A033570 -id:A033570 - cp's OEIS frontend

A282513 a(n) = floor((3*n + 2)^2/24 + 1/3).

0, 1, 3, 5, 8, 12, 17, 22, 28, 35, 43, 51, 60, 70, 81, 92, 104, 117, 131, 145, 160, 176, 193, 210, 228, 247, 267, 287, 308, 330, 353, 376, 400, 425, 451, 477, 504, 532, 561, 590, 620, 651, 683, 715, 748, 782, 817, 852, 888, 925, 963

Offset: 0

Luce ETIENNE, Feb 17 2017

List of quadruples: 2*n*(3*n+1), (2*n+1)*(3*n+1), 6*n^2+8*n+3, (n+1)*(6*n+5). These terms belong to the sequences A033580, A033570, A126587 and A049452, respectively. See links for all the permutations.
After 0, subsequence of A025767.
It seems that a(n) is the smallest number of cells that need to be painted in a (n+1) X (n+1) grid, such that it has no unpainted hexominoes (see link to Kamenetsky and Pratt). - Rob Pratt, Dmitry Kamenetsky, Aug 30 2020

			Rectangular array with four columns:
.   0,   1,   3,   5;
.   8,  12,  17,  22;
.  28,  35,  43,  51;
.  60,  70,  81,  92;
. 104, 117, 131, 145, etc.
From _Rob Pratt_, Aug 30 2020: (Start)
For n = 3, painting only 2 cells would leave an unpainted hexomino, but painting the following 3 cells avoids all unpainted hexominoes:
    . . .
    . . X
    X X .
(End)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Permutations
Dmitry Kamenetsky and Rob Pratt, 10x10 grid with no unpainted hexominoes, Puzzling StackExchange, October 2019.
Rob Pratt, Optimal solution for n=11.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A033436: floor((3*n)^2/24 + 1/3).
Cf. A000326, A000567, A025767, A033570, A033580, A049452, A064412, A126587, A222017, A269064, A274221.
Cf. A130519.
Minimum number of painted cells in other n-ominoes: A337501, A337502, A337503.

[(3*n^2+4*n+4) div 8: n in [0..50]]; // Bruno Berselli, Feb 17 2017

Table[Floor[(3 n + 2)^2/24 + 1/3], {n, 0, 50}] (* or *) CoefficientList[Series[x (1 + x + x^3)/((1 + x) (1 + x^2) (1 - x)^3), {x, 0, 50}], x] (* or *) Table[(6 n^2 + 8 n + 3 + Cos[n Pi] - 4 Cos[n Pi/2])/16, {n, 0, 50}] (* or *) Table[(3 n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n) (1 + 2 I^((n + 1) (n + 2))))/16, {n, 0, 50}] (* Michael De Vlieger, Feb 17 2017 *)
LinearRecurrence[{2,-1,0,1,-2,1},{0,1,3,5,8,12},60] (* Harvey P. Dale, Aug 10 2024 *)

a(n)=(3*n^2 + 4*n + 4)\8 \\ Charles R Greathouse IV, Feb 17 2017

G.f.: x*(1 + x + x^3)/((1 + x)*(1 + x^2)*(1 - x)^3).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>5.
a(n) = floor((3*n + 2)^2/24 + 2/3).
a(n) = (6*n^2 + 8*n + 3 + (-1)^n - 2*((-1)^((2*n - 1 + (-1)^n)/4) + (-1)^((2*n + 1 - (-1)^n)/4)))/16. Therefore:
a(2*k)   = (6*k^2 + 4*k + 1 - (-1)^k)/4,
a(2*k+1) = (k + 1)*(3*k + 2)/2.
a(n) = (6*n^2 + 8*n + 3 + cos(n*Pi) - 4*cos(n*Pi/2))/16.
a(n) = (3*n + 2)^2/24 + 1/3 + (-6 + (1 + (-1)^n)*(1 + 2*i^((n+1)*(n+2))))/16, where i=sqrt(-1).
a(n) = A130519(n+3)+A130519(n+2)+A130519(n). - R. J. Mathar, Jun 23 2021

Corrected and extended by Bruno Berselli, Feb 17 2017

A104585 a(n) = (1/2) * ( 3n^2 - n(-1)^n ).

Original entry on oeis.org

0, 2, 5, 15, 22, 40, 51, 77, 92, 126, 145, 187, 210, 260, 287, 345, 376, 442, 477, 551, 590, 672, 715, 805, 852, 950, 1001, 1107, 1162, 1276, 1335, 1457, 1520, 1650, 1717, 1855, 1926, 2072, 2147, 2301, 2380, 2542, 2625, 2795, 2882, 3060, 3151, 3337, 3432, 3626

Offset: 0

Gary W. Adamson, Mar 17 2005

nonn

Previous name was: Pentagonal wave sequence of the second kind.
Even-indexed terms are pentagonal numbers with even index in A000326. Odd-indexed terms are second pentagonal numbers with odd index in A005449.
A104584, pentagonal wave sequence of the first kind; switches odd and even applications and vice versa in A104585. The pentagonal wave triangle, A104586, has A104584 in odd columns and A104585 in even columns.
Integer values of (n+1)(2n+1)/3 in order of appearance. - Wesley Ivan Hurt, Sep 17 2013
Exponents of q in the identity 1 - Sum_{n >= 0} ( q^(3*n+2)*Product_{k = 1..n} (1 - q^(4*k-1)) ) = 1 - q^2 - q^5 + q^15 + q^22 - q^40 - q^51 + + - - .... Compare with Euler's pentagonal number theorem: Product_{n >= 1} (1 - q^n) = 1 - Sum_{n >= 1} ( q^n*Product_{k = 1..n-1} (1 - q^k) ) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15  + + - - .... - Peter Bala, Dec 03 2020

			a(5) = 40 = A005449(5), a second pentagonal number.
a(6) = 51 = A000326(6), a pentagonal number.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Pentagonal number theorem.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000326, A005449, A049452, A033570, A049453, A033568, A104586.

I:=[0, 2, 5, 15, 22]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Apr 04 2013

Table[(1/2) (3 n^2 - n (-1)^n), {n, 0, 100}] (* Vincenzo Librandi, Apr 04 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,2,5,15,22},50] (* Harvey P. Dale, Sep 14 2015 *)

a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ). - Ralf Stephan, Nov 13 2010
G.f.: x*(2+3*x+6*x^2+x^3)/(1-x)^3/(1+x)^2. - Colin Barker, Feb 13 2012
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Vincenzo Librandi, Apr 04 2013
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*log(2) - Pi/sqrt(3).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) - 3*log(3). (End)

More terms from Colin Barker, Feb 13 2012

Better name, using formula from Ralf Stephan, Joerg Arndt, Sep 17 2013

A104584 a(n) = (1/2) * ( 3n^2 + n(-1)^n ).

Original entry on oeis.org

0, 1, 7, 12, 26, 35, 57, 70, 100, 117, 155, 176, 222, 247, 301, 330, 392, 425, 495, 532, 610, 651, 737, 782, 876, 925, 1027, 1080, 1190, 1247, 1365, 1426, 1552, 1617, 1751, 1820, 1962, 2035, 2185, 2262, 2420

Offset: 0

Gary W. Adamson, Mar 17 2005

Previous name was: Pentagonal wave sequence of the first kind.
Odd-indexed terms = A033570, pentagonal numbers with odd index (1, 12, 35, 70, ...). Even-indexed terms = A049453, 2nd pentagonal numbers with even index (0, 7, 26, 57, 100, ...).
Companion sequence A104585 (Pentagonal wave sequence of the second kind), switches odd with even applications and vice versa. The pentagonal wave sequence triangle A104586 has A104584 in odd columns and A104585 in even columns.
Exponents of q in the identity Sum_{n >= 0} ( q^n*Product_{k = 1..n} (1 - q^(4*k-3)) ) = 1 + q - q^7 - q^12 + q^26 + q^35 - - + + .... Compare with Euler's pentagonal number theorem: Product_{n >= 1} (1 - q^n) = 1 - Sum_{n >= 1} ( q^n*Product_{k = 1..n-1} (1 - q^k) ) = 1 - q - q^2 + q^5 + q^7 - q^12 - q^15  + + - - .... - Peter Bala, Dec 03 2020

			a(5) = 35 = A000326(5).
a(6) = 57 = A005449(6).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Pentagonal number theorem.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000326, A005449, A049453, A033568, A104585, A104586.

I:=[0, 1, 7, 12, 26]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Apr 04 2013

Table[(1/2) (3 n^2 + n (-1)^n), {n, 0, 100}] (* Vincenzo Librandi, Apr 04 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{0,1,7,12,26},50] (* Harvey P. Dale, Feb 14 2023 *)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Vincenzo Librandi, Apr 04 2013
a(n) = (1/2) * (3*n^2 + n*(-1)^n ). - Ralf Stephan, May 20 2007
G.f. -x*(1+6*x+3*x^2+2*x^3) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Jan 10 2011
From Amiram Eldar, Feb 22 2022: (Start)
Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 4*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 3*log(3) - 6. (End)

Better name, using formula from Ralf Stephan, Joerg Arndt, Sep 17 2013

A255607 Numbers n such that both 4n+1 and 6n+1 are primes.

Original entry on oeis.org

1, 3, 7, 10, 13, 18, 25, 27, 37, 45, 58, 70, 73, 87, 100, 102, 105, 112, 115, 135, 142, 153, 165, 168, 175, 177, 192, 202, 205, 213, 220, 238, 255, 258, 277, 282, 298, 300, 312, 322, 325, 352, 357, 363, 370, 373, 417, 423, 447, 465, 472, 475, 513, 520

Offset: 1

Vincenzo Librandi, Feb 28 2015

Numbers n such that A033570(2n) is semiprime.

			10 is in this sequence because 4*10+1=41 and 6*10+1=61 are primes.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A033570, A130800, A186721.

Cf. A255584: semiprimes of the form (4*n+1)*(6*n+1).

[n: n in [1..600] | IsPrime(6*n+1) and IsPrime(4*n+1)];

A255607:=n->`if`(isprime(4*n+1) and isprime(6*n+1), n, NULL): seq(A255607(n), n=1..600); # Wesley Ivan Hurt, Feb 28 2015

Select[Range[600], PrimeQ[4 # + 1] && PrimeQ[6 # + 1] &]
Select[Range[600],AllTrue[{4#,6#}+1,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 22 2020 *)

for(n=1,10^3,if(isprime(4*n+1)&&isprime(6*n+1),print1(n,", "))) \\ Derek Orr, Mar 01 2015

select( is_A255607(n)=isprime(4*n+1)&&isprime(6*n+1), [1..555]) \\ M. F. Hasler, Dec 13 2019

a(n) = A130800(n)/2.

A104586 Pentagonal wave sequence triangle.

Original entry on oeis.org

1, 7, 2, 12, 5, 1, 26, 15, 7, 2, 35, 22, 12, 5, 1, 57, 40, 26, 15, 7, 2, 70, 51, 35, 22, 12, 5, 1, 100, 77, 57, 40, 26, 15, 7, 2

Offset: 1

Gary W. Adamson, Mar 17 2005

Row sums = A086500: 1, 9, 18, 50, 75, 147, 196...

			The first few rows are:
1;
7, 2;
12, 5, 1;
26, 15, 7, 2;
35, 22, 12, 5, 1;
57, 40, 26, 15, 7, 2;
70, 51, 35, 22, 12, 5, 1;
...

Cf. A086500, A104584, A104585, A000326, A005449, A049452, A033570, A049453, A033568.

Odd columns are terms of A104584, pentagonal wave sequence of the first kind, (starting with 1): 1, 7, 12, 26, 35, 57, 70... Even columns are terms of A104585, pentagonal wave sequence of the second kind (starting with 2): 2, 5, 15, 22, 40, 51... Odd rows are pentagonal numbers (A000326) starting with "1" at the right. Even rows are second pentagonal numbers (A005449) starting with 2 at the right. The triangle is extracted from a matrix product A * B, A = [1; 1, 2; 1, 2, 1; 1, 2, 1, 2;...], B = [1; 3, 1; 5, 3, 1; 7, 5, 3, 1;...] (both infinite lower triangular matrices, with the rest zeros).

A135712 a(n) = (4n^3 + 11n^2 + 9*n + 2)/2.

Original entry on oeis.org

1, 13, 48, 118, 235, 411, 658, 988, 1413, 1945, 2596, 3378, 4303, 5383, 6630, 8056, 9673, 11493, 13528, 15790, 18291, 21043, 24058, 27348, 30925, 34801, 38988, 43498, 48343, 53535, 59086, 65008, 71313, 78013, 85120, 92646, 100603, 109003, 117858, 127180

Offset: 0

N. J. A. Sloane, Mar 05 2008

Binomial transform yields 1,12,23,12,0,0,0,0,0,0,.. - R. J. Mathar, Apr 21 2008

J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Bisection of A002717 (odd part).

Partial sums of A033570. - Bruno Berselli, Nov 28 2013

Table[(4*n^3 + 11*n^2 + 9*n + 2)/2,{n,0,25}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1,13,48,118}, 25] (* G. C. Greubel, Oct 29 2016 *)

G.f.: (1 + 9*x + 2*x^2) / (1-x)^4. - R. J. Mathar, Apr 21 2008
From G. C. Greubel, Oct 29 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
E.g.f.: (1/2)*(2 + 24*x + 23*x^2 + 4*x^3)*exp(x). (End)
a(n) = ((2*n+1)*(2*n+3)*(4*n+3) - 1)/8 = (n+1)*(4*n^2 + 7*n + 2)/2, for  n >= 0. See the Conway and Guy reference. - Wolfdieter Lang, Apr 16 2020

A249127 a(n) = n * floor(3*n/2).

Original entry on oeis.org

0, 1, 6, 12, 24, 35, 54, 70, 96, 117, 150, 176, 216, 247, 294, 330, 384, 425, 486, 532, 600, 651, 726, 782, 864, 925, 1014, 1080, 1176, 1247, 1350, 1426, 1536, 1617, 1734, 1820, 1944, 2035, 2166, 2262, 2400, 2501, 2646, 2752, 2904, 3015, 3174, 3290, 3456, 3577, 3750, 3876, 4056, 4187, 4374, 4510

Offset: 0

Karl V. Keller, Jr., Oct 21 2014

Union of A033570, that is (2*n+1)*(3*n+1), and A033581, that is 6*n^2.

			For n=5, a(n) = 5*floor(15/2) = 5*7 = 35.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Number
Wikipedia, Pentagonal number
Wikipedia, Platonic Solid
Wolfram MathWorld, Platonic Solid
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A032766, A033581 (6*n^2), A033570 (2*n+1)*(3*n+1), A001318 (n*(3*n-1)/2).

[n*Floor(3*n/2): n in [0..60]]; // Vincenzo Librandi, Oct 22 2014

seq(n*floor(3*n/2), n=0..100); # Robert Israel, Oct 26 2014

Table[n Floor[3 n/2], {n, 0, 100}] (* Vincenzo Librandi, Oct 22 2014 *)

a(n)=3*n\2*n \\ Charles R Greathouse IV, Oct 21 2014

concat(0, Vec(-x*(2*x^3+4*x^2+5*x+1)/((x-1)^3*(x+1)^2) + O(x^100))) \\ Colin Barker, Oct 22 2014

from math import *
{print(int(n*floor(3*n/2)),end=', ') for n in range(101)}

a(n) = n * floor(3n/2) = n * A032766(n).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Colin Barker, Oct 22 2014
G.f.: -x*(2*x^3+4*x^2+5*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Oct 22 2014
a(n) = 3/2 * n^2 + ((-1)^n-1) * n/4. E.g.f.: ((3/2)*x^2+(5/4)*x)*exp(x)-(x/4)*exp(-x). - Robert Israel, Oct 26 2014

A330409 Semiprimes of the form p(6p - 1).

Original entry on oeis.org

22, 51, 145, 287, 1717, 2147, 3151, 5017, 11051, 13207, 16801, 20827, 26867, 63551, 68587, 71177, 76501, 96647, 112477, 147737, 159251, 232657, 237407, 308947, 314417, 342487, 433897, 480251, 587501, 602617, 722107, 772927, 834401, 861467, 879751, 907537, 945257, 1155887, 1177051

Offset: 1

M. F. Hasler, Dec 13 2019

nonn

			A158015(1) = 2 is the smallest prime p such that 6p - 1 = 12 - 1 = 11 is also prime, whence a(1) = A049452(2) = 2*(6*2 - 1) = 22.
prime(5) = 11 is the smallest prime not in A024898 or A158015, because 6p - 1 is not a prime, therefore A049452(11) = 11*(6*11 - 1) is not in the sequence, and idem for A049452(13).
But prime(7) = 17 is in A024898 and A158015, so a(5) = A024898(A158015(5)) = A024898(17) = 17*(6*17 - 1).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A024898 (6n-1 is prime), A158015 (primes), A049452 = {n(6n-1)}.

Complement of A255584 = A033570(A130800) (semiprimes (2n+1)(3n+1)) in A245365 (primes of the form n(3n-1)/2).

Select[Table[p(6p-1),{p,500}],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 27 2022 *)

[p*(6*p-1) | p<-primes(99), isprime(6*p-1)]

a(n) = A049452(A158015(n)) = p(6p - 1) with p = A158015(n).

A330410 a(n) = 6*prime(n) - 1.

Original entry on oeis.org

11, 17, 29, 41, 65, 77, 101, 113, 137, 173, 185, 221, 245, 257, 281, 317, 353, 365, 401, 425, 437, 473, 497, 533, 581, 605, 617, 641, 653, 677, 761, 785, 821, 833, 893, 905, 941, 977, 1001, 1037, 1073, 1085, 1145, 1157, 1181, 1193, 1265, 1337, 1361, 1373, 1397, 1433, 1445

Offset: 1

M. F. Hasler, Dec 13 2019

nonn

Composite terms are a(k) with k in {5, 6, 11, 12, 13, 18, 20, 21, ...} = indices of primes missing in A158015. Primes are A016969(A158015 - 1).

Cf. A000040 (primes), A016969 (6n+5), A024898 (6n-1 is prime), A158015 (primes in A024898), A049452 = {n(6n-1)}, A255584 = A033570(A130800) (semiprimes (2n+1)(3n+1)), A245365 (primes of the form n(3n-1)/2).

PARI
```
apply( a(n)=6*prime(n)-1, [1..99])
```
PARI
```
apply( n->6*n-1, primes(99))
```

a(n) = A016969(A000040(n)-1) = 6p - 1 with p = A000040(n) = prime(n).

cp's OEIS Frontend

A282513 a(n) = floor((3*n + 2)^2/24 + 1/3).

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A104585 a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ).

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A104584 a(n) = (1/2) * ( 3*n^2 + n*(-1)^n ).

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A255607 Numbers n such that both 4*n+1 and 6*n+1 are primes.

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A104586 Pentagonal wave sequence triangle.

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A135712 a(n) = (4*n^3 + 11*n^2 + 9*n + 2)/2.

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A249127 a(n) = n * floor(3*n/2).

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A104585 a(n) = (1/2) * ( 3n^2 - n(-1)^n ).

A104584 a(n) = (1/2) * ( 3n^2 + n(-1)^n ).

A255607 Numbers n such that both 4n+1 and 6n+1 are primes.

A135712 a(n) = (4n^3 + 11n^2 + 9*n + 2)/2.