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 A297384 Number of Eulerian cycles in the n-antiprism graph. 1
 4, 44, 372, 2932, 22484, 170196, 1279828, 9590612, 71736660, 536055124, 4003591508, 29892900180, 223162389844, 1665861735764, 12434781197652, 92816950121812, 692805066118484, 5171207088198996, 38598573880071508, 288104312443589972, 2150442403051689300 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Sequence extrapolated to n=1 and n=2 using the recurrence. - Andrew Howroyd, Jan 11 2018 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 Eric Weisstein's World of Mathematics, Antiprism Graph Eric Weisstein's World of Mathematics, Eulerian Cycle Index entries for linear recurrences with constant coefficients, signature (13, -48, 52, -16). FORMULA From Andrew Howroyd, Jan 11 2018: (Start) a(n) = 13*a(n-1) - 48*a(n-2) + 52*a(n-3) - 16*a(n-4). G.f.: 4*x*(1 - 2*x - 2*x^2)/((1 - 8*x + 4*x^2)*(1 - 4*x)*(1 - x)). (End) From Eric W. Weisstein, Jan 12 2018: (Start) a(n) = 2^n*((2 - sqrt(3))^n + (2 + sqrt(3))^n) - 2/3*(2 + 4^n). a(n) = 2^n*lucasl(2*n, sqrt(2)) - 2/3*(2 + 4^n). a(n) = A003500(n) - A039301(n-1). (End) MATHEMATICA Table[2^n ((2 - Sqrt[3])^n + (2 + Sqrt[3])^n) - 2/3 (2 + 4^n), {n, 20}] // Expand Table[2^n LucasL[2 n, Sqrt[2]] - 2/3 (2 + 4^n), {n, 20}] // Round LinearRecurrence[{13, -48, 52, -16}, {4, 44, 372, 2932}, 20] CoefficientList[Series[-4 (-1 + 2 x + 2 x^2)/(1 - 13 x + 48 x^2 - 52 x^3 + 16 x^4), {x, 0, 20}], x] PROG (PARI) Vec(4*(1 - 2*x - 2*x^2)/((1 - 8*x + 4*x^2)*(1 - 4*x)*(1 - x)) + O(x^30)) \\ Andrew Howroyd, Jan 11 2018 CROSSREFS Sequence in context: A035014 A259989 A030987 * A178294 A043039 A198962 Adjacent sequences:  A297381 A297382 A297383 * A297385 A297386 A297387 KEYWORD nonn AUTHOR Eric W. Weisstein, Dec 29 2017 EXTENSIONS a(1)-a(2) and terms a(9) and beyond from Andrew Howroyd, Jan 11 2018 STATUS approved

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Last modified October 23 22:42 EDT 2019. Contains 328378 sequences. (Running on oeis4.)
The text passage is a detailed entry from the Online Encyclopedia of Integer Sequences (OEIS) about the number of Eulerian cycles in n-antiprism graphs. It includes the sequence definition, initial terms, formulas, Mathematica and PARI code, and references.
no_question

the passage does not contain any explicit mathematical questions or word problems. it is purely a reference entry providing information about a specific integer sequence. while it includes mathematical formulas and code, these are presented as facts rather than questions to be solved.
Given a text passage, determine:
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    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

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TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
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- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A297384 Number of Eulerian cycles in the n-antiprism graph. 1
 4, 44, 372, 2932, 22484, 170196, 1279828, 9590612, 71736660, 536055124, 4003591508, 29892900180, 223162389844, 1665861735764, 12434781197652, 92816950121812, 692805066118484, 5171207088198996, 38598573880071508, 288104312443589972, 2150442403051689300 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Sequence extrapolated to n=1 and n=2 using the recurrence. - Andrew Howroyd, Jan 11 2018 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 Eric Weisstein's World of Mathematics, Antiprism Graph Eric Weisstein's World of Mathematics, Eulerian Cycle Index entries for linear recurrences with constant coefficients, signature (13, -48, 52, -16). FORMULA From Andrew Howroyd, Jan 11 2018: (Start) a(n) = 13*a(n-1) - 48*a(n-2) + 52*a(n-3) - 16*a(n-4). G.f.: 4*x*(1 - 2*x - 2*x^2)/((1 - 8*x + 4*x^2)*(1 - 4*x)*(1 - x)). (End) From Eric W. Weisstein, Jan 12 2018: (Start) a(n) = 2^n*((2 - sqrt(3))^n + (2 + sqrt(3))^n) - 2/3*(2 + 4^n). a(n) = 2^n*lucasl(2*n, sqrt(2)) - 2/3*(2 + 4^n). a(n) = A003500(n) - A039301(n-1). (End) MATHEMATICA Table[2^n ((2 - Sqrt[3])^n + (2 + Sqrt[3])^n) - 2/3 (2 + 4^n), {n, 20}] // Expand Table[2^n LucasL[2 n, Sqrt[2]] - 2/3 (2 + 4^n), {n, 20}] // Round LinearRecurrence[{13, -48, 52, -16}, {4, 44, 372, 2932}, 20] CoefficientList[Series[-4 (-1 + 2 x + 2 x^2)/(1 - 13 x + 48 x^2 - 52 x^3 + 16 x^4), {x, 0, 20}], x] PROG (PARI) Vec(4*(1 - 2*x - 2*x^2)/((1 - 8*x + 4*x^2)*(1 - 4*x)*(1 - x)) + O(x^30)) \\ Andrew Howroyd, Jan 11 2018 CROSSREFS Sequence in context: A035014 A259989 A030987 * A178294 A043039 A198962 Adjacent sequences:  A297381 A297382 A297383 * A297385 A297386 A297387 KEYWORD nonn AUTHOR Eric W. Weisstein, Dec 29 2017 EXTENSIONS a(1)-a(2) and terms a(9) and beyond from Andrew Howroyd, Jan 11 2018 STATUS approved

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Last modified October 23 22:42 EDT 2019. Contains 328378 sequences. (Running on oeis4.)
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The text passage is a detailed entry from the Online Encyclopedia of Integer Sequences (OEIS) about the number of Eulerian cycles in n-antiprism graphs. It includes the sequence definition, initial terms, formulas, Mathematica and PARI code, and references.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or word problems. It is purely a reference entry providing information about a specific integer sequence. While it includes mathematical formulas and code, these are presented as facts rather than questions to be solved.