A229136 Number of solutions to Sum_{i=1..n} x_i^2 == 1 (mod 4) with x_i in 0..3.

2, 8, 24, 64, 192, 768, 3584, 16384, 69632, 278528, 1081344, 4194304, 16515072, 66060288, 266338304, 1073741824, 4311744512, 17246978048, 68853694464, 274877906944, 1098437885952, 4393751543808, 17583596109824, 70368744177664, 281543696187392

Offset

1,1

Comments

Conjecture: a(n) = A131885(n)*2^(n-1) for n >= 1. [Corrected by Petros Hadjicostas, Dec 20 2019]

From Petros Hadjicostas, Dec 20 2019: (Start)

Since this sequence is column k = 1 of A330619, we have a(n) = 4*a(n-1) - 8*a(n-2) + 2^(2*n-3) for n >= 3. (This follows from the theory developed in A330619.) If we let b(n) = a(n)/2^(n-1) for n >= 1, we get b(n) = 2*b(n-1) - 2*b(n-2) + 2^(n-2) for n >= 3.

It follows that 2*b(n-1) = 4*b(n-2) - 4*b(n-3) + 2^(n-2) for n >= 4. Subtracting the last equation from the previous one, we get (after some algebra) b(n) = 4*b(n-1) - 6*b(n-2) + 4*b(n-3) for n >= 4. We can easily verify that b(1) = 2, b(2) = 4, and b(3) = 6, and this proves that b(n) = A131885(n) for n >= 1. This proves the above conjecture. - Petros Hadjicostas, Dec 20 2019

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (8,-24,32).

Formula

G.f.: 1/(1 - 4*x) + Q(0)/(2 - 4*x), where Q(k) = 1 + 1/(1 - 2*x*(k+1)/(2*x*(k+2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 27 2013

G.f.: -2*x*(2*x - 1)^2 / ((4*x - 1)*(8*x^2 - 4*x + 1)). - Colin Barker, Nov 10 2014

a(n) = 4*a(n-1) - 8*a(n-2) + 2^(2*n-3) for n >= 3. - Petros Hadjicostas, Dec 20 2019

Mathematica

a[1] = 2; a[2] = 8; a[3] = 24; a[n_] := a[n-1]*8 + a[n-2]*(-24) + 32*a[n - 3]; Table[a[n], {n, 15}]

Prog

(PARI) Vec(-2*x*(2*x-1)^2/((4*x-1)*(8*x^2-4*x+1)) + O(x^100)) \\ Colin Barker, Nov 10 2014

Crossrefs

Cf. A101990, A131885, A228921, A228920, A229138, A318609, A318610, A330607.

Column k = 1 of A330619.

Sequence in context: A352206 A294458 A333186 * A261452 A225524 A293006

Adjacent sequences: A229133 A229134 A229135 * A229137 A229138 A229139

Keyword

nonn,easy

Author

José María Grau Ribas, Sep 15 2013

Status

approved