Use Machine Learning/Artificial Intelligence to predict next number (n+1) in a given sequence of random increasing integers
You are solving a problem which is not designed for a ANN, neural network has difficulties when you deal with univariate, few data.

Because they will have to learn a solution based on few examples.

However, a possible solution is achievable:

import numpy as np

# Init sequence
data = [[0, 1.0], [1, 3.0], [2, 7.0], [3, 8.0],
    [4, 21.0], [5, 49.0], [6, 76.0], [7, 224.0]]

X = np.matrix(data)[:, 0]
y = np.matrix(data)[:, 1]

Reg=neural_network.MLPRegressor(solver='lbfgs',random_state=4,hidden_layer_sizes=(100,20,2),learning_rate='adaptive',verbose=True)

y2 = np.ravel(y)
F=Reg.fit(X=X,y=y2,)
F.predict(8)


Note that the random_state parameter has a very large influence when I use a Neural Network. If you move this parameter (integer) you will find that the solution has a really large range.