A271713 -id:A271713 - cp's OEIS frontend

A141631 a(n) = 3n^2 - 4n + 3.

2, 7, 18, 35, 58, 87, 122, 163, 210, 263, 322, 387, 458, 535, 618, 707, 802, 903, 1010, 1123, 1242, 1367, 1498, 1635, 1778, 1927, 2082, 2243, 2410, 2583, 2762, 2947, 3138, 3335, 3538, 3747, 3962, 4183, 4410, 4643, 4882, 5127, 5378, 5635, 5898, 6167, 6442

Offset: 1

Paul Curtz, Aug 28 2008

First bisection of A133146.

Also first bisection of A271713. - Bruno Berselli, Mar 19 2021

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000004 (third differences), A010722 (second differences).

Cf. A000290, A016969, A067998, A133146, A271713.

Table[3 n^2 - 4 n + 3, {n, 50}] (* Harvey P. Dale, Oct 28 2012 *)

a(n)=3*n^2-4*n+3 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A133146(2*n-2) = (n - 2)^2 + (n - 1)*(n + 1) + n^2.
First differences: a(n+1) - a(n) = A016969(n-1).
G.f.: x*(2 + x + 3*x^2)/(1 - x)^3. - R. J. Mathar, Oct 15 2008
a(n) = 6*n + a(n-1) - 7 for n > 1, a(1)=2. - Vincenzo Librandi, Nov 25 2010
a(n) = 2*A000290(n)^2 + A067998(n-1) = 2*n^2 + (n - 1)*(n - 3). - L. Edson Jeffery, Nov 30 2013
From Elmo R. Oliveira, Nov 13 2024: (Start)
E.g.f.: exp(x)*(3*x^2 - x + 3) - 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Edited and extended by R. J. Mathar, Oct 15 2008

A352666 Maximum number of induced copies of the claw graph K_{1,3} in an n-node graph.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 20, 40, 70, 112, 176, 261, 372, 520, 704, 935, 1220, 1560, 1976, 2464, 3038, 3710, 4480, 5376, 6392, 7548, 8856, 10320, 11970, 13800, 15840, 18095, 20580, 23320, 26312, 29601, 33176, 37072, 41300, 45875, 50830, 56160, 61920, 68096, 74732

Offset: 1

Pontus von Brömssen, Mar 26 2022

nonn

The sequence (a(n)/binomial(n,4)) is decreasing for n >= 4 and converges to 1/2, the inducibility of the claw graph.

Brown and Sidorenko (1994) prove that a bipartite optimal graph (i.e., an n-node graph with a(n) induced claw graphs) exists for all n. For n >= 2, the size k of the smallest part of an optimal bipartite graph K_{k,n-k} is one of the two integers closest to n/2 - sqrt(3*n/4-1), and a(n) = binomial(k,3)*(n-k) + binomial(n-k,3)*k. Both are optimal if and only if n is in A271713. For 7 <= n <= 10 (and, trivially, n = 3), the tripartite graph K_{1,1,n-2} is also optimal.

Jason I. Brown and Alexander Sidorenko, The inducibility of complete bipartite graphs, Journal of Graph Theory 18 (1994), 629-645.

Cf. A271713.

Maximum number of induced copies of other graphs: A028723 (4-node cycle), A111384 (3-node path), A352665 (4-node path), A352667 (paw graph), A352668 (diamond graph), A352669 (cycles).

from math import comb,isqrt
def A352666(n):
    if n <= 1: return 0
    r = isqrt(3*n-4)
    k0 = (n-r-1)//2
    return max(comb(k,3)*(n-k)+comb(n-k,3)*k for k in (k0,k0+1))

A271675 Numbers m such that 3*m + 4 is a square.

Original entry on oeis.org

0, 4, 7, 15, 20, 32, 39, 55, 64, 84, 95, 119, 132, 160, 175, 207, 224, 260, 279, 319, 340, 384, 407, 455, 480, 532, 559, 615, 644, 704, 735, 799, 832, 900, 935, 1007, 1044, 1120, 1159, 1239, 1280, 1364, 1407, 1495, 1540, 1632, 1679, 1775, 1824, 1924, 1975, 2079, 2132, 2240, 2295, 2407

Offset: 1

Juri-Stepan Gerasimov, Apr 12 2016

7 is the unique prime in this sequence. If m is in this sequence, then 3*m + 4 = k^2 for k is nonzero integer, that is, m = (k^2 - 4)/3 = (k-2)*(k+2)/3. So m can be only prime if one of divisors is prime and another one is 1. Otherwise there should be more than 1 prime divisors, that is n must be composite. - Altug Alkan, Apr 12 2016
From Ray Chandler, Apr 12 2016: (Start)
Square roots of resulting squares gives A001651 (with a different starting point).
Sequence is the union of (positive terms) in A140676 and A270710. (End)
The sequence terms are the exponents in the expansion of Sum_{n >= 0} q^n*(1 - q)*(1 - q^3)*...*(1 - q^(2*n+1)) = 1 - q^4 - q^7 + q^15 + q^20 - q^32 - q^50 + + - - .... - Peter Bala, Dec 19 2024

			a(4) = 32 because 3*32 + 4 = 100 = 10*10.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001651, A140676, A270710.

Cf. numbers n such that 3*n + k is a square: A120328 (k=-6), A271713 (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), this sequence (k=4), A100536 (k=6).

Magma
```
[n: n in [0..4000] | IsSquare(3*n+4)];
```

Select[Range[0,2500], IntegerQ@ Sqrt[3 # + 4] &] (* Michael De Vlieger, Apr 12 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{0,4,7,15,20},60] (* Harvey P. Dale, Dec 09 2016 *)

from gmpy2 import is_square
for n in range(0,10**5):
    if(is_square(3*n+4)):print(n)
# Soumil Mandal, Apr 12 2016

O.g.f.: x^2*(4 + 3*x - x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: 1 + (1 - 2*x)*exp(-x)/8 - 3*(3 - 4*x - 2*x^2)*exp(x)/8.
a(n) = A001082(n+1) - 1 = (6*n*(n+1) + (2*n + 1)*(-1)^n - 1)/8 - 1. Therefore: a(2*k+1) = k*(3*k+4), a(2*k) = (k+1)*(3*k-1).
Sum_{n>=2} 1/a(n) = 19/16 - Pi/(4*sqrt(3)). - Amiram Eldar, Jul 26 2024

Edited and extended by Bruno Berselli, Apr 12 2016

A271723 Numbers k such that 3*k - 8 is a square.

Original entry on oeis.org

3, 4, 8, 11, 19, 24, 36, 43, 59, 68, 88, 99, 123, 136, 164, 179, 211, 228, 264, 283, 323, 344, 388, 411, 459, 484, 536, 563, 619, 648, 708, 739, 803, 836, 904, 939, 1011, 1048, 1124, 1163, 1243, 1284, 1368, 1411, 1499, 1544, 1636, 1683, 1779, 1828, 1928, 1979, 2083, 2136, 2244, 2299

Offset: 1

Juri-Stepan Gerasimov, Apr 13 2016

Square roots of resulting squares gives A001651. - Ray Chandler, Apr 14 2016

			a(1) = 3 because 3*3 - 8 = 1^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A001651.

Cf. numbers n such that 3*n + k is a square: this sequence (k=-8), A120328 (k=-6), A271713 (k=-5), A056107 (k=-3), A257083 (k=-2), A033428 (k=0), A001082 (k=1), A080663 (k=3), A271675 (k=4), A100536 (k=6), A271741 (k=7), A067725 (k=9).

Magma
```
[n: n in [1..2400] | IsSquare(3*n-8)];
```

seq(seq(((3*m+k)^2+8)/3, k=1..2),m=0..50); # Robert Israel, Dec 05 2016

Select[Range@ 2400, IntegerQ@ Sqrt[3 # - 8] &] (* Bruno Berselli, Apr 14 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{3,4,8,11,19},60] (* Harvey P. Dale, Oct 02 2020 *)

from gmpy2 import is_square
[n for n in range(3000) if is_square(3*n-8)] # Bruno Berselli, Dec 05 2016

[(6*(n-1)*n-(2*n-1)*(-1)**n+23)/8 for n in range(1, 60)] # Bruno Berselli, Dec 05 2016

From Ilya Gutkovskiy, Apr 13 2016: (Start)
G.f.: x*(3 + x - 2*x^2 + x^3 + 3*x^4)/((1 - x)^3*(1 + x)^2).
a(n) = (6*(n - 1)*n - (2*n - 1)*(-1)^n + 23)/8. (End)

A271740 a(n) = 3n^2 - 2n + 2.

Original entry on oeis.org

2, 3, 10, 23, 42, 67, 98, 135, 178, 227, 282, 343, 410, 483, 562, 647, 738, 835, 938, 1047, 1162, 1283, 1410, 1543, 1682, 1827, 1978, 2135, 2298, 2467, 2642, 2823, 3010, 3203, 3402, 3607, 3818, 4035, 4258, 4487, 4722, 4963, 5210, 5463, 5722, 5987, 6258, 6535, 6818, 7107, 7402, 7703

Offset: 0

Ray Chandler, Apr 13 2016

3*a(n) - 5 is a square. - Vincenzo Librandi, Apr 13 2016

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A141631, A271713.

[3*n^2 - 2*n + 2: n in [0..50]]; // Bruno Berselli, Apr 13 2016

A271740:=n->3*n^2 - 2*n + 2: seq(A271740(n), n=0..100); # Wesley Ivan Hurt, Apr 13 2016

Mathematica
```
Table[3 n^2 - 2 n + 2, {n, 0, 50}]
```

a(n) = 3*n^2 - 2*n + 2; \\ Altug Alkan, Apr 13 2016

A271713 = A141631 union A271740.
From Bruno Berselli, Apr 13 2016: (Start)
O.g.f.: (3 + x + 2*x^2)/(1 - x)^3.
E.g.f.: (3 + 7*x + 3*x^2)*exp(x). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Wesley Ivan Hurt, Apr 13 2016

Definition changed so sequence starts one term earlier. Some formulas may need adjusting. - N. J. A. Sloane, Jun 22 2021

cp's OEIS Frontend

A141631 a(n) = 3*n^2 - 4*n + 3.

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A352666 Maximum number of induced copies of the claw graph K_{1,3} in an n-node graph.

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A271675 Numbers m such that 3*m + 4 is a square.

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A271723 Numbers k such that 3*k - 8 is a square.

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A271740 a(n) = 3*n^2 - 2*n + 2.

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Maple

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A141631 a(n) = 3n^2 - 4n + 3.

A271740 a(n) = 3n^2 - 2n + 2.