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.

User: Stephen Bartell

Stephen Bartell's wiki page.

Stephen Bartell has authored 4 sequences.

A386292 The maximum possible number of 9-cycles in an outerplanar graph on n vertices.

Original entry on oeis.org

1, 5, 11, 23, 34, 52, 68, 94, 109, 131, 151, 181, 200, 226, 250, 284, 299, 321, 341, 371, 390, 416, 440, 474, 489, 511, 531, 561, 580, 606, 630, 664, 679, 701, 721, 751, 770, 796, 820, 854, 869, 891, 911, 941, 960, 986, 1010, 1044, 1059, 1081, 1101, 1131, 1150
Offset: 9

Views

Author

Stephen Bartell, Jul 17 2025

Keywords

Comments

Excluding the first 7 terms, the first differences are 8-periodic: 15, 22, 20, 30, 19, 26, 24, 34.

Links

Crossrefs

Cf. A385869, A386226.

Formula

a(n) ~ (95/4)*n.
G.f.: x^9*(8*x^15+8*x^14+8*x^13+8*x^12+18*x^11+14*x^10+18*x^9+14*x^8+26*x^7+16*x^6 + 18*x^5 + 11*x^4+12*x^3+6*x^2+4*x+1) / ((x+1)*(x^2+1)*(x^4+1)*(x-1)^2). - Alois P. Heinz, Jul 17 2025

A386226 The maximum possible number of 8-cycles in an outerplanar graph on n vertices.

Original entry on oeis.org

1, 4, 10, 16, 27, 34, 44, 54, 69, 76, 86, 96, 111, 118, 128, 138, 153, 160, 170, 180, 195, 202, 212, 222, 237, 244, 254, 264, 279, 286, 296, 306, 321, 328, 338, 348, 363, 370, 380, 390, 405, 412, 422, 432, 447, 454, 464, 474, 489, 496, 506, 516, 531, 538, 548
Offset: 8

Views

Author

Stephen Bartell, Jul 15 2025

Keywords

Comments

For n >= 12, the first differences are 4-periodic: 7,10,10,15,7,10,10,15,...

Links

Programs

  • Mathematica
    Drop[CoefficientList[Series[x^8*(4*x^8+4*x^7+4*x^6+4*x^5+10*x^4+6*x^3+6*x^2+3*x+1)/((x+1)*(x^2+1)*(x-1)^2),{x,0,62}],x],8] (* James C. McMahon, Jul 16 2025 *)

Formula

a(n) = 10*n - 99 + 5*floor(n/4) - 3*floor((n+3)/4) for n >= 12.
a(n) ~ (21/2)*n.
G.f.: x^8*(4*x^8+4*x^7+4*x^6+4*x^5+10*x^4+6*x^3+6*x^2+3*x+1)/((x+1)*(x^2+1)*(x-1)^2). - Alois P. Heinz, Jul 15 2025

A385869 The maximum possible number of 7-cycles in an outerplanar graph on n vertices.

Original entry on oeis.org

1, 4, 7, 12, 17, 24, 27, 32, 37, 44, 47, 52, 57, 64, 67, 72, 77, 84, 87, 92, 97, 104, 107, 112, 117, 124, 127, 132, 137, 144, 147, 152, 157, 164, 167, 172, 177, 184, 187, 192, 197, 204, 207, 212, 217, 224, 227, 232, 237, 244, 247, 252, 257, 264, 267, 272, 277, 284, 287, 292, 297
Offset: 7

Views

Author

Stephen Bartell, Jul 10 2025

Keywords

Links

Programs

  • Mathematica
    Drop[CoefficientList[Series[x^7*(4*x^5+4*x^4+5*x^3+3*x^2+3*x+1)/((x+1)*(x^2+1)*(x-1)^2),{x,0,67}],x],7] (* James C. McMahon, Jul 16 2025 *)

Formula

For n >= 2, a(4n) = 20n-36, a(4n+1) = 20n-33, a(4n+2) = 20n-28, a(4n+3) = 20n-23.
For n >= 8, a(n) = 5n - 36 - 2sgn(mod(n,4))
Excluding a(7), the first differences are 4-periodic: 3,5,5,7,3,5,5,7,...
G.f.: x^7*(4*x^5+4*x^4+5*x^3+3*x^2+3*x+1)/((x+1)*(x^2+1)*(x-1)^2). - Alois P. Heinz, Jul 15 2025

A372303 Primes p such that there exists prime q < p for which p*q^2 + 1 is divisible by q^2 + p and 1 + p.

Original entry on oeis.org

7, 11, 23, 31, 41, 47, 59, 71, 79, 131, 137, 151, 167, 239, 311, 359, 443, 461, 701, 839, 911, 1021, 1039, 1367, 1721, 1847, 2207, 2351, 2551, 2861, 3191, 3719, 4019, 4691, 4759, 5039, 5167, 5279, 6971, 7481, 7853, 7919, 9311, 9619, 9689, 10607, 10739, 11447
Offset: 1

Views

Author

Stephen Bartell, May 22 2024

Keywords

Examples

			For n=4, a(4)=31 and q=17 satisfy the desired divisibilities.

Programs

  • Maple
    P:= select(isprime,[2,seq(i,i=3..100000,2)]):
nP:= nops(P);
R:= NULL:
for i from 2 to nP do
  p:= P[i];
  for j from 1 to i-1 do
    q:= P[j];
    if p*q^2 + 1 mod ilcm(p+1, q^2+p) = 0 then
      R:= R,p;
      break
    fi
od od:
R;

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