A184005 -id:A184005 - cp's OEIS frontend

A184004 a(n) = n + floor(sqrt(4n/3)); complement of A184005.

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90

Offset: 1

Clark Kimberling, Jan 08 2011

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A184005.

a=4/3; b=0;
Table[n+Floor[(a*n+b)^(1/2)],{n,80}]
Table[n-1+Ceiling[(n*n-b)/a],{n,60}]

for(n=1, 100, print1(n + floor(sqrt(4*n/3)), ", ")) \\ G. C. Greubel, Jul 22 2017

a(n) = n + sqrtint(4*n/3); \\ Michel Marcus, Oct 01 2024

from math import isqrt
def A184004(n): return n+isqrt((n<<2)//3) # Chai Wah Wu, Oct 01 2024

A270710 a(n) = 3n^2 + 2n - 1.

Original entry on oeis.org

-1, 4, 15, 32, 55, 84, 119, 160, 207, 260, 319, 384, 455, 532, 615, 704, 799, 900, 1007, 1120, 1239, 1364, 1495, 1632, 1775, 1924, 2079, 2240, 2407, 2580, 2759, 2944, 3135, 3332, 3535, 3744, 3959, 4180, 4407, 4640, 4879, 5124, 5375, 5632, 5895, 6164, 6439, 6720, 7007, 7300, 7599

Offset: 0

Ilya Gutkovskiy, Mar 22 2016

In general, the ordinary generating function for the values of quadratic polynomial p*n^2 + q*n + k, is (k + (p + q - 2*k)*x + (p - q + k)*x^2)/(1 - x)^3.
From Bruno Berselli, Mar 25 2016: (Start)
This sequence and A140676 provide all integer m such that 3*m + 4 is a square.
Numbers related to A135713 by A135713(n) = n*a(n) - Sum_{k=0..n-1} a(k).
After -1, second bisection of A184005. (End)

			a(0) = 3*0^2 + 2*0 - 1 = -1;
a(1) = 3*1^2 + 2*1 - 1 =  4;
a(2) = 3*2^2 + 2*2 - 1 = 15;
a(3) = 3*3^2 + 2*3 - 1 = 32, etc.

Ilya Gutkovskiy, Examples of the ordinary generating function for the values of quadratic polynomial.
Leo Tavares, Triple Diamond Illustration.
Eric Weisstein's World of Mathematics, Quadratic Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033428, A045944, A056105, A056109, A060747, A135713, A140676, A184005.

Cf. A000290.

List([0..50], n -> 3*n^2+2*n-1); # Bruno Berselli, Feb 16 2018

[3*n^2+2*n-1: n in [0..50]]; // Bruno Berselli, Mar 25 2016

Table[3 n^2 + 2 n - 1, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {-1, 4, 15}, 51]

makelist(3*n^2+2*n-1, n, 0, 50); /* Bruno Berselli, Mar 25 2016 */

Vec((-1 + 7*x)/(1 - x)^3 + O(x^60)) \\ Michel Marcus, Mar 22 2016

lista(nn) = {for(n=0, nn, print1(3*n^2 + 2*n - 1, ", ")); } \\ Altug Alkan, Mar 25 2016

vector(50, n, n--; 3*n^2+2*n-1) \\ Bruno Berselli, Mar 25 2016

[3*n^2+2*n-1 for n in (0..50)] # Bruno Berselli, Mar 25 2016

G.f.: (-1 + 7*x)/(1 - x)^3.
E.g.f.: exp(x)*(-1 + 5*x + 3*x^2).
a(n)  = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n)  = A033428(n) + A060747(n).
a(n)  = A045944(n) - 1 = A056109(n) - 2.
a(-n) = A140676(n-1), with A140676(-1) = -1.
Sum_{n>=0} 1/a(n) = 3*(log(3) - 2)/8 - Pi/(8*sqrt(3)) = -0.564745312278736...
a(n) = Sum_{i = n-1..2*n-1} (2*i + 1). - Bruno Berselli, Feb 16 2018
a(n) = A000290(n+1) + 2*A000290(n) - 2. - Leo Tavares, May 28 2023
Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + 3/4. - Amiram Eldar, Jul 20 2023

A179741 a(n) = (2n+1)(6*n-1).

Original entry on oeis.org

-1, 15, 55, 119, 207, 319, 455, 615, 799, 1007, 1239, 1495, 1775, 2079, 2407, 2759, 3135, 3535, 3959, 4407, 4879, 5375, 5895, 6439, 7007, 7599, 8215, 8855, 9519, 10207, 10919, 11655, 12415, 13199, 14007, 14839, 15695, 16575, 17479, 18407

Offset: 0

Paul Curtz, Jan 10 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A061037, A077591, A184005.

[(2*n+1)*(6*n-1): n in [0..50]]; // Vincenzo Librandi, Aug 04 2011

Table[12n^2+4n-1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{-1,15,55},40] (* Harvey P. Dale, Dec 17 2013 *)

a(n)=(2*n+1)*(6*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 24*n + 16.
a(n) = 2*a(n-1) - a(n-2) + 16.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A077591(n+1) + A061037(2*n-1).
From Bruno Berselli, Jan 25 2011: (Start)
G.f.: (-1 +18*x +7*x^2)/(1-x)^3.
a(n) = A184005(4*n) (n>0). (End)
E.g.f.: (-1 + 16*x + 12*x^2)*exp(x). - G. C. Greubel, Jul 22 2017
From Amiram Eldar, Oct 08 2023: (Start)
Sum_{n>=1} 1/a(n) = (3*log(3) - Pi*sqrt(3) + 4)/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = (3*Pi - 2*sqrt(3)*log(sqrt(3)+2) - 4)/16. (End)

Edited by N. J. A. Sloane, Jan 12 2011

A185918 a(n) = 12n^2 - 2n - 1.

Original entry on oeis.org

-1, 9, 43, 101, 183, 289, 419, 573, 751, 953, 1179, 1429, 1703, 2001, 2323, 2669, 3039, 3433, 3851, 4293, 4759, 5249, 5763, 6301, 6863, 7449, 8059, 8693, 9351, 10033, 10739, 11469, 12223, 13001, 13803, 14629, 15479, 16353, 17251, 18173, 19119, 20089, 21083, 22101, 23143, 24209

Offset: 0

Paul Curtz, Feb 08 2011

The second quadrisection of A184005(n-1) is A179741(n).
The first quadrisection of A184005(n-1) is a(n).
Sequence found by reading the line from -1, in the direction -1, 9, ..., in the square spiral whose vertices are -1 together with the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001082, A154106, A179741, A184005.

[-1-2*n+12*n^2: n in [0..80] ]; // Vincenzo Librandi, Feb 09 2011

A185918:=n->12*n^2-2*n-1: seq(A185918(n), n=0..60); # Wesley Ivan Hurt, Jan 31 2017

Table[12n^2-2n-1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{-1,9,43},50] (* Harvey P. Dale, May 20 2012 *)

a(n)=12*n^2-2*n-1 \\ Charles R Greathouse IV, Dec 21 2011

a(n) = A184005(4*n-1). [corrected by R. J. Mathar, Aug 24 2011]
a(n) = a(n-1) + 24*n - 14.
a(n) = 2*a(n-1) - a(n) + 24.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(1+x)*(13*x-1) / (x-1)^3. - R. J. Mathar, Aug 24 2011
a(n) = A154106(n-1) - 2, n >= 1. - Omar E. Pol, Jul 19 2012
E.g.f.: (12*x^2 + 10*x -1)*exp(x). - G. C. Greubel, Jul 22 2017

More terms from Vincenzo Librandi, Feb 09 2011

A281151 a(n) = floor(4n(n+1)/5).

Original entry on oeis.org

0, 1, 4, 9, 16, 24, 33, 44, 57, 72, 88, 105, 124, 145, 168, 192, 217, 244, 273, 304, 336, 369, 404, 441, 480, 520, 561, 604, 649, 696, 744, 793, 844, 897, 952, 1008, 1065, 1124, 1185, 1248, 1312, 1377, 1444, 1513, 1584, 1656, 1729, 1804, 1881, 1960, 2040, 2121, 2204, 2289

Offset: 0

Bruno Berselli, Jan 16 2017

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Subsequence of A047462.
Partial sums of A047486.
Cf. A184005: n^2 - floor((n-2)^2/4).
Cf. sequences with formula floor(k*n*(n+1)/(k+1)): A000217 (k=1), A143978 (k=2), A281026 (k=3), this sequence (k=4), A194275 (k=5).

Magma
```
[4*n*(n+1) div 5: n in [0..60]];
```
Mathematica
```
Table[Floor[4 n (n + 1)/5], {n, 0, 60}]
```

makelist(floor(4*n*(n+1)/5), n, 0, 60);

PARI
```
vector(60, n, n--; floor(4*n*(n+1)/5))
```
Python
```
[int(4*n*(n+1)/5) for n in range(60)]
```

[floor(4*n*(n+1)/5) for n in range(60)]

O.g.f.: x*(1 + x^2)*(1 + x)^2/((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(-n-1) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) = a(n-5) + 8*(n-2).
a(5*k+r) = 20*k^2 + 4*(2*r+1)*k + r^2, where 0 <= r <= 4. Example: for r=3, a(5*k+3) = (2*k+1)*(10*k+9), which gives: 9, 57, 145, 273, 441, 649 etc. Also, a(n) belongs to A047462, in fact: for r = 0 or 4, a(n) == 0 (mod 8); for r = 1 or 3, a(n) == 1 (mod 8); for r = 2, a(n) == 4 (mod 8).
a(n) = a(-n) + A047462(n).
a(n) = n^2 - floor((n-2)^2/5).

cp's OEIS Frontend

A184004 a(n) = n + floor(sqrt(4n/3)); complement of A184005.

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A270710 a(n) = 3*n^2 + 2*n - 1.

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A179741 a(n) = (2*n+1)*(6*n-1).

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A185918 a(n) = 12*n^2 - 2*n - 1.

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A281151 a(n) = floor(4*n*(n+1)/5).

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Magma

Mathematica

Maxima

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Sage

Formula

A270710 a(n) = 3n^2 + 2n - 1.

A179741 a(n) = (2n+1)(6*n-1).

A185918 a(n) = 12n^2 - 2n - 1.

A281151 a(n) = floor(4n(n+1)/5).