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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.)
The text passage is a detailed description of a sequence in the Online Encyclopedia of Integer Sequences (OEIS), including its definition, formula, examples, and Mathematica code for generating the sequence.
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the passage does not contain any explicit mathematical question or request for a calculation. it is purely descriptive and provides information about a specific integer sequence.
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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”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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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The text passage is a detailed description of a sequence in the Online Encyclopedia of Integer Sequences (OEIS), including its definition, formula, examples, and Mathematica code for generating the sequence.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical question or request for a calculation. It is purely descriptive and provides information about a specific integer sequence.