A066049 -id:A066049 - cp's OEIS frontend

A056220 a(n) = 2*n^2 - 1.

-1, 1, 7, 17, 31, 49, 71, 97, 127, 161, 199, 241, 287, 337, 391, 449, 511, 577, 647, 721, 799, 881, 967, 1057, 1151, 1249, 1351, 1457, 1567, 1681, 1799, 1921, 2047, 2177, 2311, 2449, 2591, 2737, 2887, 3041, 3199, 3361, 3527, 3697, 3871, 4049, 4231, 4417, 4607, 4801

Offset: 0

N. J. A. Sloane, Aug 06 2000

Image of squares (A000290) 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}. - Henry Bottomley, Dec 12 2000
Surround numbers of an n X n square. - Jason Earls, Apr 16 2001
Numbers n such that 2*n + 2 is a perfect square. - Cino Hilliard, Dec 18 2003, Juri-Stepan Gerasimov, Apr 09 2016
The sums of the consecutive integer sequences 2n^2 to 2(n+1)^2-1 are cubes, as 2n^2 + ... + 2(n+1)^2-1 = (1/2)(2(n+1)^2 - 1 - 2n^2 + 1)(2(n+1)^2 - 1 + 2n^2) = (2n+1)^3. E.g., 2+3+4+5+6+7 = 27 = 3^3, then 8+9+10+...+17 = 125 = 5^3. - Andras Erszegi (erszegi.andras(AT)chello.hu), Apr 29 2005
X values (other than 0) of solutions to the equation 2*X^3 + 2*X^2 = Y^2. To find Y values: b(n) = 2n*(2*n^2 - 1). - Mohamed Bouhamida, Nov 06 2007
Average of the squares of two consecutive terms is also a square. In fact: (2*n^2 - 1)^2 + (2*(n+1)^2 - 1)^2 = 2*(2*n^2 + 2*n + 1)^2. - Matias Saucedo (solomatias(AT)yahoo.com.ar), Aug 18 2008
Equals row sums of triangle A143593 and binomial transform of [1, 6, 4, 0, 0, 0, ...] with n > 1. - Gary W. Adamson, Aug 26 2008
Start a spiral of square tiles. Trivially the first tile fits in a 1 X 1 square. 7 tiles fit in a 3 X 3 square, 17 tiles fit in a 5 X 5 square and so on. - Juhani Heino, Dec 13 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-2, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 26 2010
For each n > 0, the recursive series, formula S(b) = 6*S(b-1) - S(b-2) - 2*a(n) with S(0) = 4n^2-4n+1 and S(1) = 2n^2, has the property that every even term is a perfect square and every odd term is twice a perfect square. - Kenneth J Ramsey, Jul 18 2010
Fourth diagonal of A154685 for n > 2. - Vincenzo Librandi, Aug 07 2010
First integer of (2*n)^2 consecutive integers, where the last integer is 3 times the first + 1. As example, n = 2: term = 7; (2*n)^2 = 16; 7, 8, 9, ..., 20, 21, 22: 7*3 + 1 = 22. - Denis Borris, Nov 18 2012
Chebyshev polynomial of the first kind T(2,n). - Vincenzo Librandi, May 30 2014
For n > 0, number of possible positions of a 1 X 2 rectangle in a (n+1) X (n+2) rectangular integer lattice. - Andres Cicuttin, Apr 07 2016
This sequence also represents the best solution for Ripà's n_1 X n_2 X n_3 dots problem, for any 0 < n_1 = n_2 < n_3 = floor((3/2)*(n_1 - 1)) + 1. - Marco Ripà, Jul 23 2018

			a(0) = 0^2-1*1 = -1, a(1) = 1^2 - 4*0 = 1, a(2) = 2^2 - 9*1 = 7, etc.
a(4) = 31 = (1, 3, 3, 1) dot (1, 6, 4, 0) = (1 + 18 + 12 + 0). - _Gary W. Adamson_, Aug 29 2008

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq., Vol. 13 (2010), Article # 10.7.8.
Mitch Phillipson, Manda Riehl, and Tristan Williams, Enumeration of Wilf classes in Sn ~ Cr for two patterns of length 3, PU. M. A., Vol. 21, No. 2 (2010), pp. 321-338.
Marco Ripà, The rectangular spiral or the n1 X n2 X ... X nk Points Problem , Notes on Number Theory and Discrete Mathematics, Vol. 20, No. 1 (2014), pp. 59-71.
Leo Tavares, Illustration: Twin Squares
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000105, A000217, A000290, A000384, A001082, A001653, A001844, A002378, A002593, A003215, A005563, A028347, A036666, A046092, A047875, A062717, A069074, A077585, A087475, A119258, A143593, A154685, A162610, A188653, A225227.
Cf. A066049 (indices of prime terms)
Column 2 of array A188644 (starting at offset 1).

List([0..50], n-> 2*n^2-1); # Muniru A Asiru, Jul 24 2018

[2*n^2-1 : n in [0..50]]; // Vincenzo Librandi, May 30 2014

A056220:=n->2*n^2-1; seq(A056220(n), n=0..50); # Wesley Ivan Hurt, Jun 16 2014

Array[2 #^2 - 1 &, 50, 0] (* Robert G. Wilson v, Jul 23 2018 *)
CoefficientList[Series[(x^2 +4x -1)/(1-x)^3, {x, 0, 50}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {-1, 1, 7}, 51] (* Robert G. Wilson v, Jul 24 2018 *)

PARI
```
a(n)=2*n^2-1;
```

[2*n^2-1 for n in (0..50)] # G. C. Greubel, Jul 07 2019

G.f.: (-1 + 4*x + x^2)/(1-x)^3. - Henry Bottomley, Dec 12 2000
a(n) = A119258(n+1,2) for n > 0. - Reinhard Zumkeller, May 11 2006
From Doug Bell, Mar 08 2009: (Start)
a(0) = -1,
a(n) = sqrt(A001844(n)^2 - A069074(n-1)),
a(n+1) = sqrt(A001844(n)^2 + A069074(n-1)) = sqrt(a(n)^2 + A069074(n-1)*2). (End)
a(n) + a(n+1) + 1 = (2n+1)^2. - Doug Bell, Mar 09 2009
a(n) = a(n-1) + 4*n - 2 (with a(0)=-1). - Vincenzo Librandi, Dec 25 2010
a(n) = A188653(2*n) for n > 0. - Reinhard Zumkeller, Apr 13 2011
a(n) = A162610(2*n-1,n) for n > 0. - Reinhard Zumkeller, Jan 19 2013
a(n) = ( Sum_{k=0..2} (C(n+k,3)-C(n+k-1,3))*(C(n+k,3)+C(n+k+1,3)) ) - (C(n+2,3)-C(n-1,3))*(C(n,3)+C(n+3,3)), for n > 3. - J. M. Bergot, Jun 16 2014
a(n) = j^2 + k^2 - 2 or 2*j*k if n >= 2 and j = n + sqrt(2)/2 and k = n - sqrt(2)/2. - Avi Friedlich, Mar 30 2015
a(n) = A002593(n)/n^2. - Bruce J. Nicholson, Apr 03 2017
a(n) = A000384(n) + n - 1. - Bruce J. Nicholson, Nov 12 2017
a(n)*a(n+k) + 2k^2 = m^2 (a perfect square), m = a(n) + (2n*k), for n>=1. - Ezhilarasu Velayutham, May 13 2019
From Amiram Eldar, Aug 10 2020: (Start)
Sum_{n>=1} 1/a(n) = 1/2 - sqrt(2)*Pi*cot(Pi/sqrt(2))/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(2)*Pi*csc(Pi/sqrt(2))/4 - 1/2. (End)
From Amiram Eldar, Feb 04 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(2))*csc(Pi/sqrt(2)).
Product_{n>=2} (1 - 1/a(n)) = (Pi/(4*sqrt(2)))*csc(Pi/sqrt(2)). (End)
a(n) = A003215(n) - A000217(n-2)*2. - Leo Tavares, Jun 29 2021
Let T(n) = n*(n+1)/2. Then a(n)^2 = T(2n-2)*T(2n+1) + n^2. - Charlie Marion, Feb 12 2023
E.g.f.: exp(x)*(2*x^2 + 2*x - 1). - Stefano Spezia, Jul 08 2023

A066436 Primes of the form 2*n^2 - 1.

Original entry on oeis.org

7, 17, 31, 71, 97, 127, 199, 241, 337, 449, 577, 647, 881, 967, 1151, 1249, 1567, 2311, 2591, 2887, 3041, 3361, 3527, 3697, 4049, 4231, 4801, 4999, 5407, 6271, 6961, 7687, 7937, 8191, 9521, 10657, 11551, 12799, 13121, 14449, 15137, 16561

Offset: 1

N. J. A. Sloane, Jan 09 2002

nonn

It is conjectured that this sequence is infinite.
Also primes p such that 8p + 8 is a square. - Cino Hilliard, Dec 18 2003
Also primes p such that 2p+2 is square; also primes p such that (p+1)/2 is square. - Ray Chandler, Sep 15 2005
Arithmetic numbers which are squares, A003601(p)=A000290(k), p prime, k integer. sigma_1(p)/sigma_0(p)=k^2; p prime, k integer. - Ctibor O. Zizka, Jul 14 2008

D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 31.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Project Euler, Problem 216: Investigating the primality of numbers of the form 2n^2-1.

See A066049 for the values of n, see A091176 for prime index.

Cf. A090697, A110558, A003601, A000290.

[ p: n in [1..100] | IsPrime(p) where p is 2*n^2-1 ]; // Klaus Brockhaus, Dec 29 2008

Select[2*Range[200]^2-1,PrimeQ] (* Harvey P. Dale, Aug 29 2016 *)

{ n=0; for (m=1, 10^9, p=2*m^2 - 1; if (isprime(p), write("b066436.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 14 2010

A118673 Positive solutions x to the equation x^2 + (x+71)^2 = y^2.

Original entry on oeis.org

0, 13, 160, 213, 280, 1113, 1420, 1809, 6660, 8449, 10716, 38989, 49416, 62629, 227416, 288189, 365200, 1325649, 1679860, 2128713, 7726620, 9791113, 12407220, 45034213, 57066960, 72314749, 262478800, 332610789, 421481416, 1529838729, 1938597916, 2456573889

Offset: 0

Mohamed Bouhamida, May 19 2006

Consider all Pythagorean triples (x,x+71,y) ordered by increasing y; sequence gives x values.
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2 the associated value in A066049, the x values are given by the sequence defined by: a(n) = 6*a(n-3) -a(n-6) + 2*p with a(0)=0, a(1)=2m+1, a(2)=6m^2-10m+4, a(3)=3p, a(4)=6m^2+10m+4, a(5)=40m^2-58m+21.
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(0)=p, b(1)=2m^2+2m+1, b(2)=10m^2-14m+5, b(3)=5p, b(4)=10m^2+14m+5, b(5)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A076296 (p=7), A118120 (p=17), A118674 (p=31), A129836 (p=97), A129992 (p=127), A129993 (p=199), A129991 (p=241), A129999 (p=337), A130004 (p=449), A130005 (p=577), A130013 (p=647), A130014 (p=881), A130017 (p=967).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(13+147*x+53*x^2-11*x^3-49*x^4-11*x^5)/((1-x)*(1 - 6*x^3 +x^6)))); // G. C. Greubel, May 07 2018

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+71)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,13,160,213,280,1113,1420},100] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,-1,0,-6,6,0,1]^n*[0;13;160;213;280;1113;1420])[1,1] \\ Charles R Greathouse IV, Apr 22 2016

x='x+O('x^30); concat([0], Vec(x*(13+147*x+53*x^2-11*x^3 -49*x^4 -11*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, May 07 2018

a(n) = 6*a(n-3) -a(n-6) +142 with a(0)=0, a(1)=13, a(2)=160, a(3)=213, a(4)=280, a(5)=1113.

O.g.f.: x*(13+147*x+53*x^2-11*x^3-49*x^4-11*x^5)/((1-x)*(1-6*x^3+x^6)). - R. J. Mathar, Jun 10 2008

Edited by R. J. Mathar, Jun 10 2008

A118120 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.

Original entry on oeis.org

0, 7, 28, 51, 88, 207, 340, 555, 1248, 2023, 3276, 7315, 11832, 19135, 42676, 69003, 111568, 248775, 402220, 650307, 1450008, 2344351, 3790308, 8451307, 13663920, 22091575, 49257868, 79639203, 128759176, 287095935, 464171332, 750463515, 1673317776

Offset: 0

Mohamed Bouhamida, May 12 2006

Also values x of Pythagorean triples (x, x+17, y).
Corresponding values y of solutions (x, y) are in A155923.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a prime number in A066436, m >= 2 the associated value in A066049, the x values are given by the sequence defined by a(n) = 6*a(n-3)-a(n-6)+2p with a(0)=0, a(1)=2m+1, a(2)=6m^2-10m+4, a(3)=3p, a(4)=6m^2+10m+4, a(5)=40m^2-58m+21 (cf. A118673).
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(0)=p, b(1)=2*m^2+2m+1, b(2)=10m^2-14m+5, b(3)=5p, b(4)=10m^2+14m+5, b(5)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A155923, A118673, A066436 (primes of the form 2*n^2-1), A066049 (2*n^2-1 is prime), A118554, A118611, A118630.

Cf. A155464 (first trisection), A155465 (second trisection), A155466 (third trisection).

[ n: n in [0..25000000] | IsSquare(2*n*(n+17)+289) ];

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+17)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,7,28,51,88,207,340},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

m=32; v=concat([0, 7, 28, 51, 88, 207], vector(m-6)); for(n=7, m, v[n]=6*v[n-3]-v[n-6]+34); v

a(n) = 6*a(n-3) -a(n-6) +34 for n > 5; a(0)=0, a(1)=7, a(2)=28, a(3)=51, a(4)=88, a(5)=207.

G. f.: x*(7 +21*x +23*x^2 -5*x^3 -7*x^4 -5*x^5)/((1-x)*(1-6*x^3+x^6)).

Edited and 248755 changed to 248775 by Klaus Brockhaus, Feb 01 2009

A143827 Numbers k such that 8*k^2 - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 11, 12, 14, 17, 18, 19, 21, 23, 25, 26, 28, 31, 32, 38, 40, 46, 49, 51, 54, 56, 59, 63, 66, 67, 70, 77, 79, 80, 82, 86, 89, 93, 94, 96, 98, 100, 102, 103, 107, 110, 114, 116, 119, 121, 124, 128, 133, 135, 137, 140, 144, 147, 150, 152, 156, 161, 166

Offset: 1

Artur Jasinski, Sep 02 2008

Contains the even terms of A066049 divided by 2. - R. J. Mathar, Sep 04 2008

Cf. A090686, A090684.

p = 8; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, x]], {x, 1, 1000}]; a

is(n)=isprime(8*n^2-1) \\ Charles R Greathouse IV, Feb 17 2017

A143828 Primes of the form 10*k^2 - 1.

Original entry on oeis.org

89, 359, 809, 1439, 4409, 8999, 10889, 12959, 20249, 23039, 35999, 47609, 51839, 56249, 65609, 75689, 98009, 116639, 123209, 129959, 136889, 143999, 151289, 158759, 166409, 234089, 272249, 282239, 302759, 313289, 334889, 404009, 416159

Offset: 1

Artur Jasinski, Sep 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A066436, A066049, A090686, A090684, A143826, A143827.

[a: n in [1..400] | IsPrime(a) where a is 10*n^2-1]; // Vincenzo Librandi, Dec 07 2011

p = 10; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, k]], {x, 1, 1000}]; a
Select[Table[10n^2-1,{n,1,800}],PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

A230351 Number of ordered ways to write n = p + q (q > 0) with p, 2p^2 - 1 and 2q^2 - 1 all prime.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 1, 1, 3, 3, 2, 1, 4, 3, 4, 2, 4, 3, 4, 5, 4, 2, 3, 6, 3, 3, 3, 5, 2, 3, 3, 3, 1, 2, 4, 2, 2, 3, 3, 1, 5, 2, 3, 3, 7, 3, 5, 4, 6, 3, 5, 6, 5, 5, 3, 6, 2, 5, 5, 3, 4, 5, 6, 2, 6, 6, 5, 1, 5, 3, 3, 3, 2, 2, 5, 6, 5, 1, 5, 6, 4, 4, 6, 6, 1, 5, 5, 4, 3, 4, 3, 3, 6, 5, 4, 1, 5, 7, 2, 4

Offset: 1

Zhi-Wei Sun, Oct 16 2013

nonn

Conjecture: a(n) > 0 for all n > 3.
We have verified this for n up to 2*10^7.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Aug 07 2023

			a(7) = 1 since 7 = 3 + 4 with 3, 2*3^2 - 1 = 17, 2*4^2 - 1 = 31 all prime.
a(40) = 1 since 40 = 2 + 38, and 2, 2*2^2 - 1 = 7 , 2*38^2 - 1 = 2887 are all prime.
a(68) = 1 since 68 = 43 + 25, and all the three numbers 43, 2*43^2 - 1 = 3697 and 2*25^2 - 1 = 1249 are prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000040, A066049, A106483, A219864, A230252, A230254, A230261.

a[n_]:=Sum[If[PrimeQ[2Prime[i]^2-1]&&PrimeQ[2(n-Prime[i])^2-1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A091176 Numbers n such that prime(n) is of the form 2*k^2 - 1.

Original entry on oeis.org

4, 7, 11, 20, 25, 31, 46, 53, 68, 87, 106, 118, 152, 163, 190, 204, 247, 344, 377, 418, 436, 474, 492, 516, 558, 580, 647, 669, 713, 816, 894, 975, 1003, 1028, 1179, 1300, 1392, 1526, 1561, 1695, 1768, 1917, 1952, 2069, 2177, 2343, 2601, 2643, 2769, 2812

Offset: 1

Ray Chandler, Dec 25 2003

nonn

A066436 indexed by A000040.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A066049, A066436, A090697, A110558.

Flatten[Position[Prime[Range[3000]],?(IntegerQ[Surd[(#+1)/2,2]]&)]] (* _Harvey P. Dale, Jan 23 2013 *)

A000040(n) = A066436(n).

A143829 Numbers n such that 10n^2 - 1 is prime.

Original entry on oeis.org

3, 6, 9, 12, 21, 30, 33, 36, 45, 48, 60, 69, 72, 75, 81, 87, 99, 108, 111, 114, 117, 120, 123, 126, 129, 153, 165, 168, 174, 177, 183, 201, 204, 207, 222, 234, 243, 252, 267, 279, 282, 285, 294, 303, 312, 315, 318, 339, 345, 348, 369, 378, 381, 384, 393, 396

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

A066436, A066049, A090686, A090684, A143826, A143827, A143828

p = 10; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, x]], {x, 1, 1000}]; a
Select[Range[500],PrimeQ[10#^2-1]&] (* Harvey P. Dale, Nov 11 2020 *)

is(n)=isprime(10*n^2-1) \\ Charles R Greathouse IV, Feb 17 2017

A230494 Number of ways to write n = x^2 + y (x, y >= 0) with 2*y^2 - 1 prime.

Original entry on oeis.org

0, 1, 2, 2, 1, 2, 3, 3, 1, 2, 4, 3, 2, 2, 3, 2, 3, 3, 4, 2, 2, 5, 2, 3, 3, 4, 3, 3, 4, 1, 3, 2, 3, 3, 2, 2, 3, 5, 3, 5, 2, 5, 6, 3, 3, 5, 5, 1, 4, 6, 4, 4, 5, 4, 3, 3, 4, 3, 5, 4, 4, 3, 4, 5, 3, 5, 4, 5, 1, 5, 4, 4, 4, 5, 4, 1, 6, 3, 3, 3, 5, 4, 2, 3, 8, 3, 4, 6, 6, 2, 4, 7, 1, 4, 4, 5, 1, 6, 5, 3

Offset: 1

Zhi-Wei Sun, Oct 20 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1. Moreover, if n > 1 is not among 2, 69, 76, then there are positive integers x and y such that x^2 + y is equal to n and 2*y^2 - 1 is prime.
(ii) Any integer n > 1 can be written as x*(x+1)/2 + y with 2*y^2 - 1 prime, where x and y are nonnegative integers. Moreover, if n is not equal to 2 or 15, then we may require additionally that x and y are both positive.
We have verified the conjecture for n up to 2*10^7.
Both conjectures verified for n up to 10^9. - Mauro Fiorentini, Aug 08 2023
See also A230351 and A230493 for similar conjectures.

			a(9) = 1 since 9 = 1^2 + 8 with 2*8^2 - 1 = 127 prime.
a(69) = 1 since 69 = 0^2 + 69 with 2*69^2 - 1 = 9521 prime.
a(76) = 1 since 76 = 0^2 + 76 with 2*76^2 - 1 = 11551 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A000040, A000290, A066049, A220272, A229166, A230351, A230493.

a[n_]:=Sum[If[PrimeQ[2(n-x^2)^2-1],1,0],{x,0,Sqrt[n]}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A056220 a(n) = 2*n^2 - 1.

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A066436 Primes of the form 2*n^2 - 1.

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A118673 Positive solutions x to the equation x^2 + (x+71)^2 = y^2.

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A118120 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.

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A143827 Numbers k such that 8*k^2 - 1 is prime.

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A143828 Primes of the form 10*k^2 - 1.

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A230351 Number of ordered ways to write n = p + q (q > 0) with p, 2*p^2 - 1 and 2*q^2 - 1 all prime.

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A230351 Number of ordered ways to write n = p + q (q > 0) with p, 2p^2 - 1 and 2q^2 - 1 all prime.