A053990 -id:A053990

Search: a053990 -id:a053990

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A072389

Numbers of the form x*(x+1) * y*(y+1) ("bipronics") with x and y nonnegative integers.

+10
7

0, 4, 12, 24, 36, 40, 60, 72, 84, 112, 120, 144, 180, 220, 240, 252, 264, 312, 336, 360, 364, 400, 420, 432, 480, 504, 540, 544, 600, 612, 660, 672, 684, 760, 792, 840, 864, 900, 924, 936, 1012, 1080, 1092, 1104, 1120, 1200, 1260, 1300, 1320

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Nonnegative numbers k = a*b = c*d, where a+b = c+d+1. - Yifan Xie, Jun 28 2024

LINKS

John Drake, Table of n, a(n) for n = 1..10000

EXAMPLE

a(3) = 1*2*2*3 = 12.

PROG

(PARI) A003056(n)=(sqrtint(8*n+1)-1)\2

list(lim)=my(v=List([0]), t); for(a=1, A003056(lim\4), t=a*(a+1); for(b=a, A003056(lim\t\2), listput(v, b*(b+1)*t))); Set(v) \\ Charles R Greathouse IV, Jul 11 2024

CROSSREFS

Cf. A053990 (sequence of bipronics with x and y distinct).

Cf. A085780 (one quarter of this).

KEYWORD

nonn,easy

AUTHOR

Stuart M. Ellerstein (ellerstein(AT)aol.com), Jul 20 2002

EXTENSIONS

More terms from James A. Sellers, Jul 23 2002

STATUS

approved

A054731

Numbers of the form x*(x + 1)*y*(y + 1)/4 where x and y are distinct.

+10
5

0, 3, 6, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 190, 198, 210, 216, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 450, 459, 465, 468

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..57.

FORMULA

a(n) = A053990(n) / 4. - Sean A. Irvine, Feb 19 2022

MATHEMATICA

With[{upto=500}, Select[Union[(#[[1]](#[[1]]+1)#[[2]](#[[2]]+1))/4&/@ Subsets[ Range[0, Floor[upto/2]], {2}]], #<=upto&]] (* Harvey P. Dale, Jan 15 2015 *)

CROSSREFS

Cf. A053990, A054734. Contains all triangular numbers >1.

KEYWORD

nonn,easy

AUTHOR

Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 22 2000

EXTENSIONS

More terms from James A. Sellers, Apr 22 2000

STATUS

approved

A054734

Numbers of the form 2*x*(x + 1)*y*(y + 1) where x and y are distinct.

+10
2

0, 24, 48, 80, 120, 144, 168, 224, 240, 288, 360, 440, 480, 504, 528, 624, 672, 720, 728, 840, 864, 960, 1008, 1080, 1088, 1200, 1224, 1320, 1344, 1368, 1520, 1584, 1680, 1728, 1848, 1872, 2024, 2160, 2184, 2208, 2240, 2400, 2520, 2600, 2640, 2808

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..46.

FORMULA

a(n) = 2 * A053990(n). - Sean A. Irvine, Feb 20 2022

CROSSREFS

Cf. A053990, A054731.

KEYWORD

nonn,easy

AUTHOR

Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 22 2000

EXTENSIONS

More terms from James A. Sellers, Apr 22 2000

STATUS

approved

A049207

Array T(m,n) of products of pronic numbers m(m+1) * n(n+1) read by antidiagonals ("bipronics").

+10
1

0, 0, 0, 0, 4, 0, 0, 12, 12, 0, 0, 24, 36, 24, 0, 0, 40, 72, 72, 40, 0, 0, 60, 120, 144, 120, 60, 0, 0, 84, 180, 240, 240, 180, 84, 0, 0, 112, 252, 360, 400, 360, 252, 112, 0, 0, 144, 336, 504, 600, 600, 504, 336, 144, 0, 0, 180, 432, 672, 840, 900, 840, 672, 432, 180, 0, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Table of n, a(n) for n=0..66.

EXAMPLE

Array begins

0 0 0 0 0 0 ...

0 4 12 24 40 ...

0 12 36 72 120 ...

0 24 72 144 240 ...

MAPLE

T := (m, n)->m*(m+1)*n*(n+1): seq(seq(T(q-p, p), p=0..q), q=0..12);

CROSSREFS

Cf. A002378, A072389, A053990.

KEYWORD

nonn,tabl,easy

AUTHOR

Stuart M. Ellerstein (ellerstein(AT)aol.com), Jul 20 2002

EXTENSIONS

Corrected and extended by Emeric Deutsch, Mar 04 2004

STATUS

approved

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