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 A229136 Number of solutions to Sum_{i=1..n} x_i^2 == 1 (mod 4) with x_i in 0..3. 8
 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 (list; graph; refs; listen; history; text; internal format)
 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: A036289 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

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Last modified September 20 16:50 EDT 2021. Contains 347586 sequences. (Running on oeis4.)