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   A266805 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(6),1,1,...], where 1^n means n ones. 5
   -14, -90, -722, -4830, -33554, -228954, -1572110, -10768122, -73825010, -505954014, -3467991794, -23769625530, -162920337422, -1116670248090, -7653777913874, -52459758093534, -359564573392850, -2464492138756122, -16891880703949070, -115778671987640634 (list; graph; refs; listen; history; text; internal format)
   OFFSET 0,1 COMMENTS See A265762 for a guide to related sequences. LINKS Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1). FORMULA a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) . G.f.:  -((2 (-7 - 10 x - 31 x^2 - 40 x^3 + 3 x^4))/(-1 + 5 x + 15 x^2 - 15 x^3 - 5 x^4 + x^5)). EXAMPLE Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction: [sqrt(6),1,1,1,...] has p(0,x) = 19-14x-13x^2+2x^3+x^4, so a(0) = -14; [1,sqrt(6),1,1,1,...] has p(1,x) = 19-90x+143x^2-90x^3+19x^4, so a(1) = -90; [1,1,sqrt(6),1,1,1...] has p(2,x) = 361-722x+527x^2-166x^3+19x^4, so a(2) = -722. MATHEMATICA u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[6]}, {{1}}]; f[n_] := FromContinuedFraction[t[n]]; t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}]; Coefficient[t, x, 0] ; (* A266804 *) Coefficient[t, x, 1];  (* A266805 *) Coefficient[t, x, 2];  (* A266806 *) Coefficient[t, x, 3];  (* A266807 *) Coefficient[t, x, 4];  (* A266804 *) CROSSREFS Cf. A265762, A266804, A266806, A266807. Sequence in context: A114242 A054487 A200191 * A301380 A047639 A202291 Adjacent sequences:  A266802 A266803 A266804 * A266806 A266807 A266808 KEYWORD easy,sign AUTHOR Clark Kimberling, Jan 09 2016 STATUS approved
  
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  Last modified December 1 22:12 EST 2021. Contains 349435 sequences. (Running on oeis4.)
explanation: The provided text is an entry from the On-Line Encyclopedia of Integer Sequences (OEIS) for sequence A266805. It describes the coefficients of x in the minimal polynomial of a specific continued fraction. The entry includes the sequence definition, initial terms, a formula, generating function, examples, and Mathematica code for generating the sequence. It does not contain a specific math question or a question and answer.
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distilabel_metadata.content: The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
  
  Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).
  
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   A266805 Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(6),1,1,...], where 1^n means n ones. 5
   -14, -90, -722, -4830, -33554, -228954, -1572110, -10768122, -73825010, -505954014, -3467991794, -23769625530, -162920337422, -1116670248090, -7653777913874, -52459758093534, -359564573392850, -2464492138756122, -16891880703949070, -115778671987640634 (list; graph; refs; listen; history; text; internal format)
   OFFSET 0,1 COMMENTS See A265762 for a guide to related sequences. LINKS Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1). FORMULA a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5) . G.f.:  -((2 (-7 - 10 x - 31 x^2 - 40 x^3 + 3 x^4))/(-1 + 5 x + 15 x^2 - 15 x^3 - 5 x^4 + x^5)). EXAMPLE Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction: [sqrt(6),1,1,1,...] has p(0,x) = 19-14x-13x^2+2x^3+x^4, so a(0) = -14; [1,sqrt(6),1,1,1,...] has p(1,x) = 19-90x+143x^2-90x^3+19x^4, so a(1) = -90; [1,1,sqrt(6),1,1,1...] has p(2,x) = 361-722x+527x^2-166x^3+19x^4, so a(2) = -722. MATHEMATICA u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[6]}, {{1}}]; f[n_] := FromContinuedFraction[t[n]]; t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}]; Coefficient[t, x, 0] ; (* A266804 *) Coefficient[t, x, 1];  (* A266805 *) Coefficient[t, x, 2];  (* A266806 *) Coefficient[t, x, 3];  (* A266807 *) Coefficient[t, x, 4];  (* A266804 *) CROSSREFS Cf. A265762, A266804, A266806, A266807. Sequence in context: A114242 A054487 A200191 * A301380 A047639 A202291 Adjacent sequences:  A266802 A266803 A266804 * A266806 A266807 A266808 KEYWORD easy,sign AUTHOR Clark Kimberling, Jan 09 2016 STATUS approved
  
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  Last modified December 1 22:12 EST 2021. Contains 349435 sequences. (Running on oeis4.)
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distilabel_metadata.raw_output_q_a_detector_0: The provided text is an entry from the On-Line Encyclopedia of Integer Sequences (OEIS) for sequence A266805. It describes the coefficients of x in the minimal polynomial of a specific continued fraction. The entry includes the sequence definition, initial terms, a formula, generating function, examples, and Mathematica code for generating the sequence. It does not contain a specific math question or a question and answer.
  
  Label: no-question