Avi Friedlich - cp's OEIS frontend

A258133 Expansion of tri-digit zeros interlaced with an arithmetic progression of positive and negative numbers.

1, 0, 0, 0, 2, -2, 2, 0, 0, 0, 3, -3, 3, 0, 0, 0, 4, -4, 4, 0, 0, 0, 5, -5, 5, 0, 0, 0, 6, -6, 6, 0, 0, 0, 7, -7, 7, 0, 0, 0, 8, -8, 8, 0, 0, 0, 9, -9, 9, 0, 0, 0, 10, -10, 10, 0, 0, 0, 11, -11, 11, 0, 0, 0, 12, -12, 12, 0, 0, 0, 13, -13, 13, 0, 0, 0, 14

Offset: 0

Avi Friedlich, May 21 2015

This sequence is observed as the second difference in the expansion of 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^6)) (A029000), a sequence noted for its interlaced and structural coordination numbers.

			G.f. = 1 + 2*x^4 - 2*x^5 + 2*x^6 + 3*x^10 - 3*x^11 + 3*x^12 + 4*x^16 + ...

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

a[ n_] := With[ {m=n-1}, If[ OddQ[ Quotient[ m, 3]], Quotient[ m+9, 6] (-1)^Mod[m, 3], 0]]; (* Michael Somos, Jun 07 2015 *)

Vec((x^9-x^7-x^6+x^4-x^3+x+1)/((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, May 24 2015

{a(n) = n--; if(n\3%2, (n+9)\6 * (-1)^(n%3), 0)}; /* Michael Somos, Jun 07 2015 */

a(n) = -a(n-1) + a(n-3) + a(n-4) + a(n-6) + a(n-7) - a(n-9) - a(n-10). a(6*k-2) = -a(6*k-1) = a(6*k) = k+1 for k >= 1.
G.f.: (x^9-x^7-x^6+x^4-x^3+x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+x+1)^2). - Colin Barker, May 24 2015
a(n) = -a(-14-n) for all n in Z. - Michael Somos, Jun 07 2015

A256376 Primes of the form 10n^2 - 90n + 163.

Original entry on oeis.org

23, 83, 163, 263, 383, 523, 683, 863, 1063, 1283, 1523, 1783, 2063, 2683, 3023, 4583, 5023, 5483, 6983, 7523, 8663, 9883, 11863, 14783, 16363, 17183, 19763, 20663, 25463, 29663, 30763, 31883, 33023, 34183, 35363, 36563, 37783, 39023, 40283, 42863, 45523, 49663, 56963, 61583, 64763

Offset: 1

Avi Friedlich, Mar 27 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

[ a: k in [7..200] | IsPrime(a) where a is 10*k^2-90*k+163];

Select[Table[10 k^2 - 90 k + 163, {k, 7, 600}], PrimeQ]

select(isprime, vector(100,n,10*n^2 + 30*n - 17)) \\ Charles R Greathouse IV, Mar 27 2015

Corrected by Vincenzo Librandi, Mar 27 2015

A256374 Primes of the form 7k^2 + 7k + 17.

Original entry on oeis.org

17, 31, 59, 101, 157, 227, 311, 409, 521, 647, 787, 941, 1109, 1291, 1487, 1697, 2411, 2677, 2957, 3251, 3559, 3881, 4217, 4567, 4931, 5309, 5701, 6961, 8837, 9341, 9859, 10391, 10937, 11497, 12071, 12659, 13877, 15809, 16481, 17167, 19309, 20051, 20807, 21577, 23159, 23971

Offset: 1

Avi Friedlich, Mar 26 2015

The values k=0 through 15 all give primes.

			For k=15 we get 1697, a prime.
For k=16 we get 1921 = 17*113, not a prime, so not a term of the sequence.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A256375.

[ a: n in [0..200] | IsPrime(a) where a is 7*n^2 + 7*n + 17 ];

Select[Table[7 n^2 + 7 n +17, {n, 0, 600}], PrimeQ]

for(k=0, 1e2, if(ispseudoprime(7*k^2+7*k+17), print1(7*k^2+7*k+17, ", "))) \\ Felix Fröhlich, Apr 01 2015

Corrected by Vincenzo Librandi, Mar 27 2015

Edited by N. J. A. Sloane, Mar 27 2015

A255847 a(n) = 2*n^2 + 16.

Original entry on oeis.org

16, 18, 24, 34, 48, 66, 88, 114, 144, 178, 216, 258, 304, 354, 408, 466, 528, 594, 664, 738, 816, 898, 984, 1074, 1168, 1266, 1368, 1474, 1584, 1698, 1816, 1938, 2064, 2194, 2328, 2466, 2608, 2754, 2904, 3058, 3216, 3378, 3544, 3714, 3888, 4066, 4248, 4434, 4624

Offset: 0

Avi Friedlich, Mar 08 2015

This is the case k=8 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.

Equivalently, numbers m such that 2*m - 32 is a square.

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

Cf. A189833.
Subsequence of A047467.
Cf. similar sequences listed in A255843.

Magma
```
[2*n^2+16: n in [0..50]];
```

Table[2 n^2 + 16, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{16,18,24},50] (* Harvey P. Dale, Nov 11 2017 *)

PARI
```
vector(50, n, n--; 2*n^2+16)
```
Sage
```
[2*n^2+16 for n in (0..50)]
```

G.f.: 2*(8 - 15*x + 9*x^2)/(1 - x)^3.
a(n) = a(-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A189833(n).
From Amiram Eldar, Mar 28 2023: (Start)
Sum_{n>=0} 1/a(n) = (1 + 2*sqrt(2)*Pi*coth(2*sqrt(2)*Pi))/32.
Sum_{n>=0} (-1)^n/a(n) = (1 + 2*sqrt(2)*Pi*cosech(2*sqrt(2)*Pi))/32. (End)
E.g.f.: 2*exp(x)*(8 + 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

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

Original entry on oeis.org

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

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

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

Original entry on oeis.org

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

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

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

User: Avi Friedlich

A258133 Expansion of tri-digit zeros interlaced with an arithmetic progression of positive and negative numbers.

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A256376 Primes of the form 10n^2 - 90n + 163.

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Magma

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Extensions

A256374 Primes of the form 7*k^2 + 7*k + 17.

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A255847 a(n) = 2*n^2 + 16.

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

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

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Magma

Mathematica

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Formula

Extensions

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

A256374 Primes of the form 7k^2 + 7k + 17.