Markiyan Hirnyk

## 7248 Reputation

12 years, 58 days

## What is the determinant...

Maple

of the matrix

`Matrix(n, (i, j) ->binomial(a*i+b*j, j) );`

? I mean an explicit formula in terms of a, b, and n which consists of n+1 summands. Of course, it should be found with Maple, not by hand.

## How to calculate the area of the set...

Maple 2017
```restart; CodeTools:-Usage(plots:-inequal({abs(abs(x-y)-abs(x+y)) >= 2*y-x+1, (x+1)^2+(y+1)^2 <= 2},
optionsimplicit = [gridrefine = 2, rational = true], x = -3 .. 1, y = -3 .. 0., scaling = constrained));

memory used=0.57GiB, alloc change=184.01MiB, cpu time=16.22s, real time=17.24s, gc time=1.58s
```

?

Mathematica 11.2.0.0 says

```113053/146523-8/169 sqrt(22)+32/289 sqrt(30)-3/169 sqrt(214-24 sqrt(22))-
2/169 sqrt(1177-132 sqrt(22))-1/289 sqrt(94-16 sqrt(30))-
4/289 sqrt(705-120 sqrt(30))+1/146523 293046(Pi)-
arcsin(1/13 sqrt(2) (3+sqrt(22)))-arcsin(1/17 sqrt(2) (1+2 sqrt(30)))```

(see Area.pdf).

## How to solve this nonstandard system?...

Maple 2017

Namely, I mean

```solve({y >= 4*x^4+4*x^2*y+1/2, sqrt((1/2)*(x-y)^2-(x-y)^4) = -2*x^2+y^2}, [x, y]);
[]
```

The answer (no solution) is not correct in view of

```eval({y >= 4*x^4+4*x^2*y+1/2, sqrt((1/2)*(x-y)^2-(x-y)^4) = -2*x^2+y^2}, [x = 0, y = 1/2]);
{(1/8)*sqrt(4) = 1/4, 1/2 <= 1/2}
eval({y >= 4*x^4+4*x^2*y+1/2, sqrt((1/2)*(x-y)^2-(x-y)^4) = -2*x^2+y^2}, [x = -1, y = -3/2]);
{(1/8)*sqrt(4) = 1/4, -3/2 <= -3/2}
```

## Seventy three, fifty six,......

Maple

We have the following sequence of natural numbers
1, 2, 4, 7, 11, 16, 67, 83, 46, 73, 47, 85, 70, 20, 16, 76, 83, 55, 73, 56, 85, 79, 119, 934, 463, 389, 1009, 9028, 8237, 7357, 7567, 7688, 8899, 10021, 12035, 53056, 65071, 17093, 39109, 90232, 23249, 94273, 37291, 19316, 61435, 53461, 16481, 18508, 80629, 92657, 75679, 97708, 80831, 13861, 16885, 58916, 62041, 14083, 38099, 99142, 24259, 95303, 30421, 12466, 66485, 58531, 13651, 15698, 89719, 91867, 76889, 98938, 84061, 16121, 12235, 53296, 69311, 11473, 37489, 98552, 25669, 96733, 33851, 15916, 62035, 53111, 11221, 12298, 89309, 90487, 78499, 99578, 87691, 19771, 17885, 58966, 67081, 18173, 37279, 97372, 27479, 97573, 37681, 18776, 67885, 58981, 19091, 19198, 89299, 99407, 70609, 90718, 81821, 12931, 14035, 53156, 65251, 15373, 37469, 96592, 29689, 98813, 32011, 11146, 64235, 53371, 17461, 16598, 89689, 98827, 73019, 91168, 86251, 15401, 10585, 58636, 63821, 12973, 38059, 95222, 22399, 99463, 36641, 14806, 60985, 59051, 15241, 14398, 89489, 98647, 74839, 93998, 90091, 19162, 26345, 54517, 71701, 10874, 47959, 96133, 33329, 92494, 49591, 19757, 75955, 56122, 22331, 13489, 98599, 99758, 85969, 97129, 92351, 15502, 20725, 52877, 78001, 10264, 46379, 97543, 34759, 95924, 43141, 14317, 71525, 52702, 20911, 12089, 98209, 90478, 87599, 99769, 96991, 20162, 26296, 69457, 75692, 29854, 46090, 9263, 3829, 9484, 5051, 1708, 8275, 5933, 3601, 1270, 929, 1138, 8521, 1469, 9853, 3802, 2297, 8137, 7534, 4574, 4972, 3013, 3323, 3454, 4765, 5897, 8209, 9253, 3755, 5800, 313, 542, 475, 805, 740, 280, 316, 848, 1084, 5038, 8543, 3697, 8203, 3269, 9865, 5932, 2639, 9607, 7315, 5384, 5083, 4054, 4754, 4825, 5536, 6608, 8320, 493, 650, 313, 571, 434, 694, 757, 1019, 9364, 4903, 3359, 9799, 10246, 64469, 96715, 52039, 93296, 69511, 11869, 97085, 58354, 45661, 16931, 14239, 93520, 2819, 9463, 3931, 1676, 7045, 5692, 3251, 1810, 469, 1253, 3811, 1474, 5033, 3598, 9247, 7724, 4573, 4051, 1802, 2380, 1132, ... .
What is the next term? Is it possible to find that with Maple? The Lagrange polynomial is not taken into account. I'd like to recall a great answer by Carl Love to a similar question.  However, the current situation seems to be different. The Predict command of Mma fails here.

## How to get asymptotic solutions?...

Maple
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