A056222 -id:A056222 - cp's OEIS frontend

A056221 Image of primes (A000040) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

-1, 4, -6, 30, -18, 42, -30, -22, 128, -112, 98, 90, -78, -70, 36, 248, -232, 158, 150, -280, 182, -142, -130, 420, 210, -198, 222, -210, -1074, 1326, -238, 560, -1092, 1212, -592, 36, 350, -310, 36, 728, -1428, 1548, -378, 402, -1966, 144, 1832, 462, -450, -442

Offset: 1

N. J. A. Sloane, Aug 06 2000

sign

a(n) > 0 if and only if n+1 is in A046868. a(n) < 0 if and only if n+1 is in A233671. - Chai Wah Wu, Sep 10 2019

Stefano Spezia, Table of n, a(n) for n = 1..10000
L. Panaitopol, On the sequence p(n)^2=p(n-1)*p(n+1), J. Inequal. Pure Appl. Math., Volume 3, Issue 4, Article 53, 2002.

Cf. A000040, A046868, A056222, A233671, A342567.

A056221 := proc(n)
        ithprime(n+1)^2-ithprime(n)*ithprime(n+2) ;
end proc:
seq(A056221(n),n=1..10) ; # R. J. Mathar, Dec 10 2011

a[n_]:=Prime[n+1]^2-Prime[n]Prime[n+2]; Array[a,50] (* Stefano Spezia, Jul 15 2024 *)

a(n) = determinant of matrix
| prime(n+1) prime(n)|
| prime(n+2) prime(n+1)|. - Zak Seidov, Jul 23 2008, indices corrected by Gary Detlefs, Dec 09 2011
a(n) = 2*A342567(n+1) for n >= 2. - Hugo Pfoertner, Jun 20 2021

A268581 a(n) = 2n^2 + 8n + 5.

Original entry on oeis.org

5, 15, 29, 47, 69, 95, 125, 159, 197, 239, 285, 335, 389, 447, 509, 575, 645, 719, 797, 879, 965, 1055, 1149, 1247, 1349, 1455, 1565, 1679, 1797, 1919, 2045, 2175, 2309, 2447, 2589, 2735, 2885, 3039, 3197, 3359, 3525, 3695, 3869, 4047, 4229, 4415, 4605

Offset: 0

Juri-Stepan Gerasimov, Apr 10 2016

Also, numbers m such that 2*m + 6 is a square.

All the terms end with a digit in {5, 7, 9}, or equivalently, are congruent to {5, 7, 9} mod 10. - Stefano Spezia, Aug 05 2021

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

Cf. numbers n such that 2*n + k is a perfect square: A093328 (k=-6), A097080 (k=-5), no sequence (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), this sequence (k=6), A059993 (k=7), A147973 (k=8), A139570 (k=9), no sequence (k=10), A222182 (k=11), A152811 (k=12), A181570 (k=13).

Magma
```
[2*n^2+8*n+5: n in [0..60]];
```
Magma
```
[n: n in [0..6000] | IsSquare(2*n+6)];
```

Table[2 n^2 + 8 n + 5, {n, 0, 50}] (* Vincenzo Librandi, Apr 13 2016 *)
LinearRecurrence[{3,-3,1},{5,15,29},50] (* Harvey P. Dale, Jan 18 2017 *)

lista(nn) = for(n=0, nn, print1(2*n^2+8*n+5, ", ")); \\ Altug Alkan, Apr 10 2016

[2*n^2 + 8*n + 5 for n in [0..46]] # Stefano Spezia, Aug 04 2021

From Vincenzo Librandi, Apr 13 2016: (Start)
G.f.: (5-x^2)/(1-x)^3.
a(n) = 2*(n+2)^2 - 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
E.g.f.: exp(x)*(5 + 10*x + 2*x^2). - Stefano Spezia, Aug 03 2021

Changed offset from 1 to 0, adapted formulas and programs by Bruno Berselli, Apr 13 2016

A271625 a(n) = = 2*(n+1)^2 - 5.

Original entry on oeis.org

3, 13, 27, 45, 67, 93, 123, 157, 195, 237, 283, 333, 387, 445, 507, 573, 643, 717, 795, 877, 963, 1053, 1147, 1245, 1347, 1453, 1563, 1677, 1795, 1917, 2043, 2173, 2307, 2445, 2587, 2733, 2883, 3037, 3195, 3357, 3523, 3693, 3867, 4045, 4227, 4413, 4603, 4797, 4995, 5197, 5403, 5613, 5827

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2016

Numbers n such that 2*n + 10 is a perfect square.

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

Cf. A000290, A201713.

Numbers h such that 2*h + k is a perfect square: A294774 (k=-9), A255843 (k=-8), A271649 (k=-7), A093328 (k=-6), A097080 (k=-5), A271624 (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), A268581 (k=6), A059993 (k=7), (-1)*A147973 (k=8), A139570 (k=9), this sequence (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).

Magma
```
[ 2*n^2 + 4*n - 3: n in [1..60]];
```

[ n: n in [1..6000] | IsSquare(2*n+10)];

Table[2 n^2 + 4 n - 3, {n, 53}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{3,13,27},60] (* Harvey P. Dale, Jun 08 2023 *)
2*Range[2,60]^2 -5 (* G. C. Greubel, Jan 21 2025 *)

x='x+O('x^99); Vec(x*(3+4*x-3*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016

def A271625(n): return 2*pow(n+1,2) - 5
print([A271625(n) for n in range(1,61)]) # G. C. Greubel, Jan 21 2025

G.f.: x*(3 + 4*x - 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 11 2016
Sum_{n>=1} 1/a(n) = 13/30 - Pi*cot(sqrt(5/2)*Pi)/(2*sqrt(10)) = 0.5627678459924... . - Vaclav Kotesovec, Apr 11 2016
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: exp(x)*(2*x^2 + 6*x - 3) + 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
a(n) = 2*A000290(n+1) - 5. - G. C. Greubel, Jan 21 2025

Name simplified by G. C. Greubel, Jan 21 2025

A271624 a(n) = 2n^2 - 4n + 4.

Original entry on oeis.org

2, 4, 10, 20, 34, 52, 74, 100, 130, 164, 202, 244, 290, 340, 394, 452, 514, 580, 650, 724, 802, 884, 970, 1060, 1154, 1252, 1354, 1460, 1570, 1684, 1802, 1924, 2050, 2180, 2314, 2452, 2594, 2740, 2890, 3044, 3202, 3364, 3530, 3700, 3874, 4052, 4234, 4420, 4610, 4804, 5002, 5204, 5410, 5620

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2016

Numbers n such that 2*n - 4 is a perfect square.

For n > 2, the number of square a(n)-gonal numbers is finite. - Muniru A Asiru, Oct 16 2016

			a(1) = 2*1^2 - 4*1 + 4 = 2.

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, numbers n such that 2*n + k is a perfect square: no sequence (k = -9), A255843 (k = -8), A271649 (k = -7), A093328 (k = -6), A097080 (k = -5), this sequence (k = -4), A051890 (k = -3), A058331 (k = -2), A001844 (k = -1), A001105 (k = 0), A046092 (k = 1), A056222 (k = 2), A142463 (k = 3), A054000 (k = 4), A090288 (k = 5), A268581 (k = 6), A059993 (k = 7), (-1)*A147973 (k = 8), A139570 (k = 9), A271625 (k = 10), A222182 (k = 11), A152811 (k = 12), A181510 (k = 13), A161532 (k = 14), no sequence (k = 15).

Cf. A005893, A160457.

Magma
```
[ 2*n^2 - 4*n + 4: n in [1..60]];
```

[ n: n in [1..6000] | IsSquare(2*n-4)];

Table[2 n^2 - 4 n + 4, {n, 54}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{2,4,10},60] (* Harvey P. Dale, Jul 18 2023 *)

x='x+O('x^99); Vec(2*x*(1-x+2*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016

a(n)=2*n^2-4*n+4 \\ Charles R Greathouse IV, Apr 11 2016

a(n) = 2*A002522(n-1).
G.f.: 2*x*(1 - x + 2*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 11 2016
Sum_{n>=1} 1/a(n) = (1 + Pi*coth(Pi))/4 = 1.038337023734290587067... . - Vaclav Kotesovec, Apr 11 2016
a(n) = A005893(n-1), n > 1. - R. J. Mathar, Apr 12 2016
a(n) = 2 + 2*(n-1)^2. - Tyler Skywalker, Jul 21 2016
From Elmo R. Oliveira, Nov 17 2024: (Start)
E.g.f.: 2*(exp(x)*(x^2 - x + 2) - 2).
a(n) = 2*A160457(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A271649 a(n) = 2*(n^2 - n + 2).

Original entry on oeis.org

4, 8, 16, 28, 44, 64, 88, 116, 148, 184, 224, 268, 316, 368, 424, 484, 548, 616, 688, 764, 844, 928, 1016, 1108, 1204, 1304, 1408, 1516, 1628, 1744, 1864, 1988, 2116, 2248, 2384, 2524, 2668, 2816, 2968, 3124, 3284, 3448, 3616, 3788, 3964, 4144, 4328, 4516, 4708, 4904, 5104, 5308, 5516

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2016

Numbers n such that 2*n - 7 is a perfect square.

Galois numbers for three-dimensional vector space, defined as the total number of subspaces in a three-dimensional vector space over GF(n-1), when n-1 is a power of a prime. - Artur Jasinski, Aug 31 2016, corrected by Robert Israel, Sep 23 2016

			a(1) = 2*(1^2 - 1 + 2) = 4.

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

Cf. A000124, A014206, A137882.

Numbers h such that 2*h + k is a perfect square: no sequence (k=-9), A255843 (k=-8), this sequence (k=-7), A093328 (k=-6), A097080 (k=-5), A271624 (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), A268581 (k=6), A059993 (k=7), (-1)*A147973 (k=8), A139570 (k=9), A271625 (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).

Magma
```
[ 2*n^2 - 2*n + 4: n in [1..60]];
```

[ n: n in [1..6000] | IsSquare(2*n-7)];

A271649:=n->2*(n^2-n+2): seq(A271649(n), n=1..60); # Wesley Ivan Hurt, Aug 31 2016

Table[2 (n^2 - n + 2), {n, 53}] (* or *)
Select[Range@ 5516, IntegerQ@ Sqrt[2 # - 7] &] (* or *)
Table[SeriesCoefficient[(-4 (1 - x + x^2))/(-1 + x)^3, {x, 0, n}], {n, 0, 52}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{4,8,16},60] (* Harvey P. Dale, Jun 14 2022 *)

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

a(n) = 4*A000124(n).
a(n) = 2*A014206(n).
a(n) = A137882(n), n > 1. - R. J. Mathar, Apr 12 2016
Sum_{n>=1} 1/a(n) = tanh(sqrt(7)*Pi/2)*Pi/(2*sqrt(7)). - Amiram Eldar, Jul 30 2024
From Elmo R. Oliveira, Nov 18 2024: (Start)
G.f.: 4*x*(1 - x + x^2)/(1 - x)^3.
E.g.f.: 2*(exp(x)*(x^2 + 2) - 2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A381374 Little Hankel transform of A317614: a(n) = A317614(n+1)^2 - A317614(n)*A317614(n+2).

Original entry on oeis.org

1, 1, 97, 49, 769, 289, 2977, 961, 8161, 2401, 18241, 5041, 35617, 9409, 63169, 16129, 104257, 25921, 162721, 39601, 242881, 58081, 349537, 82369, 487969, 113569, 663937, 152881, 883681, 201601, 1153921, 261121, 1481857, 332929, 1875169, 418609, 2342017, 519841, 2891041

Offset: 1

Stefano Spezia, Feb 21 2025

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A317614.

Cf. A056221, A056222, A239607, A374668.

LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{1,97,49,769,289,2977,961,8161,2401,18241},38]

a(n) = (10 + 6*(-1)^n + 4*n*(n + 2)*(3*(n + 1)^2 + (-1)^n*(2*n^2 + 4*n + 5)))/16.
a(n) = 5*a(n-2) - 10*a(n-4) + 10*a(n-6) - 5*a(n-8) + a(n-10) for n > 10.
G.f.: (1 + x + 92*x^2 + 44*x^3 + 294*x^4 + 54*x^5 + 92*x^6 - 4*x^7 + x^8 + x^9)/(1 - x^2)^5.
E.g.f.: ((4 + 3*x + 123*x^2 + 10*x^3 + 5*x^4)*cosh(x) + (1 + 69*x + 21*x^2 + 50*x^3 + x^4)*sinh(x))/4.
a(2*n) = A239607(n).

cp's OEIS Frontend

A056221 Image of primes (A000040) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

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A268581 a(n) = 2*n^2 + 8*n + 5.

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A271625 a(n) = = 2*(n+1)^2 - 5.

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A271624 a(n) = 2*n^2 - 4*n + 4.

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A271649 a(n) = 2*(n^2 - n + 2).

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Maple

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A381374 Little Hankel transform of A317614: a(n) = A317614(n+1)^2 - A317614(n)*A317614(n+2).

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Mathematica

A268581 a(n) = 2n^2 + 8n + 5.

A271624 a(n) = 2n^2 - 4n + 4.