A094626 -id:A094626 - cp's OEIS frontend

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

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

A191595 Order of smallest n-regular graph of girth 5.

Original entry on oeis.org

5, 10, 19, 30, 40, 50

Offset: 2

N. J. A. Sloane, Jun 07 2011

Current upper bounds for a(8)..a(20) are 80, 96, 124, 154, 203, 230, 288, 312, 336, 448, 480, 512, 576. - Corrected from "Lower" to "Upper" and updated, from Table 4 of the Dynamic cage survey, by Jason Kimberley, Dec 29 2012

Current lower bounds for a(8)..a(20) are 67, 86, 103, 124, 147, 174, 199, 230, 259, 294, 327, 364, 403. - from Table 4 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012

M. Abreu et al., A family of regular graphs of girth 5, Discrete Math., 308 (2008), 1810-1815.
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).
G. Royle, Cages of higher valency

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), this sequence (n,5).

Moore lower bound on the orders of (k,g) cages: 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), A188377 (g=7). - Jason Kimberley, Nov 02 2011

a(n) >= A002522(n) with equality if and only if n = 2, 3, 7 or possibly 57. - Jason Kimberley, Nov 02 2011

a(2) = 5 prepended by Jason Kimberley, Jan 02 2013

A094623 Expansion of g.f. x(1+10x)/((1-x)(1-10x^2)).

Original entry on oeis.org

0, 1, 11, 21, 121, 221, 1221, 2221, 12221, 22221, 122221, 222221, 1222221, 2222221, 12222221, 22222221, 122222221, 222222221, 1222222221, 2222222221, 12222222221, 22222222221, 122222222221, 222222222221, 1222222222221

Offset: 0

Paul Barry, May 15 2004

Previous name was: Sequence whose n-th term digits sum to n.

n-th term digits are reversals of A094624(n).

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A094624, A094625, A094626.

LinearRecurrence[{1, 10, -10}, {0, 1, 11}, 30] (* Paolo Xausa, Feb 22 2024 *)

G.f.: x*(1+10*x)/((1-x)*(1-10*x^2)).
a(n) = (10^(n/2)/2)*(11/9 + 2*sqrt(10)/9 - (2*sqrt(10)/9 - 11/9)*(-1)^n) - 11/9.
E.g.f.: (11*(cosh(sqrt(10)*x) - cosh(x)) + 2*sqrt(10)*sinh(sqrt(10)*x) - 11*sinh(x))/9. - Stefano Spezia, Feb 21 2024

A094624 Expansion of g.f. x(1+11x+x^2)/((1-x)(1+x)(1-10*x^2)).

Original entry on oeis.org

0, 1, 11, 12, 121, 122, 1221, 1222, 12221, 12222, 122221, 122222, 1222221, 1222222, 12222221, 12222222, 122222221, 122222222, 1222222221, 1222222222, 12222222221, 12222222222, 122222222221, 122222222222, 1222222222221, 1222222222222, 12222222222221

Offset: 0

Paul Barry, May 15 2004

Previous name: "Sequence whose n-th term digits sum to n."

n-th term digits are reversals of A094623(n).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,11,0,-10).

Cf. A094623, A094625, A094626.

LinearRecurrence[{0, 11, 0, -10}, {0, 1, 11, 12}, 30] (* Paolo Xausa, Feb 22 2024 *)

concat(0, Vec(x*(1+11*x+x^2)/((1-x)*(1+x)*(1-10*x^2)) + O(x^40))) \\ Colin Barker, Dec 01 2015

a(n) = 10^(n/2)*(11/18 + 11*sqrt(10)/180 - (11*sqrt(10)/180 - 11/18)(-1)^n) - 13/18 - (-1)^n/2.
From Colin Barker, Dec 01 2015: (Start)
a(n) = 11*a(n-2) - 10*a(n-4) for n > 3.
G.f.: x*(1+11*x+x^2) / ((1-x)*(1+x)*(1-10*x^2)). (End)
E.g.f.: (110*(cosh(sqrt(10)*x) - cosh(x)) + 11*sqrt(10)*sinh(sqrt(10)*x) - 20*sinh(x))/90. - Stefano Spezia, Feb 21 2024

A094627 Expansion of (1+x)^2/((1-x)(1-10x^2)).

Original entry on oeis.org

1, 3, 14, 34, 144, 344, 1444, 3444, 14444, 34444, 144444, 344444, 1444444, 3444444, 14444444, 34444444, 144444444, 344444444, 1444444444, 3444444444, 14444444444, 34444444444, 144444444444, 344444444444, 1444444444444

Offset: 0

Paul Barry, May 15 2004

The digital sum of the n-th term is 2n+1.

a(n) = floor(10^floor(n/2)*(2-(-1)^n+4/9)) = 1,3,14,34,144,344,... (i.e. 2-(-1)^n = 1 for even n, 3 for odd n, followed by floor(n/2) digits '4'.) - M. F. Hasler

			(x^2 + 2*x + 1)/(10*x^3 - 10*x^2 - x + 1) = 1 + 3*x + 14*x^2 + 34*x^3 + 144*x^4 + 344*x^5 + 1444*x^6 + 3444*x^7 + 14444*x^8 + ...

Index entries for linear recurrences with constant coefficients, signature (1,10,-10).

Cf. A094626.

A011557aer := proc(n) if type(n,'odd') then 0 ; else 10^(n/2) ; end if; end proc:
A094627 := proc(n) (13*A011557aer(n)+31*A011557aer(n-1)-4)/9 ; end proc:
seq(A094627(n),n=0..10) ; # R. J. Mathar, Nov 16 2010

sr[n_,nn_]:=Table[FromDigits[PadRight[{n},i,4]],{i,nn}]; With[{nn=20}, Sort[ Join[ sr[ 1,nn],sr[3,nn]]]] (* Harvey P. Dale, May 25 2014 *)

a(n) = 10^(n/2)*( 31*sqrt(10)/180 +13/18 -(31*sqrt(10)/180-13/18)*(-1)^n )-4/9.

a(n) = (13*b(n)+31*b(n-1)-4)/9 with b(n) = 1,0,10,0,100,0,1000,.. (aerated A011557) [R. J. Mathar, Nov 26 2010]

Swapped the generic comment and the specific definition; added Maple prog.

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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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A191595 Order of smallest n-regular graph of girth 5.

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A094623 Expansion of g.f. x*(1+10*x)/((1-x)*(1-10*x^2)).

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A094624 Expansion of g.f. x*(1+11*x+x^2)/((1-x)*(1+x)*(1-10*x^2)).

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A094627 Expansion of (1+x)^2/((1-x)*(1-10*x^2)).

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A094623 Expansion of g.f. x(1+10x)/((1-x)(1-10x^2)).

A094624 Expansion of g.f. x(1+11x+x^2)/((1-x)(1+x)(1-10*x^2)).

A094627 Expansion of (1+x)^2/((1-x)(1-10x^2)).