A348776 The numbers >= 2 with 3 repeated.

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, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 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

Offset: 1

Author

N. J. A. Sloane, Nov 07 2021, following a suggestion from L. Guyot and T. Y. Lam

Keywords

• nonn
• easy

Comments

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], ...

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

References

• T. Y. Lam, Excursions in Ring Theory, in preparation, 2021. See Section 24.

Links

• F. Grunewald, J. Mennicke, and L. Vaserstein, On the groups SL_2(Z[x]) and SL_2(k[x, y]), Israel J. Math., 86(1-3):157-193, 1994.
• Luc Guyot, The stable rank of Z[x] is 3, arXiv:2111.02965 [math.AC], 2021-2025.
• MathOverflow, Bass' stable range of Z[X].
• L. N. Vaseršteĭn and Andrey Aleksandrovich Suslin, Serre's Problem on Projective Modules over Polynomial Rings, and Algebraic K-theory, Mathematics of the USSR-Izvestiya 10.5 (1976): 937 (Russian version).
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Programs

Python

def A348776(n): return n+int(n<3) # Chai Wah Wu, Aug 09 2022

Formula

a(n) = n for n >= 3.

From Chai Wah Wu, Aug 09 2022: (Start)

a(n) = 2*a(n-1) - a(n-2) for n > 4.

G.f.: x*(x^3 - x^2 - x + 2)/(x - 1)^2. (End)

E.g.f.: x*(2*(1 + exp(x)) + x)/2. - Stefano Spezia, Apr 25 2025