A062318 -id:A062318 - cp's OEIS frontend

A188377 a(n) = n^3 - 4n^2 + 6n - 2.

7, 22, 53, 106, 187, 302, 457, 658, 911, 1222, 1597, 2042, 2563, 3166, 3857, 4642, 5527, 6518, 7621, 8842, 10187, 11662, 13273, 15026, 16927, 18982, 21197, 23578, 26131, 28862, 31777, 34882, 38183, 41686, 45397, 49322, 53467, 57838, 62441, 67282, 72367

Offset: 3

Adeniji, Adenike & Makanjuola, Samuel (somakanjuola(AT)unilorin.edu.ng) Apr 14 2011

Number of nilpotent elements in the identity difference partial one - one transformation semigroup, denoted by N(IDI_n). For n=3, #N(IDI_n) = 7.

a(n+1) is also the Moore lower bound on the order of an (n,7)-cage. - Jason Kimberley, Oct 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023.
Gordon Royle, Cages of higher valency
R. P. Sullivan, Semigroups generated by nilpotent transformations, Journal of Algebra 110 (1987), 324-345.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A188716, A188947.

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), this sequence (g=7). - Jason Kimberley, Oct 30 2011

[n^3 - 4*n^2 + 6*n - 2: n in [3..50]]; // Vincenzo Librandi, May 01 2011

[SequenceToInteger([2^^3,1],n-2):n in [5..50]]; // Jason Kimberley, Oct 20 2011

Table[n^3 - 4*n^2 + 6*n - 2, {n, 3, 80}] (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)
LinearRecurrence[{4,-6,4,-1},{7,22,53,106},50] (* Harvey P. Dale, May 29 2019 *)

a(n)=n^3-4*n^2+6*n-2 \\ Charles R Greathouse IV, Apr 06 2012

a(n+1) = (n+1)^3 - 4*(n+1)^2 + 6*(n+1) - 2
       = (n-1)^3 + 2*(n-1)^2 + 2*(n-1) + 2
       = 1222 read in base n-1.
- Jason Kimberley, Oct 20 2011
From Colin Barker, Apr 06 2012: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x^3*(7 - 6*x + 7*x^2 - 2*x^3)/(1-x)^4. (End)
E.g.f.: 2 - x - x^2 + exp(x)*(x^3 - x^2 + 3*x - 2). - Stefano Spezia, Apr 09 2022

Edited by N. J. A. Sloane, Apr 23 2011

A094626 Expansion of x(1+x)/((1-x)(1-10*x^2)).

Original entry on oeis.org

0, 1, 2, 12, 22, 122, 222, 1222, 2222, 12222, 22222, 122222, 222222, 1222222, 2222222, 12222222, 22222222, 122222222, 222222222, 1222222222, 2222222222, 12222222222, 22222222222, 122222222222, 222222222222, 1222222222222, 2222222222222, 12222222222222

Offset: 0

Paul Barry, May 15 2004

Previous name: Sequence whose n-th term digits sum to n.
a(n) is the smallest integer with digits from {0,1,2} having digit sum n. Namely the base-10 reading of the ternary string of A062318. - Jason Kimberley, Nov 01 2011
a(n) is the Moore lower bound on the order of an (11,n)-cage. - Jason Kimberley, Oct 18 2011

Colin Barker, Table of n, a(n) for n = 0..1000
G. Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A094623, A094624.

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), this sequence (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Nov 01 2011

LinearRecurrence[{1, 10, -10}, {0, 1, 2}, 30] (* Paolo Xausa, Feb 21 2024 *)

concat(0, Vec(x*(1+x)/((1-x)*(1-10*x^2)) + O(x^30))) \\ Colin Barker, Mar 17 2017

G.f.: x*(1+x)/((1-x)*(1-10*x^2)).
a(n) = 10^(n/2)*(11*sqrt(10)/180 + 1/9 - (11*sqrt(10)/180 - 1/9)*(-1)^n) - 2/9.
From Colin Barker, Mar 17 2017: (Start)
a(n) = 2*(10^(n/2) - 1)/9 for n even.
a(n) = (11*10^((n-1)/2) - 2)/9 for n odd. (End)
E.g.f.: (20*(cosh(sqrt(10)*x) - cosh(x) - sinh(x)) + 11*sqrt(10)*sinh(sqrt(10)*x))/90. - Stefano Spezia, Apr 09 2022

A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

Original entry on oeis.org

3, 4, 4, 5, 6, 5, 6, 8, 10, 6, 7, 10, 17, 14, 7, 8, 12, 26, 26, 22, 8, 9, 14, 37, 42, 53, 30, 9, 10, 16, 50, 62, 106, 80, 46, 10, 11, 18, 65, 86, 187, 170, 161, 62, 11, 12, 20, 82, 114, 302, 312, 426, 242, 94, 12, 13, 22, 101, 146, 457, 518, 937, 682, 485, 126, 13

Offset: 1

Jason Kimberley, Oct 27 2011

k >= 2; g >= 3.
The base k-1 reading of the base 10 string of A094626(g).
Exoo and Jajcay Theorem 1: M(k,g) <= A054760(k,g) with equality if and only if: k = 2 and g >= 3; g = 3 and k >= 2; g = 4 and k >= 2; g = 5 and k = 2, 3, 7 or possibly 57; or g = 6, 8, or 12, and there exists a symmetric generalized n-gon of order k - 1.

			This is the table formed from the antidiagonals for k+g = 5..20:
3   4   5   6    7    8    9     10    11    12    13    14    15   16  17 18
4   6  10  14   22   30    46    62    94   126   190   254   382  510 766
5   8  17  26   53   80   161   242   485   728  1457  2186  4373 6560
6  10  26  42  106  170   426   682  1706  2730  6826 10922 27306
7  12  37  62  187  312   937  1562  4687  7812 23437 39062
8  14  50  86  302  518  1814  3110 10886 18662 65318
9  16  65 114  457  800  3201  5602 22409 39216
10 18  82 146  658 1170  5266  9362 42130
11 20 101 182  911 1640  8201 14762
12 22 122 222 1222 2222 12222
13 24 145 266 1597 2928
14 26 170 314 2042
15 28 197 366
16 30 226
17 32
18

E. Bannai and T. Ito, On finite Moore graphs, J. Fac. Sci. Tokyo, Sect. 1A, 20 (1973) 191-208.
R. M. Damerell, On Moore graphs, Proc. Cambridge Phil. Soc. 74 (1973) 227-236.

Jason Kimberley, Table of n, a(n) for n = 1..20100 (k+g = 5..204)
Jason Kimberley, Table of n, k+g, k, g, M(k,g)=a(n) for k+g = 5..204 (n = 1..20100)
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency

Moore lower bound on the order of a (k,g) cage: this sequence (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7), 2*A053698 (g=8), 2*A053699 (g=10), 2*A053700 (g=12), 2*A053716 (g=14), 2*A053716 (g=16), 2*A102909 (g=18), 2*A103623 (g=20), 2*A060885 (g=22), 2*A105067 (g=24), 2*A060887 (g=26), 2*A104376 (g=28), 2*A104682 (g=30), 2*A105312 (g=32).

Cf. A054760 (the actual order of a (k,g)-cage).

ExtendedStringToInt:=func;
M:=func;
k_:=2;g_:=3;
anti:=func;
[anti(kg):kg in[5..15]];

Table[Function[g, FromDigits[#, k - 1] &@ IntegerDigits@ SeriesCoefficient[x (1 + x)/((1 - x) (1 - 10 x^2)), {x, 0, g}]][n - k + 3], {n, 2, 12}, {k, n, 2, -1}] // Flatten (* Michael De Vlieger, May 15 2017 *)

M(k,2i) = 2 sum_{j=0}^{i-1}(k-1)^j =  string "2"^i read in base k-1.
M(k,2i+1) = (k-1)^i +  2 sum_{j=0}^{i-1}(k-1)^j = string "1"*"2"^i read in base k-1.
Recurrence:
M(k,3) = k + 1,
M(k,2i) = M(k,2i-1) + (k-1)^(i-1),
M(k,2i+1) = M(k,2i) + (k-1)^i.

A198306 Moore lower bound on the order of a (6,g)-cage.

Original entry on oeis.org

7, 12, 37, 62, 187, 312, 937, 1562, 4687, 7812, 23437, 39062, 117187, 195312, 585937, 976562, 2929687, 4882812, 14648437, 24414062, 73242187, 122070312, 366210937, 610351562, 1831054687, 3051757812, 9155273437, 15258789062

Offset: 3

Jason Kimberley, Oct 30 2011

Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), this sequence (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

LinearRecurrence[{1,5,-5},{7,12,37},30] (* Harvey P. Dale, Jun 28 2015 *)

a(2*i) = 2*Sum_{j=0..i-1} 5^j = string "2"^i read in base 5.
a(2*i+1) = 5^i + 2*Sum_{j=0..i-1} 5^j = string "1"*"2"^i read in base 5.
a(n) <= A218554(n). - Jason Kimberley, Dec 21 2012
a(n) = a(n-1)+5*a(n-2)-5*a(n-3). G.f.: -x^3*(10*x^2-5*x-7) / ((x-1)*(5*x^2-1)). - Colin Barker, Feb 01 2013
From Colin Barker, Nov 25 2016: (Start)
a(n) = (5^(n/2) - 1)/2 for n>2 and even.
a(n) = (3*5^((n-1)/2) - 1)/2 for n>2 and odd. (End)
E.g.f.: (5*cosh(sqrt(5)*x) - 5*cosh(x) - 5*sinh(x) + 3*sqrt(5)*sinh(sqrt(5)*x) - 10*x*(1 + x))/10. - Stefano Spezia, Apr 07 2022

A198307 Moore lower bound on the order of a (7,g)-cage.

Original entry on oeis.org

8, 14, 50, 86, 302, 518, 1814, 3110, 10886, 18662, 65318, 111974, 391910, 671846, 2351462, 4031078, 14108774, 24186470, 84652646, 145118822, 507915878, 870712934, 3047495270, 5224277606, 18284971622, 31345665638, 109709829734, 188073993830, 658258978406

Offset: 3

Jason Kimberley, Oct 30 2011

Colin Barker, Table of n, a(n) for n = 3..1000
Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,6,-6).

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), this sequence (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

DeleteCases[CoefficientList[Series[2 x^3*(4 + 3 x - 6 x^2)/((1 - x) (1 - 6 x^2)), {x, 0, 31}], x], 0] (* Michael De Vlieger, Mar 17 2017 *)
LinearRecurrence[{1,6,-6},{8,14,50},30] (* or *) CoefficientList[ Series[ -((2 (-4-3 x+6 x^2))/(1-x-6 x^2+6 x^3)),{x,0,30}],x] (* Harvey P. Dale, Aug 03 2021 *)

Vec(2*x^3*(4 + 3*x - 6*x^2) / ((1 - x)*(1 - 6*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

a(2*i) = 2*Sum_{j=0..i-1}6^j =  string "2"^i read in base 6.
a(2*i+1) = 6^i +  2*Sum_{j=0..i-1}6^j = string "1"*"2"^i read in base 6.
a(n) <= A218555(n).
From Colin Barker, Feb 01 2013: (Start)
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) for n>5.
G.f.: 2*x^3*(4 + 3*x - 6*x^2) / ((1 - x)*(1 - 6*x^2)). (End)
From Colin Barker, Mar 17 2017: (Start)
a(n) = 2*(6^(n/2) - 1)/5 for n>2 and even.
a(n) = (7*6^(n/2-1/2) - 2)/5 for n>2 and odd. (End)
E.g.f.: (12*(cosh(sqrt(6)*x) - cosh(x)) + 7*sqrt(6)*sinh(sqrt(6)*x) - 12*sinh(x) - 30*x*(1 + x))/30. - Stefano Spezia, Apr 07 2022

A198308 Moore lower bound on the order of an (8,g)-cage.

Original entry on oeis.org

9, 16, 65, 114, 457, 800, 3201, 5602, 22409, 39216, 156865, 274514, 1098057, 1921600, 7686401, 13451202, 53804809, 94158416, 376633665, 659108914, 2636435657, 4613762400, 18455049601, 32296336802, 129185347209, 226074357616, 904297430465, 1582520503314

Offset: 3

Jason Kimberley, Oct 30 2011

Colin Barker, Table of n, a(n) for n = 3..1000
Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,7,-7).

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), this sequence (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

LinearRecurrence[{1,7,-7},{9,16,65},40] (* Harvey P. Dale, Oct 14 2019 *)

Vec(x^3*(9 + 7*x - 14*x^2) / ((1 - x)*(1 - 7*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

a(2*i) = 2 Sum_{j=0..i-1} 7^j = string "2"^i read in base 7.
a(2*i+1) = 7^i + 2 Sum_{j=0..i-1} 7^j = string "1"*"2"^i read in base 7.
From Colin Barker, Feb 01 2013: (Start)
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) for n>5.
G.f.: x^3*(9 + 7*x - 14*x^2) / ((1 - x)*(1 - 7*x^2)). (End)
From Colin Barker, Mar 17 2017: (Start)
a(n) = (7^(n/2) - 1)/3 for n even.
a(n) = (4*7^(n/2-1/2) - 1)/3 for n odd. (End)
E.g.f.: (7*(cosh(sqrt(7)*x) - cosh(x) - sinh(x)) + 4*sqrt(7)*sinh(sqrt(7)*x) - 21*x*(1 + x))/21. - Stefano Spezia, Apr 09 2022

A198309 Moore lower bound on the order of a (9,g)-cage.

Original entry on oeis.org

10, 18, 82, 146, 658, 1170, 5266, 9362, 42130, 74898, 337042, 599186, 2696338, 4793490, 21570706, 38347922, 172565650, 306783378, 1380525202, 2454267026, 11044201618, 19634136210, 88353612946, 157073089682, 706828903570, 1256584717458, 5654631228562

Offset: 3

Jason Kimberley, Oct 30 2011

Colin Barker, Table of n, a(n) for n = 3..1000
Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,8,-8).

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), this sequence (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

LinearRecurrence[{1,8,-8},{10,18,82},30] (* Harvey P. Dale, Apr 03 2015 *)

Vec(2*x^3*(5 + 4*x - 8*x^2) / ((1 - x)*(1 - 8*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

a(2*i) = 2 Sum_{j=0..i-1} 8^j =  string "2"^i read in base 8.
a(2*i+1) = 8^i + 2 Sum_{j=0..i-1} 8^j = string "1"*"2"^i read in base 8.
From Colin Barker, Feb 01 2013: (Start)
a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) for n>5.
G.f.: 2*x^3*(5 + 4*x - 8*x^2) / ((1 - x)*(1 - 8*x^2)). (End)
From Colin Barker, Mar 17 2017: (Start)
a(n) = 2*(2^(3*n/2) - 1)/7 for n even.
a(n) = (9*2^((3*(n-1))/2) - 2)/7 for n odd. (End)
E.g.f.: (8*(cosh(2*sqrt(2)*x) - cosh(x) - sinh(x)) + 9*sqrt(2)*sinh(2*sqrt(2)*x) - 28*x*(1 + x))/28. - Stefano Spezia, Apr 09 2022

A198310 Moore lower bound on the order of a (10,g)-cage.

Original entry on oeis.org

11, 20, 101, 182, 911, 1640, 8201, 14762, 73811, 132860, 664301, 1195742, 5978711, 10761680, 53808401, 96855122, 484275611, 871696100, 4358480501, 7845264902, 39226324511, 70607384120, 353036920601, 635466457082, 3177332285411

Offset: 3

Jason Kimberley, Oct 30 2011

Paolo Xausa, Table of n, a(n) for n = 3..1000
Gordon Royle, Cages of higher valency
Index entries for linear recurrences with constant coefficients, signature (1,9,-9).

Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), this sequence (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7).

A198310[n_] := (-3-(-3)^n+4*3^n)/12; Array[A198310, 30, 3] (* or *)
LinearRecurrence[{1, 9, -9}, {11, 20, 101}, 30] (* Paolo Xausa, Feb 21 2024 *)

a(n)=(-3-(-3)^n+4*3^n)/12 \\ Charles R Greathouse IV, Jul 06 2017

a(2i) = 2*Sum_{j=0..i-1} 9^j =  string "2"^i read in base 9.
a(2i+1) = 9^i +  2*Sum_{j=0..i-1} 9^j = string "1"*"2"^i read in base 9.
From Colin Barker, Feb 01 2013: (Start)
a(n) = (-3-(-3)^n+4*3^n)/12.
a(n) = a(n-1)+9*a(n-2)-9*a(n-3).
G.f.: -x^3*(18*x^2-9*x-11) / ((x-1)*(3*x-1)*(3*x+1)). (End)
E.g.f.: (3*(cosh(3*x) - cosh(x) - sinh(x)) + 5*sinh(3*x))/12 - x - x^2. - Stefano Spezia, Apr 09 2022

A180516 Numbers of the form i*4^j-1 (i=1..3, j >= 0).

Original entry on oeis.org

0, 1, 2, 3, 7, 11, 15, 31, 47, 63, 127, 191, 255, 511, 767, 1023, 2047, 3071, 4095, 8191, 12287, 16383, 32767, 49151, 65535, 131071, 196607, 262143, 524287, 786431, 1048575, 2097151, 3145727, 4194303, 8388607, 12582911, 16777215, 33554431, 50331647, 67108863, 134217727, 201326591, 268435455, 536870911, 805306367, 1073741823, 2147483647, 3221225471

Offset: 1

N. J. A. Sloane, Jan 25 2011

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,4,-4).

Smallest number whose base b sum of digits is n: A000225 (b=2), A062318 (b=3), this sequence (b=4), A181287 (b=5), A181288 (b=6), A181303 (b=7), A165804 (b=8), A140576 (b=9), A051885 (b=10). - Jason Kimberley, Nov 02 2011

LinearRecurrence[{1, 0, 4, -4}, Range[0, 3], 50] (* Paolo Xausa, Aug 27 2024 *)

From Colin Barker, Feb 01 2013: (Start)
a(n) = a(n-1) + 4*a(n-3) - 4*a(n-4).
G.f.: x^2*(x^2+x+1) / ((x-1)*(4*x^3-1)). (End)

A037233 Order of (4,n) cage, i.e., minimal order of 4-regular graph of girth n.

Original entry on oeis.org

5, 8, 19, 26, 67, 80

Offset: 3

Erich Friedman

a(9) <= 275, a(10) <= 384, a(12) = 728. - From Royle's page via Jason Kimberley, Dec 26 2012

Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency
Eric Weisstein's World of Mathematics, Cage Graph (claims too much)

Orders of cages: A054760 (n,k), A000066 (3,n), this sequence (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).

a(n) >= A062318(n+1). - Jason Kimberley, Dec 21 2012

Extended by Jason Kimberley, Apr 25 2010

cp's OEIS Frontend

A188377 a(n) = n^3 - 4n^2 + 6n - 2.

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A094626 Expansion of x*(1+x)/((1-x)*(1-10*x^2)).

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A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

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A198306 Moore lower bound on the order of a (6,g)-cage.

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A198307 Moore lower bound on the order of a (7,g)-cage.

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A198308 Moore lower bound on the order of an (8,g)-cage.

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A198309 Moore lower bound on the order of a (9,g)-cage.

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A198310 Moore lower bound on the order of a (10,g)-cage.

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A094626 Expansion of x(1+x)/((1-x)(1-10*x^2)).