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User: Arturo Magidin

Arturo Magidin has authored 2 sequences.

A213506 Number of nonisomorphic 2-generator p-groups of class at most 2 and order p^n.

1, 1, 2, 4, 6, 8, 13, 17, 23, 31, 40, 50, 65, 79, 97, 119, 143, 169, 203, 237, 277, 323, 373, 427, 492, 558, 633, 717, 807, 903, 1014, 1128, 1254, 1392, 1539, 1695, 1870, 2050, 2246, 2458, 2682, 2918, 3178, 3446, 3734, 4042, 4366, 4706, 5075, 5455, 5860

Offset: 0

Arturo Magidin, Jun 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Ahmad, A. Magidin and R. F. Morse, Two generator p-groups of nilpotency class 2 and their conjugacy classes, Publ. Math. Debrecen 81 (2012), no. 1-2, 145-166.
C. Voll, Enumerating finite class-2-nilpotent groups on 2 generators, C. R. Math. Acad. Sci. Paris 347 (2009), no. 23-24, 1347-1350.

A213506 := proc(n)
        a := 0 ;
        for t from 0 to n do
                for s from t to n-t do
                        r := n-s-t ;
                        if r >= s then
                                m := min(t,r-s) ;
                                a := a+t+1+m*(2*t+1-m)/2 ;
                        end if;
                end do:
        end do:
        return a;
end proc:
seq(A213506(n),n=0..70) ; # R. J. Mathar, Jun 26 2012

CoefficientList[Series[1/((1 - x)*(1 - x^2)*(1 - x^3)^2*(1 - x^4)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 28 2014 *)

a(n)=floor((n^4+26*n^3+234*n^2+(909-64*(n%3))*n+1701)/1728+(n+1)*(-1)^n/64) \\ Tani Akinari, Jun 28 2014

Vec( 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)) + O(x^100)) \\ Michel Marcus, Jun 28 2014

a(n) = Sum_{r+s+t=n, r >= s >= t >= 0}( (t+1)+(1/2)*min{t,r-s}*(2*t+1-min{t,r-s}) ).
G.f.: 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)). - Tani Akinari, Jun 28 2014
a(n) = floor( (n^4+26*n^3+234*n^2+(909-64*(n mod 3))*n+1701)/1728+(n+1)*(-1)^n/64 ). [Tani Akinari, Jun 28 2014 - see PARI code]

A111138 Let b(n) denote the number of nontriangular numbers less than or equal to n. Then a(n) = b(n-1) + a(b(n-1)), with a(1) = a(2) = 0, a(3) = 1.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 4, 5, 7, 10, 10, 11, 13, 16, 20, 20, 21, 23, 26, 30, 35, 35, 36, 38, 41, 45, 50, 56, 56, 57, 59, 62, 66, 71, 77, 84, 84, 85, 87, 90, 94, 99, 105, 112, 120, 120, 121, 123, 126, 130, 135, 141, 148, 156, 165, 165, 166, 168, 171, 175, 180, 186, 193, 201

Offset: 1

Arturo Magidin, Oct 17 2005; definition corrected Feb 01 2006

For a subgroup H of order p^n (p an odd prime) of the subgroup generated by all commutators [x_j,x_i] in the relatively free group F of class three and exponent p, freely generated by x_1, x_2,..., x_k, (k sufficiently large) the minimum size of the subgroup of [H,F] of F_3 is p^{kn - a(n)}.
The sequence arises when finding a purely numerical sufficient condition for the capability of p-groups of class two and exponent p, where p is an odd prime.
Partial sums of A002262. - Gionata Neri, Sep 04 2015

			a(31) = b(30) + a(b(30)) = 23 + a(23) = 23 + b(22) + a(b(22)) = 23 + 16 + a(16) = 39 + b(15) + a(b(15)) = 39 + 10 + a(10) = 49 + b(9) + a(b(9)) = 49 + 6 + a(6) = 55 + b(5) + a(b(5)) = 55 + 3 + a(3) = 58 + 1 = 59.

Arturo Magidin, Capable groups of prime exponent and class two II, arXiv:math/0506578 [math.GR], 2005.

Cf. A083920.

a[1] = a[2] = 0; a[3] = 1; a[n_] := a[n] = b[n - 1] + a[b[n - 1]]; b[n_] := n - Floor[(Sqrt[8n + 1] - 1)/2]; Array[a, 64] (* Robert G. Wilson v, Feb 01 2006 *)

a(n) = my(r,m=sqrtint(n<<1,&r)); if(r>1,2); \\ Kevin Ryde, Oct 26 2024

If we write n = (m choose 2) + s, 0<=s<=m, then a(n)=(m choose 3) + (s choose 2).

a(N) = Comb(T,2)+Comb(R,3) where R:=Round(Sqrt(2*N)) and T:=N-Comb(R,2). - Gerald Hillier, Nov 18 2017

More terms from Robert G. Wilson v, Feb 01 2006

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User: Arturo Magidin

A213506 Number of nonisomorphic 2-generator p-groups of class at most 2 and order p^n.

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A111138 Let b(n) denote the number of nontriangular numbers less than or equal to n. Then a(n) = b(n-1) + a(b(n-1)), with a(1) = a(2) = 0, a(3) = 1.

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