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%I #47 Aug 09 2022 09:05:03
%S 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 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,
%U 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 The numbers >= 2 with 3 repeated.
%C 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 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 T. Y. Lam, Excursions in Ring Theory, in preparation, 2021. See Section 24.
%H 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 Luc Guyot, <a href="https://arxiv.org/abs/2111.02965">The stable rank of Z[x] is 3</a>, arXiv:2111.02965 [math.AC], November 2021.
%H MathOverFlow, <a href="https://mathoverflow.net/questions/132839/bass-stable-range-of-mathbf-zx">Bass' stable range of 𝐙[𝑋]</a>
%H 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = n for n >= 3.
%F From _Chai Wah Wu_, Aug 09 2022: (Start)
%F a(n) = 2*a(n-1) - a(n-2) for n > 4.
%F G.f.: x*(x^3 - x^2 - x + 2)/(x - 1)^2. (End)
%o (Python)
%o def A348776(n): return n+int(n<3) # _Chai Wah Wu_, Aug 09 2022
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, Nov 07 2021, following a suggestion from L. Guyot and T. Y. Lam.
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