This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A324129 #41 Jan 05 2025 19:51:41 %S A324129 1,1,3,6,13,25,49,91,169,306,551,979,1729,3029,5279,9150,15793,27149, %T A324129 46513,79439,135301,229866,389643,659111,1112833,1875625,3156219, %U A324129 5303286,8898709,14912641,24961201,41734339,69705889,116311074,193898159,322961275 %N A324129 a(n) = n*Fibonacci(n) + ((-1)^n + 1)/2. %C A324129 Equals A324128(n)/2. %C A324129 This sequence is distantly related to (one-half) the number of losing strings using a binary alphabet in the "same game" described by Burns and Purcell (2007) and Kurz (2001). - _Petros Hadjicostas_, Sep 01 2019 %H A324129 Colin Barker, <a href="/A324129/b324129.txt">Table of n, a(n) for n = 0..1000</a> %H A324129 Chris Burns and Benjamin Purcell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/45-3/burns.pdf">Counting the number of winning strings in the 1-dimensional same game</a>, Fibonacci Quarterly, 45(3) (2007), 233-238. %H A324129 Sascha Kurz, <a href="/A035617/a035617.pdf">Polynomials in "same game"</a>, 2001. %H A324129 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-4,-2,2,1). %F A324129 From _Chai Wah Wu_, Feb 20 2019: (Start) %F A324129 a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3) - 2*a(n-4) + 2*a(n-5) + a(n-6) for n > 5. %F A324129 G.f.: (x^5 - x^4 - 2*x^3 + x^2 + x - 1)/((x - 1)*(x + 1)*(x^2 + x - 1)^2). (End) %F A324129 a(n) = A309874(n)/2 + A099920(n-1) = 2^(n-1) - A035615(n)/2 + A099920(n-1) = A323812(n) + A099920(n-1) for n >= 2. [Sequence A309874 counts the losing strings while A035615 counts the winning strings using a binary alphabet in the "same game". See Burns and Purcell (2007) and Kurz (2001).] - _Petros Hadjicostas_, Sep 01 2019 %t A324129 A324129[n_]:=Fibonacci[n]n+((-1)^n+1)/2;Array[A324129,50,0] (* _Paolo Xausa_, Nov 15 2023 *) %o A324129 (PARI) Vec((1 - x - x^2 + 2*x^3 + x^4 - x^5) / ((1 - x)*(1 + x)*(1 - x - x^2)^2) + O(x^40)) \\ _Colin Barker_, Mar 03 2019 %o A324129 (Magma) [n*Fibonacci(n)+((-1)^n+1)/2:n in [0..35]]; // _Marius A. Burtea_, Aug 29 2019 %Y A324129 Cf. A000045, A035615, A099920, A323812, A324128. %K A324129 nonn,easy %O A324129 0,3 %A A324129 _N. J. A. Sloane_, Feb 20 2019