A255844 -id:A255844 - cp's OEIS frontend

A255843 a(n) = 2*n^2 + 4.

4, 6, 12, 22, 36, 54, 76, 102, 132, 166, 204, 246, 292, 342, 396, 454, 516, 582, 652, 726, 804, 886, 972, 1062, 1156, 1254, 1356, 1462, 1572, 1686, 1804, 1926, 2052, 2182, 2316, 2454, 2596, 2742, 2892, 3046, 3204, 3366, 3532, 3702, 3876, 4054, 4236, 4422

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=2 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.

Equivalently, numbers m such that 2*m - 8 is a square.

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

Cf. A059100.
Cf. unsigned A147973: numbers of the form 2*m^2-4.
Cf. sequences of the form 2*m^2+2*k: A005893 (k=1), this sequence (k=2), A255844 (k=3), A155966 (k=4), A255845 (k=5), A255842 (k=6), A255846 (k=7), A255847 (k=8), A255848 (k=9).

Magma
```
[2*n^2+4: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 4, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+4)
```
Sage
```
[2*n^2+4 for n in (0..50)]
```

G.f.: 2*(2 - 3*x + 3*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A059100(n).
a(n) = a(n-1) + 4n - 2. - Bob Selcoe, Mar 25 2020
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(2)*Pi*coth(sqrt(2)*Pi))/8.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(2)*Pi*cosech(sqrt(2)*Pi))/8. (End)
E.g.f.: 2*exp(x)*(2 + x + x^2). - Stefano Spezia, Aug 07 2024

Edited by Bruno Berselli, Mar 13 2015

A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

Original entry on oeis.org

3, 3, 2, 6, 3, 11, 6, 18, 11, 27, 18, 38, 27, 51, 38, 66, 51, 83, 66, 102, 83, 123, 102, 146, 123, 171, 146, 198, 171, 227, 198, 258, 227, 291, 258, 326, 291, 363, 326, 402, 363, 443, 402, 486, 443, 531, 486, 578, 531, 627, 578, 678, 627, 731, 678, 786, 731

Offset: 0

Paul Curtz, Jan 22 2016

Trisections:
3,  6,  6,  27, 27, 66,  66, ... = 3*(1, 2, 2, 9, 9, 22, 22, ... ). See A056105.
3,  3,  18, 18, 51, 51, 102, ... = 3*(1, 1, 6, 6, 17, 17, ... ). See A056109.
2, 11,  11, 38, 38, 83,  83, ...   (== 2 (mod 9)).
The trisections also have the signature (1,2,-2,-1,1). The corresponding main sequence is 0, 0, 0, 0, 1, 1, 3, 3, ... = A161680(n) with each term duplicated.

			a(0) = (2+13)/5, a(1) = (13+2)/5, a(2) = (5+5)/5, a(3) = (29+1)/5, ... (using first formula).

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

Cf. A000290, A007395, A010701, A022998, A056105, A056109, A059100, A161680, A174474, A193356, A255844, A261327.

&cat [[(n-1)^2+2, (n+1)^2+2]: n in [0..50]]; // Vincenzo Librandi, Jan 23 2016

Flatten[Table[{n^2 - 2 n + 3, n^2 + 2 n + 3}, {n, 0, 30}]] (* Vincenzo Librandi, Jan 23 2016 *)
CoefficientList[Series[(3 - 7 x^2 + 4 x^3 + 2 x^4)/((1 - x)^3 (1 + x)^2), {x, 0, 56}], x] (* Michael De Vlieger, Jan 24 2016 *)

Vec((3-7*x^2+4*x^3+2*x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 22 2016

a(n) = (A261327(n+2) + A261327(n-3))/5.
a(n+1) = a(n) + (-1)^n * A022998(n), a(0)=3.
a(n+3) = a(n) + 3*A193356(n), a(0)=a(1)=3, a(2)=2.
a(n) = 3 + A174474(n).
a(2n) + a(2n+1) = A255844(n).
From Colin Barker, Jan 22 2016: (Start)
a(n) = (2*n^2 - 6*(-1)^n*n - 2*n + 3*(-1)^n + 21)/8.
a(n) = (n^2 - 4*n + 12)/4 for n even.
a(n) = (n^2 + 2*n + 9)/4 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
G.f.: (3 - 7*x^2 + 4*x^3 + 2*x^4) / ((1-x)^3*(1+x)^2).
(End)

More terms from Colin Barker, Jan 22 2016

cp's OEIS Frontend

A255843 a(n) = 2*n^2 + 4.

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