A188161 -id:A188161 - cp's OEIS frontend

A224790 a(n) = 3*9^n + 8.

11, 35, 251, 2195, 19691, 177155, 1594331, 14348915, 129140171, 1162261475, 10460353211, 94143178835, 847288609451, 7625597484995, 68630377364891, 617673396283955, 5559060566555531, 50031545098999715, 450283905890997371, 4052555153018976275

Offset: 0

Eric M. Schmidt, Apr 18 2013

Number of conjugacy classes in Ree group 2G2(3*9^n).

Eric M. Schmidt, 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 (10,-9).

Cf. A188161.

List([0..20], n-> 8 + 3^(2*n+1)); # G. C. Greubel, Nov 12 2019

[8 + 3^(2*n+1): n in [0..20]]; // G. C. Greubel, Nov 12 2019

seq(8+3^(2*n+1), n=0..20); # G. C. Greubel, Nov 12 2019

8 + 3^(2*Range[21]-1) (* G. C. Greubel, Nov 12 2019 *)

a(n)=3*9^n+8 \\ Charles R Greathouse IV, Sep 24 2015

Sage
```
[3*9^n+8 for n in [0..20]]
```

G.f.: 8/(1-x) + 3/(1-9*x).
a(n) = 10*a(n-1) - 9*a(n-2).
E.g.f.: 8*exp(x) + 3*exp(9*x). - G. C. Greubel, Nov 12 2019

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.

A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

Original entry on oeis.org

4, 10, 34, 130, 514, 2050, 8194, 32770, 131074, 524290, 2097154, 8388610, 33554434, 134217730, 536870914, 2147483650, 8589934594, 34359738370, 137438953474, 549755813890, 2199023255554, 8796093022210, 35184372088834, 140737488355330, 562949953421314

Offset: 0

Peter M. Chema, Feb 28 2017

Number of vertices required to make a Sierpinski tetrahedron or tetrix of side length 2^n. The sum of the vertices (balls) plus line segments (rods) of one tetrix equals the vertices of its larger, adjacent iteration. See formula.
Equivalently, the number of vertices in the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 17 2017
Also the independence number of the (n+2)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 29 2021
Final digit alternates 4 and 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph
Eric Weisstein's World of Mathematics, Tetrix
Eric Weisstein's World of Mathematics, Vertex Count
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Subsequence of A016957.
Cf. A052539, A279511, A279512.
First bisection of A052548, A087288; second bisection of A049332, A133140, A135440.
Cf. A002023 (edge count).

Table[2 4^n + 2, {n, 0, 30}] (* Bruno Berselli, Feb 28 2017 *)
2 (4^Range[0, 20] + 1) (* Eric W. Weisstein, Aug 17 2017 *)
LinearRecurrence[{5, -4}, {4, 10}, 20] (* Eric W. Weisstein, Aug 17 2017 *)
CoefficientList[Series[-((2 (-2 + 5 x))/(1 - 5 x + 4 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 17 2017 *)

a(n)=2*4^n+2 \\ Charles R Greathouse IV, Feb 28 2017

Vec(2*(2 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Mar 02 2017

def a(n): return 2*4**n + 2
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 29 2021

G.f.: 2*(2 - 5*x)/((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
a(n+1) = a(n) + A002023(n).
a(n) = 2*A052539(n) = A188161(n) - 1 = A087289(n) + 1 = A056469(2*n+2) = A261723(4*n+1).
E.g.f.: 2*(exp(4*x) + exp(x)). - G. C. Greubel, Aug 17 2017

Entry revised by Editors of OEIS, Mar 01 2017

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.

A195464 a(n) = 2^(4n + 3) + 24^n + 3.

Original entry on oeis.org

13, 139, 2083, 32899, 524803, 8390659, 134225923, 2147516419, 34359869443, 549756338179, 8796095119363, 140737496743939, 2251799847239683, 36028797153181699, 576460752840294403, 9223372039002259459, 147573952598266347523, 2361183241469182345219

Offset: 0

Brad Clardy, Sep 19 2011

Binary numbers of form 1j1i11 where j and i are the number of 0's, n is the index, i = 2*n+1, j = 2*n+3.

			Terms starting from n=1 written in binary are 10001011, 100000100011, 1000000010000011, 10000000001000000011.

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A188161.

[2^(4*n + 3) + 2*4^n + 3: n in [0..20]]; // Vincenzo Librandi, Sep 30 2011

a(n) = 2^(4*n+3) + A188161(n).
From Alexander R. Povolotsky, Sep 19 2011: (Start)
a(n+2) = 20*a(n+1) - 64*a(n) + 135.
G.f.: (-13 + 134*x - 256*x^2)/(-1 + 21*x - 84*x^2 + 64*x^3). (End)
a(n) = 3 + A026244(n+1). - Bruno Berselli, Sep 19 2011

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).

cp's OEIS Frontend

A224790 a(n) = 3*9^n + 8.

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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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A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

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

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A195464 a(n) = 2^(4*n + 3) + 2*4^n + 3.

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

A195464 a(n) = 2^(4n + 3) + 24^n + 3.