A066436 -id:A066436 - cp's OEIS frontend

A066049 Numbers n such that 2*n^2 - 1 is a prime.

2, 3, 4, 6, 7, 8, 10, 11, 13, 15, 17, 18, 21, 22, 24, 25, 28, 34, 36, 38, 39, 41, 42, 43, 45, 46, 49, 50, 52, 56, 59, 62, 63, 64, 69, 73, 76, 80, 81, 85, 87, 91, 92, 95, 98, 102, 108, 109, 112, 113, 115, 118, 125, 126, 127, 132, 134, 137, 140, 141, 143, 153, 154, 155

Offset: 1

N. J. A. Sloane, Jan 09 2002

nonn

It is conjectured that this sequence is infinite.

A066436 gives resulting primes p such that (p+1)/2 is square. - Ray Chandler

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

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A028870, A066436, A090697, A091176, A110558.

Select[Range[200],PrimeQ[2#^2-1]&] (* Harvey P. Dale, Jun 14 2011 *)

{ n=0; for (m=1, 10^9, if (isprime(2*m^2 - 1), write("b066049.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 08 2009

a(n) = A090697(n)/2 = A110558(n)/4. - Ray Chandler, Sep 15 2005

a(n) = A160697(n+1). - Reinhard Zumkeller, May 24 2009

Extended by Ray Chandler, Sep 15 2005

A118674 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2.

Original entry on oeis.org

0, 9, 60, 93, 140, 429, 620, 893, 2576, 3689, 5280, 15089, 21576, 30849, 88020, 125829, 179876, 513093, 733460, 1048469, 2990600, 4274993, 6111000, 17430569, 24916560, 35617593, 101592876, 145224429, 207594620, 592126749, 846430076

Offset: 1

Mohamed Bouhamida, May 19 2006

Also values x of Pythagorean triples (x, x+31, y).
Corresponding values y of solutions (x, y) are in A157646.
For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (33 + 8*sqrt(2))/31 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1539 + 850*sqrt(2))/31^2 for n mod 3 = 0.

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

cf. A157646, A066436 (primes of the form 2*n^2-1), A118673, A129836, A001652, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3 + 2*sqrt(2)), A157647 (decimal expansion of (33 + 8*sqrt(2))/31), A157648 (decimal expansion of (1539 + 850*sqrt(2))/31^2).

I:=[0,9,60,93,140,429,620]; [n le 7 select I[n] else Self(n-1) - 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // G. C. Greubel, Mar 31 2018

ClearAll[a]; Evaluate[Array[a, 6]] = {0, 9, 60, 93, 140, 429}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 62; Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Dec 27 2011, after given formula *)
LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,9,60,93,140,429,620}, 50] (* G. C. Greubel, Mar 31 2018 *)

{forstep(n=0, 850000000, [1, 3], if(issquare(2*n^2+62*n+961), print1(n, ",")))};

a(n) = 6*a(n-3) - a(n-6) + 62 for n > 6; a(1)=0, a(2)=9, a(3)=60, a(4)=93, a(5)=140, a(6)=429.
G.f.: x*(9 + 51*x + 33*x^2 - 7*x^3 - 17*x^4 - 7*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 31*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Mar 11 2009

A129836 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 97)^2 = y^2.

Original entry on oeis.org

0, 15, 228, 291, 368, 1575, 1940, 2387, 9416, 11543, 14148, 55115, 67512, 82695, 321468, 393723, 482216, 1873887, 2295020, 2810795, 10922048, 13376591, 16382748, 63658595, 77964720, 95485887, 371029716, 454411923, 556532768

Offset: 1

Mohamed Bouhamida, May 21 2007

Also values x of Pythagorean triples (x, x + 97, y).
Corresponding values y of solutions (x, y) are in A157469.
For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436, the x values are given by the sequence defined by a(n) = 6*a(n-3) - a(n-6) + 2p with a(1)=0, a(2) = 2m + 1, a(3) = 6m^2 - 10m + 4, a(4) = 3p, a(5) = 6m^2 + 10m + 4, a(6) = 40m^2 - 58m + 21 (cf. A118673).
Pairs (p, m) are (7, 2), (17, 3), (31, 4), (71, 6), (97, 7), (127, 8), (199, 10), (241, 11), (337, 13), (449, 15), (577, 17), (647, 18), (881, 21), (967, 22), ...
lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (99 + 14*sqrt(2))/97 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (19491 + 12070*sqrt(2))/97^2 for n mod 3 = 0.
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(1) = p, b(2) = 2m^2 + 2m + 1, b(3) = 10m^2 - 14m + 5, b(4) = 5p, b(5) = 10m^2 + 14m + 5, b(6) = 58m^2 - 82m + 29. - Mohamed Bouhamida, Sep 09 2009

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

Cf. A157469, A066436 (primes of the form 2*n^2 - 1), A001652, A118673, A118674, A156035 (decimal expansion of 3 + 2*sqrt(2)), A157470 (decimal expansion of (99 + 14*sqrt(2))/97), A157471 (decimal expansion of (19491 + 12070*sqrt(2))/97^2).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(15+213*x+63*x^2-13*x^3-71*x^4-13*x^5)/((1-x)*(1-6*x^3 + x^6)))); // G. C. Greubel, May 07 2018

ClearAll[a]; Evaluate[Array[a, 6]] = {0, 15, 228, 291, 368, 1575}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 194; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Dec 27 2011, after given formula *)
LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,15,228,291,368,1575,1940}, 50] (* G. C. Greubel, May 07 2018 *)

forstep(n=0, 600000000, [3, 1], if(issquare(2*n^2+194*n+9409), print1(n, ",")))

a(n) = 6*a(n-3) - a(n-6) + 194 for n > 6; a(1)=0, a(2)=15, a(3)=228, a(4)=291, a(5)=368, a(6)=1575.
G.f.: x*(15 + 213*x + 63*x^2 - 13*x^3 - 71*x^4 - 13*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 97*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Mar 12 2009

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

A207059 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+289)^2 = y^2.

Original entry on oeis.org

119, 231, 300, 476, 867, 1496, 2120, 2511, 3519, 5780, 9435, 13067, 15344, 21216, 34391, 55692, 76860, 90131, 124355, 201144, 325295, 448671, 526020, 725492, 1173051, 1896656, 2615744, 3066567, 4229175, 6837740, 11055219, 15246371, 17873960, 24650136

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

For the generic case x^2 + (x + p^2)^2 = y^2 with p = 2*m^2 - 1 a prime number in A066436, m>=3, (0; p^2) and (4*m^3 + 2*m^2 - 2*m - 1; 4*m^4 + 4*m^3 - 2*m - 1) are solutions. - Mohamed Bouhamida, Aug 24 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein, Diophantine equation (MathWorld).
Wikipedia, Diophantine equation
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,6,-6,0,0,0,-1,1).

Cf. A204765, A205644, A205672, A207058.

LinearRecurrence[ {1, 0, 0, 0, 6, -6, 0, 0, 0, -1, 1}, {119, 231, 300, 476, 867, 1496, 2120, 2511, 3519, 5780, 9435}, 60]

Vec(x*(85*x^9+48*x^8+23*x^7+48*x^6+85*x^5-391*x^4-176*x^3-69*x^2-112*x-119)/((x-1)*(x^10-6*x^5+1))+O(x^60)) \\ Stefano Spezia, Aug 24 2019

G.f.: x*(85*x^9+48*x^8+23*x^7+48*x^6+85*x^5-391*x^4-176*x^3-69*x^2-112*x-119)/((x-1)*(x^10-6*x^5+1)). - Colin Barker, Aug 05 2012

a(n) = 6*a(n-5) - a(n-10) + 578 with a(1) = 119, a(2) = 231, a(3) = 300, a(4) = 476, a(5) = 867, a(6) = 1496, a(7) = 2120, a(8) = 2511, a(9) = 3519, a(10) = 5780. - Mohamed Bouhamida, Aug 24 2019

A129993 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+199)^2 = y^2.

Original entry on oeis.org

0, 21, 504, 597, 704, 3441, 3980, 4601, 20540, 23681, 27300, 120197, 138504, 159597, 701040, 807741, 930680, 4086441, 4708340, 5424881, 23818004, 27442697, 31619004, 138821981, 159948240, 184289541, 809114280, 932247141, 1074118640

Offset: 1

Mohamed Bouhamida, Jun 14 2007

Also values x of Pythagorean triples (x, x+199, y).
Corresponding values y of solutions (x, y) are in A159548.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (201+20*sqrt(2))/199 for n mod 3 = {1, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 0.

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

Cf. A159548, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

I:=[0,21,504,597,704,3441,3980]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // G. C. Greubel, Mar 31 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,21,504,597,704,3441,3980},30] (* Harvey P. Dale, Jun 03 2012 *)

{forstep(n=0, 500000000, [1, 3], if(issquare(2*n^2+398*n+39601), print1(n, ",")))};

a(n) = 6*a(n-3) - a(n-6) + 398 for n > 6; a(1)=0, a(2)=21, a(3)=504, a(4)=597, a(5)=704, a(6)=3441.
G.f.: x*(21+483*x+93*x^2-19*x^3-161*x^4-19*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 199*A001652(k) for k >= 0.
a(1)=0, a(2)=21, a(3)=504, a(4)=597, a(5)=704, a(6)=3441, a(7)=3980, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Jun 03 2012

Edited and two terms added by Klaus Brockhaus, Apr 14 2009

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 *)

A076296 Consider all Pythagorean triples (X,X+7,Z); sequence gives X values.

Original entry on oeis.org

-3, 0, 5, 8, 21, 48, 65, 140, 297, 396, 833, 1748, 2325, 4872, 10205, 13568, 28413, 59496, 79097, 165620, 346785, 461028, 965321, 2021228, 2687085, 5626320, 11780597, 15661496, 32792613, 68662368, 91281905, 191129372, 400193625, 532029948, 1113983633

Offset: 0

Henry Bottomley, Oct 05 2002

First two terms included for consistency with A076293.
From Klaus Brockhaus, Feb 18 2009: (Start)
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (9+4*sqrt(2))/7 for n mod 3 = {1, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 3 = 0. (End)
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 x values are given by the sequence defined by: a(n) = 6*a(n-3) - a(n-6) + 2p with a(1)=0, a(2)=2m+1, a(3)=6m^2-10m+4, a(4)=3p, a(5)=6m^2+10m+4, a(6)=40m^2-58m+21. Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=2m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009
For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a prime number, m>=2, the first three consecutive solutions are: (0;p), (2*m+1; 2*m^2+2*m+1), (6*m^2-10*m+4; 10*m^2-14*m+5) and the other solutions are defined by: (X(n); Y(n))= (3*X(n-3)+2*Y(n-3)+p; 4*X(n-3)+3*Y(n-3)+2*p). - Mohamed Bouhamida, Aug 20 2019

			8 is in the sequence as the shorter leg of the (8,15,17) triangle.

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. A001652, A076293, A076294, A076295.

Cf. A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7). - Klaus Brockhaus, Feb 18 2009

I:=[-3,0,5,8,21,48,65]; [n le 7 select I[n] else Self(n-1) +6*Self(n-3) -6*Self(n-4) -Self(n-6) +Self(n-7): n in [1..30]]; // G. C. Greubel, May 04 2018

CoefficientList[Series[(3-3x-5x^2-21x^3+5x^4+3x^5+4x^6)/(-1+x+6x^3-6x^4-x^6+x^7),{x,0,50}],x] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)
LinearRecurrence[{1,0,6,-6,0,-1,1}, {-3,0,5,8,21,48,65}, 50] (* T. D. Noe, Feb 07 2012 *)

x='x+O('x^30); Vec((-3+3*x+5*x^2+21*x^3-5*x^4-3*x^5-4*x^6)/((1-x)*(1-6*x^3 +x^6))) \\ G. C. Greubel, May 04 2018

a(n) = 6a(n-3) - a(n-6) + 14 = (A076293(n) - 7)/2.
a(n) = sqrt(A076294(n)^2 - A076295(n)^2) = A076295(n) - 7.
a(3*n+1) = 7*A001652(n).
From Mohamed Bouhamida, Jul 06 2007: (Start)
a(n) = 5*(a(n-3) + a(n-6)) - a(n-9) + 28.
a(n) = 7*(a(n-3) - a(n-6)) + a(n-9). (End)
G.f.: (-3 + 3*x + 5*x^2 + 21*x^3 - 5*x^4 - 3*x^5 - 4*x^6)/((1-x)*(1 - 6*x^3 + x^6)). - Klaus Brockhaus, Feb 18 2009

More terms from Klaus Brockhaus, Feb 18 2009

A129992 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+127)^2 = y^2.

Original entry on oeis.org

0, 17, 308, 381, 468, 2117, 2540, 3045, 12648, 15113, 18056, 74025, 88392, 105545, 431756, 515493, 615468, 2516765, 3004820, 3587517, 14669088, 17513681, 20909888, 85498017, 102077520, 121872065, 498319268, 594951693, 710322756, 2904417845, 3467632892

Offset: 1

Mohamed Bouhamida, Jun 14 2007

nonn

Also values x of Pythagorean triples (x, x+127, y).
Corresponding values y of solutions (x, y) are in A159466.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (129+16*sqrt(2))/127 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (34947+21922*sqrt(2))/127^2 for n mod 3 = 0.

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

Cf. A159466, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159467 (decimal expansion of (129+16*sqrt(2))/127), A159468 (decimal expansion of (34947+21922*sqrt(2))/127^2).

I:=[0,17,308,381,468,2117,2540]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // G. C. Greubel, Mar 31 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,17,308,381,468,2117,2540},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 500000000, [1, 3], if(issquare(2*n^2+254*n+16129), print1(n, ",")))};

a(n) = 6*a(n-3) - a(n-6) + 254 for n > 6; a(1)=0, a(2)=17, a(3)=308, a(4)=381, a(5)=468, a(6)=2117.
G.f.: x*(17+291*x+73*x^2-15*x^3-97*x^4-15*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 127*A001652(k) for k >= 0.

Edited and two more terms added by Klaus Brockhaus, Apr 13 2009

cp's OEIS Frontend

A066049 Numbers n such that 2*n^2 - 1 is a prime.

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A118674 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2.

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A129836 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 97)^2 = y^2.

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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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A207059 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+289)^2 = y^2.

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A129993 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+199)^2 = y^2.

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