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 A348776 #58 Jul 09 2025 04:57:22 %S A348776 2,3,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26, %T A348776 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49, %U A348776 50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83 %N A348776 The numbers >= 2 with 3 repeated. %C A348776 This sequence, 2, 3, 3, 4, 5, 6, 7, ..., gives the stable range of the polynomial rings Z, Z[x_1], Z[x_1, x_2], Z[x_1, x_2, x_3], ... %C A348776 A note on terminology: "stable range" and "stable rank" are the same thing. In the English-speaking world, people have always used the term "stable range", which was what Bass had invented in the early '60s. When Russian workers wrote on this theme, of course they used a Russian translation of the term "stable range". When the term was translated back into English, it became "stable rank"! - T. Y. Lam, Nov 07 2021 %D A348776 T. Y. Lam, Excursions in Ring Theory, in preparation, 2021. See Section 24. %H A348776 F. Grunewald, J. Mennicke, and L. Vaserstein, <a href="https://doi.org/10.1007/BF02773676">On the groups SL_2(Z[x]) and SL_2(k[x, y])</a>, Israel J. Math., 86(1-3):157-193, 1994. %H A348776 Luc Guyot, <a href="https://arxiv.org/abs/2111.02965">The stable rank of Z[x] is 3</a>, arXiv:2111.02965 [math.AC], 2021-2025. %H A348776 MathOverflow, <a href="https://mathoverflow.net/questions/132839/bass-stable-range-of-mathbf-zx">Bass' stable range of Z[X]</a>. %H A348776 L. N. Vaseršteĭn and Andrey Aleksandrovich Suslin, <a href="https://iopscience.iop.org/article/10.1070/IM1976v010n05ABEH001822">Serre's Problem on Projective Modules over Polynomial Rings, and Algebraic K-theory</a>, Mathematics of the USSR-Izvestiya 10.5 (1976): 937 (<a href="http://mi.mathnet.ru/eng/izv2227">Russian version</a>). %H A348776 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1). %F A348776 a(n) = n for n >= 3. %F A348776 From _Chai Wah Wu_, Aug 09 2022: (Start) %F A348776 a(n) = 2*a(n-1) - a(n-2) for n > 4. %F A348776 G.f.: x*(x^3 - x^2 - x + 2)/(x - 1)^2. (End) %F A348776 E.g.f.: x*(2*(1 + exp(x)) + x)/2. - _Stefano Spezia_, Apr 25 2025 %o A348776 (Python) %o A348776 def A348776(n): return n+int(n<3) # _Chai Wah Wu_, Aug 09 2022 %K A348776 nonn,easy %O A348776 1,1 %A A348776 _N. J. A. Sloane_, Nov 07 2021, following a suggestion from L. Guyot and T. Y. Lam