A085930 -id:A085930 - cp's OEIS frontend

A085931 Leading diagonal of A085930.

2, 4, 12, 17, 23, 39, 48, 58, 69, 95, 109, 124, 140, 157, 195, 215, 236, 258, 281, 305, 357, 384, 412, 441, 471, 502, 534, 602, 637, 673, 710, 748, 787, 827, 868, 954, 998, 1043, 1089, 1136, 1184, 1233, 1283, 1334, 1440, 1494, 1549, 1605, 1662, 1720, 1779

Offset: 1

Jason Earls and Amarnath Murthy, Jul 14 2003

Gheorghe Coserea, Table of n, a(n) for n = 1..10000
T. Forbes, Prime k-tuplets

Cf. A085930.

a(n) = my(x = (sqrtint(1+8*n)-1)\2); (x+n)*(x+n+1)/2 - n;
vector(51, n, a(n))  \\ Gheorghe Coserea, Mar 25 2016

Offset corrected by Gheorghe Coserea, Mar 25 2016

A334563 a(n) is the maximum number of 4-cycles possible in an n-vertex planar graph.

Original entry on oeis.org

0, 0, 0, 0, 3, 9, 16, 24, 33, 43, 54, 66, 79, 93, 108, 124, 141, 159, 178, 198, 219, 241, 264, 288, 313, 339, 366, 394, 423, 453, 484, 516, 549, 583, 618, 654, 691, 729, 768, 808, 849, 891, 934, 978, 1023, 1069, 1116, 1164, 1213, 1263, 1314, 1366, 1419, 1473, 1528

Offset: 0

Stefano Spezia, May 06 2020

For n > 1, the parity changes every two terms like in A000217 for n > 0.

Debarun Ghosh, Ervin Győri, Oliver Janzer, Addisu Paulos, Nika Salia, Oscar Zamora, The maximum number of induced C_5's in a planar graph, arXiv:2004.01162 [math.CO], 2020.
S. L. Hakimi and E. F. Schmeichel, On the number of cycles of length k in a maximal planar graph, J. Graph Theory 3 (1979): 69-86.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A079566, A085930, A328180.

I:=[0, 0, 0, 0, 3, 9, 16]; [n le 7 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..55]];

Join[{0, 0, 0, 0}, Table[(n^2+3n-22)/2, {n, 4, 54}]]

my(x='x + O('x^55)); concat([0, 0, 0, 0], Vec(serlaplace(11 + 9*x + 3*x^2 + x^3/3 + exp(x)*(x^2 + 4*x - 22)/2)))

(x^4*(3 - 2*x^2)/(1 - x)^3).series(x, 55).coefficients(x, sparse=False)

O.g.f.: x^4*(3 - 2*x^2)/(1 - x)^3.
E.g.f.: 11 + 9*x + 3*x^2 + x^3/3 + exp(x)*(x^2 + 4*x - 22)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 6.
a(n) = (n^2 + 3*n - 22)/2 for n > 3 and 0 otherwise (see Theorem 2 in Hakimi and Schmeichel).
a(n) = a(n-1) + n + 1 for n > 4.
a(n) = A000217(n) + n - 11 for n > 3.
a(n) = A085930(12, n-3) for 3 < n < 16. - Michel Marcus, Jun 01 2020

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A085931 Leading diagonal of A085930.

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A334563 a(n) is the maximum number of 4-cycles possible in an n-vertex planar graph.

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