cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-15 of 15 results.

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

Views

Author

Jason Kimberley, Oct 30 2011

Keywords

Links

Crossrefs

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).

Programs

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

Formula

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

Views

Author

Jason Kimberley, Oct 30 2011

Keywords

Links

Crossrefs

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).

Programs

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

Formula

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

Views

Author

Jason Kimberley, Oct 30 2011

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    LinearRecurrence[{1,7,-7},{9,16,65},40] (* Harvey P. Dale, Oct 14 2019 *)
  • PARI
    Vec(x^3*(9 + 7*x - 14*x^2) / ((1 - x)*(1 - 7*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

Formula

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

Views

Author

Jason Kimberley, Oct 30 2011

Keywords

Links

Crossrefs

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).

Programs

  • Mathematica
    LinearRecurrence[{1,8,-8},{10,18,82},30] (* Harvey P. Dale, Apr 03 2015 *)
  • PARI
    Vec(2*x^3*(5 + 4*x - 8*x^2) / ((1 - x)*(1 - 8*x^2)) + O(x^40)) \\ Colin Barker, Mar 17 2017

Formula

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

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

Original entry on oeis.org

5, 10, 19, 30, 40, 50
Offset: 2

Views

Author

N. J. A. Sloane, Jun 07 2011

Keywords

Comments

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

Links

Crossrefs

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

Formula

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

Extensions

a(2) = 5 prepended by Jason Kimberley, Jan 02 2013
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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