There's a curious article by Roger Penrose in Puzzler's Tribute. Find the missing number in the sequence

7, 9, 12, ?, 24, 36, 56, 90

Answer: 24·ln(2)!

Why? Evaluate the formula f(n) = 24(2ⁿ - 1)/n for n = -3, -2, -1, 1, 2, 3, 4. The value f(0) is missing, but could be found via the l'Hôpital rule!

7 = 7
9 = 3·3
12 = 3·2·2
???
24 = 2·2·2·3
36 = 2·2·3·3
56 = 2·2·2·7
90 = 2·3·3·5

Note that multiplying the 4 numbers below the unknown number by 1, 2, 3, 4 makes all of them multiples of 3 and 8:

24·1 = 2·2·2·3
36·2 = 2·2·2·3·3
56·3 = 2·2·2·3·7
90·4 = 2·2·2·3·3·5

24·1 = 24·1
36·2 = 24·3
56·3 = 24·7
90·4 = 24·15

Numbers 1, 3, 7, 15 have the suspicious look of powers of 2 decreased by 1

24·1 = 24·(2 - 1)
36·2 = 24·(4 - 1)
56·3 = 24·(8 - 1)
90·4 = 24·(16 - 1)

24 = 24·(2 - 1)/1
36 = 24·(4 - 1)/2
56 = 24·(8 - 1)/3
90 = 24·(16 - 1)/4

Does the pattern hold for the numbers above the unknown number?

12 = 24·(1/2 - 1)/(-1)
9 = 24·(1/4 - 1)/(-2)
7 = 24·(1/8 - 1)/(-3)

Yes it does.

a_n = 24·(2^n-3 - 1)/(n - 3), n ¹ 3

Extend the range of function a_n to real numbers x ¹ 3 and calculate the limit for x ® 3 using the l'Hospital rule

a(x) = 24·(2^x-3 - 1)/(x - 3)

lim a(x) = 24·lim{(2^x-3 - 1)/(x - 3)} = 24·lim {(2^x-3 - 1)} / lim (x - 3) =
= 24·lim {ln(2)·2^x-3} / lim 1 = 24·ln(2) » 16.6355

If I really needed the missing number, I would go about it differently. First, I would plot the sequence in Excel. This is not playing smart after the fact, I did that for the bizarre sequence 1, 2, 4, 8, 16, 31, 37, 43, 67, ... as well. Our sequence 7, 9, 12, ?, 24, 36, 56, 90 looks like a smooth function of natural numbers and I would interpolate using a natural spline.

Splines (particulary natural splines) are very useful and for those not familiar with splines I present a few details. A spline S_m(x) of the m-th order with connection points x₁ < x₂ < ... < x_n is a function given in each interval (-¥, x₁), (x₁, x₂), ..., (x_n, ¥) by a polynomial of at most m-th degree and the derivatives of S_m are continuous up to the (m - 1)-th order. A spline of the m-th order can be expressed as

S_m(x) = P_m(x) + Sⁿ_i=1 c_i(x - x_i)₊^m

where P_m(x) is a polynomial of at most m-th degree and the ()₊ operator means "the positive part of". For a given table f(x₁), f(x₂), ..., f(x_n) of function values, the general spline is not unique, which is unpleasant. Uniqueness is guaranteed for natural splines. A natural spline of the (2k + 1)-th order (only odd order natural splines do exist) is a spline that can be expressed as

N_2k+1(x) = P_k(x) + Sⁿ_i=1 c_i(x - x_i)₊^2k+1

where P_k(x) is a polynomial of at most k-th degree. To get the k + 1 coefficients of the polynomial P_k(x) and the n coefficients c₁, c₂, ... c_n for a given table f(x₁), f(x₂), ..., f(x_n) of function values, we have n linear equations

N_2k+1(x_i) = f(x_i), i = 1, 2, ..., n

and k + 1 linear equations

Sⁿ_i=1 c_i^p = 0, p = 0, 1, ..., k

The last k + 1 equations result from limiting the dergee of the polynomial P_2k+1(x) for a general odd order spline to the polynomial P_k(x) for the natural spline.

A natural spline of the first order for a given table of function values is a piecewise linear function inside the table that turns to constants outside the table. Which is the "natural" thing to do. Natural splines of higher orders are a generalization of this concept. Since a spline is a piecewise polynomial function, its integral can be calculated directly. Starting with the 3-rd order, splines have continuous first and second derivatives, which means that common methods of successive approximations can be used to find their roots and extrema.

To find the natural spline of the 3-rd order for the sequence 7, 9, 12, ?, 24, 36, 56, 90 we form a 9´9 matrix (n = 7, k = 1, n + k + 1 = 9) for the above 9 linear equations:

½   0   0   0   0   0   0   0   1   0½ ½c1½     ½ 7½
½   1   0   0   0   0   0   0   1   1½ ½c2½     ½ 9½
½   8   1   0   0   0   0   0   1   2½ ½c3½     ½12½
½  64  27   8   0   0   0   0   1   4½ ½c4½     ½24½
½ 125  64  27   1   0   0   0   1   5½ ½c5½  =  ½36½
½ 216 125  64   8   1   0   0   1   6½ ½c6½     ½56½
½ 343 216 125  27   8   1   0   1   7½ ½c7½     ½90½
½   1   1   1   1   1   1   1   0   0½ ½a0½     ½ 0½
½   0   1   2   4   5   6   7   0   0½ ½a1½     ½ 0½

Calculating the inverse matrix (using the Excel MINVERSE array function) and multiplying it by the right side column, we get the 7 coeeficients c₁, c₂, ..., c₇ of the 3-rd order natural spline N₃(x) and the 2 coefficients a₀, a₁ of the polynomial P₁(x). Calculating the value of the natural spline for x = 3:

N₃(3) » 16.5363

which is pretty close to 24·ln(2) » 16.6355 (0.60% less). Using a natural spline of the 5-th order would yield a slightly better result - 16.5684 (0.40% less than 24·ln(2)). The plots show how well a 3-rd order natural spline approximates a smooth function given by a table of a few function values inside the table.

Blue lines represent the function a(x) = 24·(2^x-3 - 1)/(x - 3) and the red lines the 3-rd order natural spline N₃(x) through the 7 given points.

Regards, Vladimir

By any chance, do you remember how you solved the "bizarre sequence" 1,2,4,8,16,31,37,43,67 ????

You would be of a great help !!!

Thanks in advance !!!

The answer I got is 38. Is it correct? I'm waiting for your reply.