text: The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
  The OEIS is supported by the many generous donors to the OEIS Foundation.
  
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   A282079 Number of n-element subsets of [n+2] having an even sum. 2
   1, 1, 2, 6, 9, 9, 12, 20, 25, 25, 30, 42, 49, 49, 56, 72, 81, 81, 90, 110, 121, 121, 132, 156, 169, 169, 182, 210, 225, 225, 240, 272, 289, 289, 306, 342, 361, 361, 380, 420, 441, 441, 462, 506, 529, 529, 552, 600, 625, 625, 650, 702, 729, 729, 756, 812, 841 (list; graph; refs; listen; history; text; internal format)
   OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1). FORMULA G.f.: -(x^4-2*x^3+4*x^2-2*x+1)/((x^2+1)^2*(x-1)^3). a(n) = A282011(n+2,n). a(n) = (2*(1+n)*(2+n) - i*(-i)^n*((1+2*i)+(1+i)*n) + i^n*((2+i)+(1+i)*n))/8 where i=sqrt(-1). - Colin Barker, Feb 06 2017 EXAMPLE a(3) = 6: {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}. a(4) = 9: {1,2,3,4}, {1,2,3,6}, {1,2,4,5}, {1,2,5,6}, {1,3,4,6}, {1,4,5,6}, {2,3,4,5}, {2,3,5,6}, {3,4,5,6}. PROG (PARI) Vec(-(x^4-2*x^3+4*x^2-2*x+1) / ((x^2+1)^2*(x-1)^3) + O(x^90)) \\ Colin Barker, Feb 06 2017 CROSSREFS Cf. A282011. Sequence in context: A263178 A198230 A263495 * A121248 A268677 A108370 Adjacent sequences: A282076 A282077 A282078 * A282080 A282081 A282082 KEYWORD nonn,easy AUTHOR Alois P. Heinz, Feb 05 2017 STATUS approved
  
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  Last modified May 22 09:57 EDT 2024. Contains 372745 sequences. (Running on oeis4.)
explanation: The provided text is an entry from the On-Line Encyclopedia of Integer Sequences (OEIS) for sequence A282079. It describes the number of n-element subsets of [n+2] having an even sum. The entry includes the sequence definition, a generating function, a formula, examples, and a PARI program to generate the sequence. However, it does not contain a specific math question or a question and answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
  The OEIS is supported by the many generous donors to the OEIS Foundation.
  
   Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
   A282079 Number of n-element subsets of [n+2] having an even sum. 2
   1, 1, 2, 6, 9, 9, 12, 20, 25, 25, 30, 42, 49, 49, 56, 72, 81, 81, 90, 110, 121, 121, 132, 156, 169, 169, 182, 210, 225, 225, 240, 272, 289, 289, 306, 342, 361, 361, 380, 420, 441, 441, 462, 506, 529, 529, 552, 600, 625, 625, 650, 702, 729, 729, 756, 812, 841 (list; graph; refs; listen; history; text; internal format)
   OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1). FORMULA G.f.: -(x^4-2*x^3+4*x^2-2*x+1)/((x^2+1)^2*(x-1)^3). a(n) = A282011(n+2,n). a(n) = (2*(1+n)*(2+n) - i*(-i)^n*((1+2*i)+(1+i)*n) + i^n*((2+i)+(1+i)*n))/8 where i=sqrt(-1). - Colin Barker, Feb 06 2017 EXAMPLE a(3) = 6: {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}. a(4) = 9: {1,2,3,4}, {1,2,3,6}, {1,2,4,5}, {1,2,5,6}, {1,3,4,6}, {1,4,5,6}, {2,3,4,5}, {2,3,5,6}, {3,4,5,6}. PROG (PARI) Vec(-(x^4-2*x^3+4*x^2-2*x+1) / ((x^2+1)^2*(x-1)^3) + O(x^90)) \\ Colin Barker, Feb 06 2017 CROSSREFS Cf. A282011. Sequence in context: A263178 A198230 A263495 * A121248 A268677 A108370 Adjacent sequences: A282076 A282077 A282078 * A282080 A282081 A282082 KEYWORD nonn,easy AUTHOR Alois P. Heinz, Feb 05 2017 STATUS approved
  
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  Last modified May 22 09:57 EDT 2024. Contains 372745 sequences. (Running on oeis4.)
distilabel_metadata.role: user
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 A282079. It describes the number of n-element subsets of [n+2] having an even sum. The entry includes the sequence definition, a generating function, a formula, examples, and a PARI program to generate the sequence. However, it does not contain a specific math question or a question and answer.
  
  Label: no-question