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%I #36 Jan 05 2023 22:02:28
%S 1,1,2,1,2,10,1,5,250,106,1138,2,25,146,298,5,17,1,97,253970,2,226,
%T 3034,9148450,2050,10,157,126890,1,14341370,5,110671282,986,7586,530,
%U 130,173,5129602,11068353370,21685,694966754,17883410,5528222698,17,41,11248618,60037,10,242718010,24514292738
%N Twice the coefficient of sqrt(q) in e^h, where e is the fundamental unit and h is the class number of Q(sqrt(q)), q prime and congruent to 1 mod 4. (The coefficient lies in (1/2)Z, so twice it is an integer.)
%C This also arises in the relation satisfied by Euler classes in the connective K-theory of the classifying space of the group of order pq, where p=(q-1)/2. See p. 39 in Bruner and Greenlees, cited below. Take an irreducible representation of the cyclic group of order q which generates the representations as a ring, induce it up to the group of order pq, and let z be its Euler class in ku^{2p}(BG_{pq}). Then z satisfies the relation z^3 -2bq z^2 + qz = 0. This follows from the arithmetic fact that in Q(sort(q)) we have the relation e^h = a + b sqrt(q), as shown on pp. 39-42 of Bruner and Greenlees.
%C This is closely related to the subsequence of A078357 containing those entries such that the corresponding entry in A077426 is prime. However, a(22) = 226 (corresponding to e^3 = 1710 + 113*sqrt(229)) does not occur in A078357, and more such terms appear after this.
%C For the n-th Pythagorean prime q=A002144(n), a(n) is also -1/q of the coefficient of term x in the minimal polynomial of A=Product_{a} 2*sin(a*Pi/q) (where the index runs through all quadratic residues in {1,2,...,q-1}) and B=Product_{b} 2*sin(b*Pi/q) (where the index runs through all quadratic nonresidues in {1,2,...,q-1}). It is easy to show that A*B = p. By the class number formula of real quadratic number fields, one obtains B/A = e^(+-2h), so A+B = sqrt(q)*(e^h+e^(-h)) is exactly q*a(n). - _Zichang Wang_, Dec 15 2022
%D R. R. Bruner and J. P. C. Greenlees, The Connective K-theory of Finite Groups, Memoirs AMS, Vol. 165, No. 785, 2003.
%D T. Mitsuhiro, T. Nakahara and T. Uehara, The Class Number Formula of a Real Quadratic Field and an Estimate of the Value of a Unit, Canadian Mathematical Bulletin, 38(1)(1995), 98-103.
%H Zichang Wang, <a href="/A225432/b225432.txt">Table of n, a(n) for n = 1..2000</a>
%H R. R. Bruner and J. P. C. Greenlees, <a href="https://www.semanticscholar.org/paper/The-Connective-K-Theory-of-Finite-Groups-Bruner-Greenlees/ad7f06aae2f629052f1472f94959ba941a4835c6">The Connective K-theory of Finite Groups</a>, Semantic Scholar.
%H T. Mitsuhiro, T. Nakahara and T. Uehara, <a href="https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/class-number-formula-of-a-real-quadratic-field-and-an-estimate-of-the-value-of-a-unit/693D31C6D873028486E2D1A32ED34970">The Class Number Formula of a Real Quadratic Field and an Estimate of the Value of a Unit</a>.
%t (* e.g., first 270 terms *)
%t Lq = Select[4*Range[1000] + 1, PrimeQ[#] &];
%t Lh = NumberFieldClassNumber[Sqrt[Lq]];
%t Le = NumberFieldFundamentalUnits[Sqrt[Lq]];
%t Transpose[RootReduce[(Le^(2 Lh) + 1)/(Sqrt[Lq] Le^Lh)]][[1]]
%t (* _Zichang Wang_, Dec 15 2022 *)
%o # Magma code to generate all terms for which the prime q is less than or equal to 4N+1 (an initial segment of the sequence). (Note that the brute force computation of the fundamental unit is very inefficient, and will have trouble producing the 39th term.)
%o pr := [4*n+1 : n in [1..N] | IsPrime(4*n+1)];
%o for i in [1..#pr] do
%o q := pr[i];
%o Q<s> := QuadraticField(q);
%o h := ClassNumber(Q);
%o x := 1;
%o while not IsSquare(x*x*q-4) do
%o x := x+1;
%o end while;
%o x := x/2;
%o tr,y := IsSquare(x*x*q-1);
%o e := y + x*s;
%o eh := e^h;
%o b := (eh-Trace(eh)/2)/s;
%o print i,2*b;
%o end for;
%Y Cf. A002144, A077426, A078357.
%K nonn
%O 1,3
%A _Robert R. Bruner_, May 07 2013
%E a(39) onward from _Zichang Wang_, Dec 15 2022
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