A255843 -id:A255843 - cp's OEIS frontend

A255844 a(n) = 2*n^2 + 6.

6, 8, 14, 24, 38, 56, 78, 104, 134, 168, 206, 248, 294, 344, 398, 456, 518, 584, 654, 728, 806, 888, 974, 1064, 1158, 1256, 1358, 1464, 1574, 1688, 1806, 1928, 2054, 2184, 2318, 2456, 2598, 2744, 2894, 3048, 3206, 3368, 3534, 3704, 3878, 4056, 4238, 4424, 4614

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=3 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + 1)^3 + (n - 1)^3.
Equivalently, numbers m such that 2*m-12 is a square.
For n = 0..16, 3*a(n)-1 is prime (see A087370); for n = 0..12, 3*a(n)-5 is prime (see A107303).

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

Cf. A016825 (first differences), A087370, A107303, A114949, A117950.
Cf. A152811: nonnegative numbers of the form 2*m^2-6.
Subsequence of A000378.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+6: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 6, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+6)
```
Sage
```
[2*n^2+6 for n in (0..50)]
```

G.f.: 2*(3-5*x+4*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A117950(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/12.
Sum_{n>=0} (-1)^n/a(n) = (1 + (sqrt(3)*Pi)*cosech(sqrt(3)*Pi))/12. (End)
E.g.f.: 2*exp(x)*(3 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Corrected and extended by Bruno Berselli, Mar 11 2015

A294774 a(n) = 2n^2 + 2n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset: 0

Bruno Berselli, Nov 08 2017

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)

Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

A255842 a(n) = 2*n^2 + 12.

Original entry on oeis.org

12, 14, 20, 30, 44, 62, 84, 110, 140, 174, 212, 254, 300, 350, 404, 462, 524, 590, 660, 734, 812, 894, 980, 1070, 1164, 1262, 1364, 1470, 1580, 1694, 1812, 1934, 2060, 2190, 2324, 2462, 2604, 2750, 2900, 3054, 3212, 3374, 3540, 3710, 3884, 4062, 4244, 4430

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=6 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + sqrt(2))^3 + (n - sqrt(2))^3.
Equivalently, numbers m such that 2*m - 24 is a square.
For n = 0..10, a(n) - 1 is prime (see A092968).

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

Cf. A016825 (first differences), A092968, A114949.
Subsequence of A047238 and A047406.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+12: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 12, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+12)
```
Sage
```
[2*n^2+12 for n in (0..50)]
```

G.f.: 2*(6 - 11*x + 7*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A114949(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/24.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/24. (End)
E.g.f.: 2*exp(x)*(6 + x + x^2). - Elmo R. Oliveira, Jan 24 2025

Edited by Bruno Berselli, Mar 11 2015

A255845 a(n) = 2*n^2 + 10.

Original entry on oeis.org

10, 12, 18, 28, 42, 60, 82, 108, 138, 172, 210, 252, 298, 348, 402, 460, 522, 588, 658, 732, 810, 892, 978, 1068, 1162, 1260, 1362, 1468, 1578, 1692, 1810, 1932, 2058, 2188, 2322, 2460, 2602, 2748, 2898, 3052, 3210, 3372, 3538, 3708, 3882, 4060, 4242, 4428

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=5 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.

Equivalently, numbers m such that 2*m - 20 is a square.

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

Cf. A016825 (first differences), A117951.
Subsequence of A047463.
Cf. similar sequences listed in A255843.

[2*n^2+10: n in [0..50]]; // Vincenzo Librandi, Mar 08 2015

Mathematica
```
Table[2 n^2 + 10, {n, 0, 50}]
```

a(n)=2*n^2+10 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*A117951(n).
From Vincenzo Librandi, Mar 08 2015: (Start)
G.f.: 2*(5 - 9*x + 6*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(5)*Pi*coth(sqrt(5)*Pi))/20.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(5)*Pi*cosech(sqrt(5)*Pi))/20. (End)
E.g.f.: 2*exp(x)*(5 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Edited by Bruno Berselli, Mar 13 2015

A255846 a(n) = 2*n^2 + 14.

Original entry on oeis.org

14, 16, 22, 32, 46, 64, 86, 112, 142, 176, 214, 256, 302, 352, 406, 464, 526, 592, 662, 736, 814, 896, 982, 1072, 1166, 1264, 1366, 1472, 1582, 1696, 1814, 1936, 2062, 2192, 2326, 2464, 2606, 2752, 2902, 3056, 3214, 3376, 3542, 3712, 3886, 4064, 4246, 4432

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=7 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.

Equivalently, numbers m such that 2*m - 28 is a square.

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

Cf. A117619.
Subsequence of A047235 and A047451.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+14: n in [0..50]];
```
Mathematica
```
Table[2 n^2 + 14, {n, 0, 50}]
```
PARI
```
vector(50, n, n--; 2*n^2+14)
```
Sage
```
[2*n^2+14 for n in (0..50)]
```

G.f.: 2*(7 - 13*x + 8*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A117619(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(7)*Pi*coth(sqrt(7)*Pi))/28.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(7)*Pi*cosech(sqrt(7)*Pi))/28. (End)
E.g.f.: 2*exp(x)*(7 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Edited by Bruno Berselli, Mar 13 2015

A255848 a(n) = 2*n^2 + 18.

Original entry on oeis.org

18, 20, 26, 36, 50, 68, 90, 116, 146, 180, 218, 260, 306, 356, 410, 468, 530, 596, 666, 740, 818, 900, 986, 1076, 1170, 1268, 1370, 1476, 1586, 1700, 1818, 1940, 2066, 2196, 2330, 2468, 2610, 2756, 2906, 3060, 3218, 3380, 3546, 3716, 3890, 4068, 4250, 4436

Offset: 0

Avi Friedlich, Mar 08 2015

For n>3, the sequence gives the 6th diagonal of triangle in A055096.
Also, this is the case k=9 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. It is noted that a(n)*n = (n + sqrt(3))^3 + (n - sqrt(3))^3.
Equivalently, numbers m such that 2*m-36 is a square.

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

Cf. A016825 (first differences), A055096, A189834.
Subsequence of A047463.
Cf. similar sequences listed in A255843.

[2*n^2+18: n in [0..50]]; // Vincenzo Librandi, Mar 08 2015

f[n_] := 2 n^2 + 18; Array[f, 50, 0] (* Robert G. Wilson v, Mar 08 2015 *)
CoefficientList[Series[(18 - 34 x + 20 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 08 2015 *)
LinearRecurrence[{3,-3,1},{18,20,26},50] (* Harvey P. Dale, Aug 20 2021 *)

vector(50, n, 2*n^2+18) \\ Derek Orr, Mar 09 2015

[2*n^2+18 for n in (0..50)] # Bruno Berselli, Mar 11 2015

a(n) = 2*A189834(n).
From Vincenzo Librandi, Mar 08 2015: (Start)
G.f.: 2*(9 - 17*x + 10*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + 3*Pi*coth(3*Pi))/36.
Sum_{n>=0} (-1)^n/a(n) = (1 + 3*Pi*cosech(3*Pi))/36. (End)
E.g.f.: 2*exp(x)*(9 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Edited by Bruno Berselli, Mar 11 2015

cp's OEIS Frontend

A255844 a(n) = 2*n^2 + 6.

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A294774 a(n) = 2*n^2 + 2*n + 5.

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A255842 a(n) = 2*n^2 + 12.

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A255845 a(n) = 2*n^2 + 10.

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A255846 a(n) = 2*n^2 + 14.

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Magma

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PARI

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A255848 a(n) = 2*n^2 + 18.

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Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

Sage

A294774 a(n) = 2n^2 + 2n + 5.