A224790 -id:A224790 - cp's OEIS frontend

A188161 a(n) = 2*4^n + 3.

5, 11, 35, 131, 515, 2051, 8195, 32771, 131075, 524291, 2097155, 8388611, 33554435, 134217731, 536870915, 2147483651, 8589934595, 34359738371, 137438953475, 549755813891, 2199023255555, 8796093022211, 35184372088835, 140737488355331, 562949953421315, 2251799813685251

Offset: 0

Brad Clardy, Mar 22 2011

For n > 0, binary representation of a(n) is 1X11 where X is 2*n-1 zeros.

Number of conjugacy classes in Suzuki group Sz(2*4^n). - Eric M. Schmidt, Apr 18 2013

			The first seven terms written in binary are 101, 1011, 100011, 10000011, 1000000011, 100000000011, and 10000000000011.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A141725 (4^(n+1)-3), A224790.

[2*4^n+3: n in [1..100]]; // Vincenzo Librandi, Mar 29 2011
(Decimal BASIC)
FOR n=0 TO 1000
   PRINT n; 2*4^n+3
NEXT n
END ! /* Bruno Berselli, Apr 28 2011 */

2 4^Range[0,30]+3  (* Harvey P. Dale, Apr 02 2011 *)

a(n)=2*4^n+3 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = A141725(n) - 2*A141725(n-1) for n > 0.
G.f.: (5-14*x)/((1-4*x)*(1-x)). - R. J. Mathar, Apr 09 2011
a(n) = 5*a(n-1) - 4*a(n-2). - Joerg Arndt, Apr 09 2011
From Felix P. Muga II, Mar 19 2014: (Start)
a(n) = a(n-1) + 6*4^(n-1) for n > 0, a(0)=5.
a(n) = a(n-1) + 12*a(n-2) - 36 for n > 1, a(0)=5, a(1)=11. (End)
E.g.f.: exp(x)*(2*exp(3*x) + 3). - Elmo R. Oliveira, Mar 08 2025

A225928 a(n) = 416^n + 84^n + 17.

Original entry on oeis.org

29, 113, 1169, 16913, 264209, 4202513, 67141649, 1073872913, 17180393489, 274880004113, 4398054899729, 70368777732113, 1125900041060369, 18014399046352913, 288230378299195409, 4611686027017322513, 73786976329197944849, 1180591620854850256913

Offset: 0

Eric M. Schmidt, May 21 2013

Number of conjugacy classes in Ree group 2F4(2*4^n).

Eric M. Schmidt, Table of n, a(n) for n = 0..500
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A188161, A224790, A225929 - A225938.

LinearRecurrence[{21,-84,64},{29,113,1169},20] (* Harvey P. Dale, May 06 2016 *)

a(n)=4*16^n+8*4^n+17 \\ Charles R Greathouse IV, May 22 2013

Sage
```
[4*16^n + 8*4^n + 17 for n in [0..20]]
```

G.f.: 17/(1-x) + 8/(1-4x) + 4/(1-16x).

a(n) = 21*a(n-1) - 84*a(n-2) + 64*a(n-3) for n > 2. - Wesley Ivan Hurt, Oct 08 2017

A225938 Number of conjugacy classes in Chevalley group E_8(q) as q runs through the prime powers (A246655).

Original entry on oeis.org

1156, 12825, 97154, 519071, 6906102, 19543486, 49150839, 238045722, 889575240, 4600759094, 7439557452, 17980383618, 82034207430, 159213167411, 293713437009, 518754968088, 882274298862, 1136129443366, 3612770425152, 8189556710532, 11973138177210, 24340206797502

Offset: 1

Eric M. Schmidt, May 21 2013

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.
Index entries for sequences related to groups

Cf. A188161, A224790, A225928-A225937.

A225938 := proc(n)
    local q ;
    q := A246655(n) ;
    if modp(q,2) = 0 then
        q^8 + q^7 + 2*q^6 + 3*q^5 +  9*q^4 + 14*q^3 + 32*q^2 + 47*q +  70;
    elif modp(q,3) = 0 then
        q^8 + q^7 + 2*q^6 + 3*q^5 + 10*q^4 + 16*q^3 + 39*q^2 + 65*q + 102 ;
    elif modp(q,5) = 0 then
        q^8 + q^7 + 2*q^6 + 3*q^5 + 10*q^4 + 16*q^3 + 40*q^2 + 67*q + 111 ;
    else
        q^8 + q^7 + 2*q^6 + 3*q^5 + 10*q^4 + 16*q^3 + 40*q^2 + 67*q + 112 ;
    end if;
end proc: # R. J. Mathar, Jan 09 2017

qmax = 100;
Reap[For[q = 2, q < qmax, q++, If[PrimePowerQ[q], cc = q^8 + q^7 + 2 q^6 + 3 q^5 + Which[Mod[q, 2] == 0, 9 q^4 + 14 q^3 + 32 q^2 + 47 q + 70, Mod[q, 3] == 0, 10 q^4 + 16 q^3 + 39 q^2 + 65 q + 102, Mod[q, 5] == 0, 10 q^4 + 16 q^3 + 40 q^2 + 67 q + 111, True, 10 q^4 + 16 q^3 + 40 q^2 + 67 q + 112]; Sow[cc]]]][[2, 1]] (* Jean-François Alcover, Mar 24 2020 *)

def A225938(q) : return q^8 + q^7 + 2*q^6 + 3*q^5 + (9*q^4 + 14*q^3 + 32*q^2 + 47*q + 70 if q%2==0 else 10*q^4 + 16*q^3 + 39*q^2 + 65*q + 102 if q%3==0 else 10*q^4 + 16*q^3 + 40*q^2 + 67*q + 111 if q%5==0 else 10*q^4 + 16*q^3 + 40*q^2 + 67*q + 112)

Let q be the n-th prime power. Then a(n) is
q^8 + q^7 + 2q^6 + 3q^5 +  9q^4 + 14q^3 + 32q^2 + 47q +  70, q==0(mod 2);
q^8 + q^7 + 2q^6 + 3q^5 + 10q^4 + 16q^3 + 39q^2 + 65q + 102, q==0(mod 3);
q^8 + q^7 + 2q^6 + 3q^5 + 10q^4 + 16q^3 + 40q^2 + 67q + 111, q==0(mod 5);
q^8 + q^7 + 2q^6 + 3q^5 + 10q^4 + 16q^3 + 40q^2 + 67q + 112, otherwise.

A225929 Number of conjugacy classes in Chevalley group G_2(q) as q runs through the prime powers.

Original entry on oeis.org

16, 23, 32, 44, 72, 88, 107, 152, 204, 296, 332, 408, 584, 684, 791, 908, 1032, 1096, 1452, 1772, 1944, 2312, 2508, 2924, 3608, 3852, 4232, 4632, 5192, 5484, 6408, 6731, 7064, 8108, 9612, 10412, 10824, 11672, 12108, 13004, 14892, 15884, 16392, 16648, 17432

Offset: 1

Eric M. Schmidt, May 21 2013

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.

Cf. A188161, A224790, A225928 - A225938.

def A225929(q) : return q^2 + 2*q + (9 if q%6 in [1,5] else 8)

Let q be the n-th prime power. Then a(n) = q^2 + 2q + c, where c = 9 if q == 1, 5 (mod 6) and c = 8 otherwise.

A225937 Number of conjugacy classes in adjoint Chevalley group E_7(q) as q runs through the prime powers.

Original entry on oeis.org

531, 4569, 24553, 105644, 992834, 2447517, 5477205, 21674822, 68494004, 287617189, 437805224, 946620206, 3567305234, 6369359984, 10879403385, 17889596996, 28462405562, 35505127221, 97646355404, 199751157632, 278452165094, 517886829194, 692659723976

Offset: 1

Eric M. Schmidt, May 21 2013

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.

Cf. A188161, A224790, A225928 - A225938.

def A225937(q) : return q^7 + q^6 + 2*q^5 + (4*q^4 + 10*q^3 + 15*q^2 + 25*q + 21 if q%2==0 else 5*q^4 + 13*q^3 + 24*q^2 + 46*q + 57 if q%3==0 else 5*q^4 + 13*q^3 + 24*q^2 + 47*q + 59)

Let q be the n-th prime power. Then, a(n) is
q^7 + q^6 + 2q^5 + 4q^4 + 10q^3 + 15q^2 + 25q + 21 if q == 0 (mod 2);
q^7 + q^6 + 2q^5 + 5q^4 + 13q^3 + 24q^2 + 46q + 57 if q == 0 (mod 3);
q^7 + q^6 + 2q^5 + 5q^4 + 13q^3 + 24q^2 + 47q + 59 otherwise.

A225930 Number of conjugacy classes in twisted Chevalley group 3D4(q) as q runs through the prime powers.

Original entry on oeis.org

35, 126, 345, 786, 2806, 4685, 7386, 16110, 30946, 69909, 88746, 137566, 292566, 406906, 551886, 732546, 954310, 1082405, 1926226, 2896410, 3500206, 4985766, 5884906, 8042226, 12326286, 14076610, 17043525, 20456446, 25774710, 28792666, 39449446, 43584810, 48037086

Offset: 1

Eric M. Schmidt, May 21 2013

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Eric M. Schmidt)
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.

Cf. A000961 (without 1), A188161, A224790, A225928-A225938.

Map[(#^2 + 1)*(# + 1)*# + 5 + Mod[#, 2] &, Select[Range[100], PrimePowerQ]] (* Paolo Xausa, Jan 16 2025 *)

apply(x->(x^4 + x^3 + x^2 + x + 5 + (x%2)), select(isprimepower, [1..100])) \\ Michel Marcus, Jan 16 2025

Let q be the n-th prime power. Then, a(n) = q^4 + q^3 + q^2 + q + c, where c = 5 if q is even and c = 6 if q is odd.

A225931 Number of conjugacy classes in Chevalley group F_4(q) as q runs through the prime powers.

Original entry on oeis.org

95, 273, 539, 1156, 3566, 5603, 8751, 18346, 34364, 75443, 95656, 146882, 308254, 426656, 576345, 762412, 990326, 1120595, 1985636, 2976016, 3591434, 5103526, 6017672, 8208724, 12553402, 14326796, 17326739, 20785106, 26163886, 29214704, 39981062, 44156775

Offset: 1

Eric M. Schmidt, May 21 2013

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.

Cf. A188161, A224790, A225928 - A225938.

def A225931(q) : return q^4 + 2*q^3 + (6*q^2 + 10*q + 19 if q%2==0 else 7*q^2 + 15*q + 30 if q%3==0 else 7*q^2 + 15*q + 31)

Let q be the n-th prime power.
a(n) = q^4 + 2q^3 + 6q^2 + 10q + 19 if q == 0 mod 2.
a(n) = q^4 + 2q^3 + 7q^2 + 15q + 30 if q == 0 mod 3.
a(n) = q^4 + 2q^3 + 7q^2 + 15q + 31 otherwise.

A225932 Number of conjugacy classes in simply connected Chevalley group E_6(q) as q runs through the prime powers.

Original entry on oeis.org

180, 1269, 6116, 20454, 140886, 304548, 605685, 1965462, 5262486, 17969012, 25736406, 49802214, 155060070, 254728710, 402876885, 616803846, 918054582, 1109465220, 2638941366, 4871761782, 6475396806, 11018543046, 14135564454, 22598655270, 42920128086

Offset: 1

Eric M. Schmidt, May 21 2013

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.

Cf. A188161, A224790, A225928 - A225938.

def A225932(q) : return q^6 + q^5 + 2*q^4 + 2*q^3 + [15*q^2 + 21*q + 60, 6*q^2 + 4*q + 4, 7*q^2 + 5*q + 3, 14*q^2 + 20*q + 52, 7*q^2 + 5*q + 4][q%6-1]

Let q be the n-th prime power.
a(n) = q^6 + q^5 + 2q^4 + 2q^3 + 15q^2 + 21q + 60 if q == 1 (mod 6).
a(n) = q^6 + q^5 + 2q^4 + 2q^3 +  6q^2 +  4q +  4 if q == 2 (mod 6).
a(n) = q^6 + q^5 + 2q^4 + 2q^3 +  7q^2 +  5q +  3 if q == 3 (mod 6).
a(n) = q^6 + q^5 + 2q^4 + 2q^3 + 14q^2 + 20q + 52 if q == 4 (mod 6).
a(n) = q^6 + q^5 + 2q^4 + 2q^3 +  7q^2 +  5q +  4 if q == 5 (mod 6).

A225933 Number of conjugacy classes in adjoint Chevalley group E_6(q) as q runs through the prime powers.

Original entry on oeis.org

180, 1269, 5940, 20454, 140470, 304548, 605685, 1965462, 5261278, 17967252, 25736406, 49799782, 155060070, 254724622, 402876885, 616803846, 918048406, 1109465220, 2638932670, 4871761782, 6475385158, 11018543046, 14135549422, 22598655270, 42920128086

Offset: 1

Eric M. Schmidt, May 21 2013

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.

Cf. A188161, A224790, A225928 - A225938.

def A225933(q) : return q^6 + q^5 + 2*q^4 + 2*q^3 + [9*q^2 + 9*q + 22, 6*q^2 + 4*q + 4, 7*q^2 + 5*q + 3, 8*q^2 + 8*q + 20, 7*q^2 + 5*q + 4][q%6-1]

Let q be the n-th prime power.
a(n) = q^6 + q^5 + 2q^4 + 2q^3 + 9q^2 + 9q + 22 if q == 1 (mod 6).
a(n) = q^6 + q^5 + 2q^4 + 2q^3 + 6q^2 + 4q +  4 if q == 2 (mod 6).
a(n) = q^6 + q^5 + 2q^4 + 2q^3 + 7q^2 + 5q +  3 if q == 3 (mod 6).
a(n) = q^6 + q^5 + 2q^4 + 2q^3 + 8q^2 + 8q + 20 if q == 4 (mod 6).
a(n) = q^6 + q^5 + 2q^4 + 2q^3 + 7q^2 + 5q +  4 if q == 5 (mod 6).

A225934 Number of conjugacy classes in simply connected twisted Chevalley group 2E6(q) as q runs through the prime powers.

Original entry on oeis.org

346, 1389, 6102, 21182, 141262, 306574, 607533, 1969886, 5266030, 17975982, 25750142, 49814254, 155091326, 254757166, 402919341, 616863422, 918109966, 1109543806, 2639036782, 4871920766, 6475547950, 11018778302, 14135789614, 22598987966, 42920581982

Offset: 1

Eric M. Schmidt, May 21 2013

nonn

Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Frank Luebeck, Numbers of Conjugacy Classes in Finite Groups of Lie Type.

Cf. A188161, A224790, A225928 - A225938.

def A225934(q) : return q^6 + q^5 + 2*q^4 + 4*q^3 + [11*q^2 + 11*q + 16, 18*q^2 + 26*q + 62, 11*q^2 + 11*q + 15, 10*q^2 + 10*q + 14, 19*q^2 + 27*q + 72][q%6-1]

Let q be the n-th prime power.
a(n) = q^6 + q^5 + 2q^4 + 11q^2 + 11q + 16 if q == (1 mod 6).
a(n) = q^6 + q^5 + 2q^4 + 18q^2 + 26q + 62 if q == (2 mod 6).
a(n) = q^6 + q^5 + 2q^4 + 11q^2 + 11q + 15 if q == (3 mod 6).
a(n) = q^6 + q^5 + 2q^4 + 10q^2 + 10q + 14 if q == (4 mod 6).
a(n) = q^6 + q^5 + 2q^4 + 19q^2 + 27q + 72 if q == (5 mod 6).

cp's OEIS Frontend

A188161 a(n) = 2*4^n + 3.

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A225928 a(n) = 4*16^n + 8*4^n + 17.

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A225938 Number of conjugacy classes in Chevalley group E_8(q) as q runs through the prime powers (A246655).

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A225929 Number of conjugacy classes in Chevalley group G_2(q) as q runs through the prime powers.

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Sage

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A225937 Number of conjugacy classes in adjoint Chevalley group E_7(q) as q runs through the prime powers.

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Sage

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A225930 Number of conjugacy classes in twisted Chevalley group 3D4(q) as q runs through the prime powers.

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Mathematica

PARI

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A225931 Number of conjugacy classes in Chevalley group F_4(q) as q runs through the prime powers.

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Sage

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A225932 Number of conjugacy classes in simply connected Chevalley group E_6(q) as q runs through the prime powers.

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A225928 a(n) = 416^n + 84^n + 17.