0] -> Nothing)/4 // Flatten (* Michael De Vlieger, Apr 14 2016 *)

Edited by Charles R Greathouse IV, Nov 11 2009

cp's OEIS Frontend

A160651 a(n) is the number of triangular nonnegative integers that are each equal to n(n+1)/2 - m(m+1)/2, for some m's where 0 <= m <= n.

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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.

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A120476 Triangle read by rows: a(n,m)=(2*n-1)*(n-m)*(n+m+1)/2, where n is the column and m the row index.

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Crossrefs

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Mathematica

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A132169 Irregular triangle read by rows. A141616(n)/4.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A120476 Triangle read by rows: a(n,m)=(2n-1)(n-m)*(n+m+1)/2, where n is the column and m the row index.