A156035 -id:A156035 - cp's OEIS frontend

A159466 Positive numbers y such that y^2 is of the form x^2 + (x+127)^2 with integer x.

113, 127, 145, 533, 635, 757, 3085, 3683, 4397, 17977, 21463, 25625, 104777, 125095, 149353, 610685, 729107, 870493, 3559333, 4249547, 5073605, 20745313, 24768175, 29571137, 120912545, 144359503, 172353217, 704729957, 841388843, 1004548165

Offset: 1

Klaus Brockhaus, Apr 13 2009

(-15, a(1)) and (A129992(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2 + (x+127)^2 = y^2.
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 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (34947 + 21922*sqrt(2))/127^2 for n mod 3 = 1.

			(-15, a(1)) = (-15, 113) is a solution: (-15)^2 + (-15+127)^2 = 225 + 12544 = 12769 = 113^2.
(A129992(1), a(2)) = (0, 127) is a solution: 0^2 + (0+127)^2 = 16129 = 127^2.
(A129992(3), a(4)) = (308, 533) is a solution: 308^2 + (308+127)^2 = 94864 + 189225 = 284089 = 533^2.

G. C. Greubel, Table of n, a(n) for n = 1..3900
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A129992, A001653, 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:=[113,127,145,533,635,757]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Jun 15 2018

LinearRecurrence[{0,0,6,0,0,-1},{113,127,145,533,635,757},50] (* Harvey P. Dale, Feb 06 2015 *)

{forstep(n=-16, 500000000, [1, 3], if(issquare(2*n^2+254*n+16129, &k), print1(k, ",")))}

a(n) = 6*a(n-3) - a(n-6)for n > 6; a(1)=113, a(2)=127, a(3)=145, a(4)=533, a(5)=635, a(6)=757.
G.f.: (1-x)*(113+240*x+385*x^2+240*x^3+113*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 127*A001653(k) for k >= 1.

A159468 Decimal expansion of (34947+21922*sqrt(2))/127^2.

Original entry on oeis.org

4, 0, 8, 8, 8, 7, 0, 3, 4, 0, 0, 2, 9, 9, 4, 5, 4, 1, 8, 8, 0, 0, 3, 3, 6, 0, 5, 3, 8, 2, 3, 8, 5, 7, 7, 2, 6, 9, 7, 6, 5, 2, 3, 4, 5, 7, 8, 7, 1, 7, 4, 9, 9, 9, 4, 4, 3, 1, 0, 9, 7, 6, 0, 1, 6, 0, 1, 6, 3, 9, 1, 2, 1, 6, 3, 4, 7, 1, 4, 5, 2, 0, 7, 8, 1, 0, 8, 9, 6, 8, 4, 8, 8, 6, 2, 6, 4, 4, 0, 3, 0, 9, 3, 6, 1

Offset: 1

Klaus Brockhaus, Apr 13 2009

lim_{n -> infinity} b(n)/b(n-1) = (34947+21922*sqrt(2))/127^2 for n mod 3 = 0, b = A129992.

lim_{n -> infinity} b(n)/b(n-1) = (34947+21922*sqrt(2))/127^2 for n mod 3 = 1, b = A159466.

			(34947+21922*sqrt(2))/127^2 = 4.08887034002994541880...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A129992, A159466, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A159467 (decimal expansion of (129+16*sqrt(2))/127).

(34947 + 21922*Sqrt(2))/127^2; // G. C. Greubel, Mar 30 2018

with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((34947+21922*sqrt(2))/127^2))[1]; # Muniru A Asiru, Mar 31 2018

RealDigits[(34947+21922*Sqrt[2])/127^2,10,120][[1]] (* Harvey P. Dale, May 11 2012 *)

(34947+21922*sqrt(2))/127^2 \\ G. C. Greubel, Mar 30 2018

Equals (226+97*sqrt(2))/(226-97*sqrt(2)).

Equals (3+2*sqrt(2))*(16-sqrt(2))^2/(16+sqrt(2))^2.

A159589 Positive numbers y such that y^2 is of the form x^2+(x+449)^2 with integer x.

Original entry on oeis.org

421, 449, 481, 2045, 2245, 2465, 11849, 13021, 14309, 69049, 75881, 83389, 402445, 442265, 486025, 2345621, 2577709, 2832761, 13671281, 15023989, 16510541, 79682065, 87566225, 96230485, 464421109, 510373361, 560872369, 2706844589

Offset: 1

Klaus Brockhaus, Apr 18 2009

(-29,a(1)) and (A130004(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (451+30*sqrt(2))/449 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 1.

			(-29, a(1)) = (-29, 421) is a solution: (-29)^2+(-29+449)^2 = 841+176400 = 177241 = 421^2.
(A130004(1), a(2)) = (0, 449) is a solution: 0^2+(0+449)^2 = 201601 = 449^2.
(A130004(3), a(4)) = (1204, 2045) is a solution: 1204^2+(1204+449)^2 = 1449616+2732409 = 4182025 = 2045^2.

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

Cf. A130004, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159590 (decimal expansion of (451+30*sqrt(2))/449), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).

I:=[421,449,481,2045,2245,2465]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 08 2018

LinearRecurrence[{0,0,6,0,0,-1}, {421,449,481,2045,2245,2465}, 50] (* G. C. Greubel, May 08 2018 *)

{forstep(n=-32, 50000000, [3, 1], if(issquare(2*n^2+898*n+201601, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4)/(1- 6*x^3+x^6)) \\ G. C. Greubel, May 08 2018

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=421, a(2)=449, a(3)=481, a(4)=2045, a(5)=2245, a(6)=2465.
G.f.: (1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 449*A001653(k) for k >= 1.

A159690 Positive numbers y such that y^2 is of the form x^2+(x+881)^2 with integer x.

Original entry on oeis.org

841, 881, 925, 4121, 4405, 4709, 23885, 25549, 27329, 139189, 148889, 159265, 811249, 867785, 928261, 4728305, 5057821, 5410301, 27558581, 29479141, 31533545, 160623181, 171817025, 183790969, 936180505, 1001423009, 1071212269

Offset: 1

Klaus Brockhaus, Apr 21 2009

(-41,a(1)) and (A130014(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (883+42*sqrt(2))/881 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 1.

			(-41, a(1)) = (-41, 841) is a solution: (-41)^2+(-41+881)^2 = 1681+705600 = 707281 = 841^2.
(A130014(1), a(2)) = (0, 881) is a solution: 0^2+(0+881)^2 = 776161 = 881^2.
(A130014(3), a(4)) = (2440, 4121) is a solution: 2440^2+(2440+881)^2 = 5953600+11029041 = 16982641 = 4121^2.

G. C. Greubel, Table of n, a(n) for n = 1..3501
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A130014, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159691 (decimal expansion of (883+42*sqrt(2))/881), A159692 (decimal expansion of (2052963+1343918*sqrt(2))/881^2).

I:=[841, 881, 925, 4121, 4405, 4709]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Jun 02 2018

CoefficientList[Series[(1 - x)*(841 + 1722*x + 2647*x^2 + 1722*x^3 + 841*x^4)/(1 - 6*x^3 + x^6), {x,0,50}], x] (* or *) LinearRecurrence[{0, 0,6,0,0,-1}, {841, 881, 925, 4121, 4405, 4709}, 30] (* G. C. Greubel, Jun 02 2018 *)

{forstep(n=-44, 10000000, [3, 1], if(issquare(2*n^2+1762*n+776161, &k), print1(k, ",")))}

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=841, a(2)=881, a(3)=925, a(4)=4121, a(5)=4405, a(6)=4709.
G.f.: (1-x)*(841+1722*x+2647*x^2+1722*x^3+841*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 881*A001653(k) for k >= 1.

A159701 Positive numbers y such that y^2 is of the form x^2+(x+967)^2 with integer x.

Original entry on oeis.org

925, 967, 1013, 4537, 4835, 5153, 26297, 28043, 29905, 153245, 163423, 174277, 893173, 952495, 1015757, 5205793, 5551547, 5920265, 30341585, 32356787, 34505833, 176843717, 188589175, 201114733, 1030720717, 1099178263

Offset: 1

Klaus Brockhaus, Apr 21 2009

(-43, a(1)) and (A130017(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+967)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (969+44*sqrt(2))/967 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (2487411+1629850*sqrt(2))/967^2 for n mod 3 = 1.

			(-43, a(1)) = (-43, 925) is a solution: (-43)^2+(-43+967)^2 = 1849+853776 = 855625 = 925^2.
(A130017(1), a(2)) = (0, 967) is a solution: 0^2+(0+967)^2 = 935089 = 967^2.
(A130017(3), a(4)) = (2688, 4537) is a solution: 2688^2+(2688+967)^2 = 7225344+13359025 = 20584369 = 4537^2.

G. C. Greubel, Table of n, a(n) for n = 1..3895
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A130017, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159702 (decimal expansion of (969+44*sqrt(2))/967), A159703 (decimal expansion of (2487411+1629850*sqrt(2))/967^2).

I:=[925,967,1013,4537,4835,5153]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 22 2018

LinearRecurrence[{0,0,6,0,0,-1}, {925,967,1013,4537,4835,5153}, 40] (* G. C. Greubel, May 22 2018 *)

{forstep(n=-44, 10000000, [1, 3], if(issquare(2*n^2+1934*n+935089, &k), print1(k, ",")))};

x='x+O('x^30); Vec((1-x)*(925+1892*x+2905*x^2+1892*x^3+925*x^4)/( 1-6*x^3+x^6)) \\ G. C. Greubel, May 22 2018

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=925, a(2)=967, a(3)=1013, a(4)=4537, a(5)=4835, a(6)=5153.
G.f.: (1-x)*(925+1892*x+2905*x^2+1892*x^3+925*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 967*A001653(k) for k >= 1.

A159750 Positive numbers y such that y^2 is of the form x^2+(x+47)^2 with integer x.

Original entry on oeis.org

37, 47, 65, 157, 235, 353, 905, 1363, 2053, 5273, 7943, 11965, 30733, 46295, 69737, 179125, 269827, 406457, 1044017, 1572667, 2369005, 6084977, 9166175, 13807573, 35465845, 53424383, 80476433, 206710093, 311380123, 469051025, 1204794713

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-12, a(1)) and (A118675(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (51+14*sqrt(2))/47 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 for n mod 3 = 1.
For the generic case x^2+(x+p)^2=y^2 with p= m^2 -2 a prime number in A028871, m>=2, the x values are given by the sequence defined by: a(n)= 6*a(n-3) -a(n-6) +2*p with a(1)=0, a(2)= 2*m +2, a(3)= 3*m^2 -10*m +8, a(4)= 3*p, a(5)= 3*m^2 +10*m +8, a(6)= 20*m^2 -58*m +42. Y values are given by the sequence defined by: b(n)= 6*b(n-3) -b(n-6) with b(1)=p, b(2)= m^2 +2*m +2, b(3)= 5*m^2 -14*m +10, b(4)= 5*p, b(5)= 5*m^2 +14*m +10, b(6)= 29*m^2 -82*m +58. - Mohamed Bouhamida, Sep 09 2009

			(-12, a(1)) = (-12, 37) is a solution: (-12)^2+(-12+47)^2 = 144+1225 = 1369 = 37^2.
(A118675(1), a(2)) = (0, 47) is a solution: 0^2+(0+47)^2 = 2209 = 47^2.
(A118675(3), a(4)) = (85, 157) is a solution: 85^2+(85+47)^2 = 7225+17424 = 24649 = 157^2.

G. C. Greubel, Table of n, a(n) for n = 1..3900
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A118675, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).

I:=[37,47,65,157,235,353]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 22 2018

LinearRecurrence[{0,0,6,0,0,-1}, {37,47,65,157,235,353}, 50] (* G. C. Greubel, May 22 2018 *)

{forstep(n=-12, 100000000, [1, 3], if(issquare(2*n^2+94*n+2209, &k), print1(k, ",")))};

x='x+O('x^30); Vec((1-x)*(37+84*x+149*x^2+84*x^3+37*x^4)/(1 -6*x^3 +x^6)) \\ G. C. Greubel, May 22 2018

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=37, a(2)=47, a(3)=65, a(4)=157, a(5)=235, a(6)=353.
G.f.: (1-x)*(37+84*x+149*x^2+84*x^3+37*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 47*A001653(k) for k >= 1.

A159758 Positive numbers y such that y^2 is of the form x^2+(x+79)^2 with integer x.

Original entry on oeis.org

65, 79, 101, 289, 395, 541, 1669, 2291, 3145, 9725, 13351, 18329, 56681, 77815, 106829, 330361, 453539, 622645, 1925485, 2643419, 3629041, 11222549, 15406975, 21151601, 65409809, 89798431, 123280565, 381236305, 523383611, 718531789

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-16, a(1)) and (A118676(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+79)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (83+18*sqrt(2))/79 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (10659+6110*sqrt(2))/79^2 for n mod 3 = 1.
For the generic case x^2 + (x+p)^2 = y^2 with p = m^2 - 2 a prime number in A028871, m >= 5, the x values are given by the sequence defined by a(n) = 6*a(n-3) - a(n-6) + 2*p with a(1)=0, a(2) = 2*m + 2, a(3) = 3*m^2 - 10*m + 8, a(4) = 3*p, a(5) = 3*m^2 + 10*m + 8, a(6) = 20*m^2 - 58*m + 42. Y values are given by the sequence defined by b(n) = 6*b(n-3) - b(n-6) with b(1)=p, b(2) = m^2 + 2*m + 2, b(3) = 5*m^2 - 14*m + 10, b(4)= 5*p, b(5) = 5*m^2 + 14*m + 10, b(6) = 29*m^2 - 82*m + 58. - Mohamed Bouhamida, Sep 09 2009

			(-16, a(1)) = (-16, 65) is a solution: (-16)^2 + (-16+79)^2 = 256+3969 = 4225 = 65^2.
(A118676(1), a(2)) = (0, 79) is a solution: 0^2 + (0+79)^2 = 6241 = 79^2.
(A118676(3), a(4)) = (161, 289) is a solution: 161^2 + (161+79)^2 = 25921 + 57600 = 83521 = 289^2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A118676, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159759 (decimal expansion of (83+18*sqrt(2))/79), A159760 (decimal expansion of (10659+6110*sqrt(2))/79^2).

I:=[65,79,101,289,395,541]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 22 2018

RecurrenceTable[{a[1]==65,a[2]==79,a[3]==101,a[4]==289,a[5]==395, a[6]== 541, a[n]==6a[n-3]-a[n-6]},a[n],{n,30}] (* or *) LinearRecurrence[ {0,0,6,0,0,-1},{65,79,101,289,395,541},30] (* Harvey P. Dale, Oct 03 2011 *)

{forstep(n=-16, 10000000, [1, 3], if(issquare(2*n^2+158*n+6241, &k), print1(k, ",")))}

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=65, a(2)=79, a(3)=101, a(4)=289, a(5)=395, a(6)=541.
G.f.: (1-x)*(65+144*x+245*x^2+144*x^3+65*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 79*A001653(k) for k >= 1.

A159777 Positive numbers y such that y^2 is of the form x^2+(x+167)^2 with integer x.

Original entry on oeis.org

145, 167, 197, 673, 835, 1037, 3893, 4843, 6025, 22685, 28223, 35113, 132217, 164495, 204653, 770617, 958747, 1192805, 4491485, 5587987, 6952177, 26178293, 32569175, 40520257, 152578273, 189827063, 236169365, 889291345, 1106393203

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-24, a(1)) and (A130608(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (171+26*sqrt(2))/167 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (56211+34510*sqrt(2))/167^2 for n mod 3 = 1.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2 - 2 a prime number in A028871, m >= 5, the x values are given by the sequence defined by: a(n) = 6*a(n-3) - a(n-6) + 2*p with a(1)=0, a(2) = 2*m + 2, a(3) = 3*m^2 - 10*m + 8, a(4) = 3*p, a(5) = 3*m^2 + 10*m + 8, a(6) = 20*m^2 - 58*m + 42. Y values are given by the sequence defined by: b(n) = 6*b(n-3) - b(n-6) with b(1) = p, b(2) = m^2 + 2*m + 2, b(3) = 5*m^2 - 14*m + 10, b(4) = 5*p, b(5) = 5*m^2 + 14*m + 10, b(6) = 29*m^2 - 82*m + 58. - Mohamed Bouhamida, Sep 09 2009

			(-24, a(1)) = (-24, 145) is a solution: (-24)^2 + (-24+167)^2 = 576 + 20449 = 21025 = 145^2.
(A130608(1), a(2)) = (0, 167) is a solution: 0^2 + (0+167)^2 = 27889 = 167^2.
(A130608(3), a(4)) = (385, 673) is a solution: 385^2 + (385+167)^2 = 148225 + 304704 = 452929 = 673^2.

G. C. Greubel, Table of n, a(n) for n = 1..3900
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A130608, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159778 (decimal expansion of (171+26*sqrt(2))/167), A159779 (decimal expansion of (56211+34510*sqrt(2))/167^2).

I:=[145,167,197,673,835,1037]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 21 2018

LinearRecurrence[{0,0,6,0,0,-1}, {145,167,197,673,835,1037}, 50] (* G. C. Greubel, May 21 2018 *)

{forstep(n=-24, 10000000, [1, 3], if(issquare(2*n^2+334*n+27889, &k), print1(k, ",")))};

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=145, a(2)=167, a(3)=197, a(4)=673, a(5)=835, a(6)=1037.
G.f.: (1-x)*(145+312*x+509*x^2+312*x^3+145*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 167*A001653(k) for k >= 1.

A159809 Positive numbers y such that y^2 is of the form x^2+(x+223)^2 with integer x.

Original entry on oeis.org

197, 223, 257, 925, 1115, 1345, 5353, 6467, 7813, 31193, 37687, 45533, 181805, 219655, 265385, 1059637, 1280243, 1546777, 6176017, 7461803, 9015277, 35996465, 43490575, 52544885, 209802773, 253481647, 306254033, 1222820173, 1477399307

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-28, a(1)) and (A130609(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (227+30*sqrt(2))/223 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (105507+65798*sqrt(2))/223^2 for n mod 3 = 1.
For the generic case x^2 + (x+p)^2 = y^2 with p = m^2 - 2 a prime number in A028871, m >= 5, the x values are given by the sequence defined by a(n) = 6*a(n-3) - a(n-6) + 2*p with a(1)=0, a(2) = 2*m + 2, a(3) = 3*m^2 - 10*m + 8, a(4) = 3*p, a(5) = 3*m^2 + 10*m + 8, a(6) = 20*m^2 - 58*m + 42. Y values are given by the sequence defined by b(n) = 6*b(n-3) - b(n-6) with b(1) = p, b(2) = m^2 + 2*m + 2, b(3) = 5*m^2 - 14*m + 10, b(4) = 5*p, b(5) = 5m^2 + 14*m + 10, b(6) = 29*m^2 - 82*m + 58. - Mohamed Bouhamida, Sep 09 2009

			(-28, a(1)) = (-28, 197) is a solution: (-28)^2 + (-28+223)^2 = 784 + 38025 = 38809 = 197^2.
(A130609(1), a(2)) = (0, 223) is a solution: 0^2 + (0+223)^2 = 49729 = 223^2.
(A130609(3), a(4)) = (533, 925) is a solution: 533^2 + (533+223)^2 = 284089 + 571536 = 855625 = 925^2.

G. C. Greubel, Table of n, a(n) for n = 1..3900
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A130609, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A130610 (decimal expansion of (227+30*sqrt(2))/223), A130611 (decimal expansion of (105507+65798*sqrt(2))/223^2).

I:=[197,223,257,925,1115,1345]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 21 2018

LinearRecurrence[{0,0,6,0,0,-1}, {197,223,257,925,1115,1345}, 50] (* G. C. Greubel, May 21 2018 *)

{forstep(n=-28, 10000000, [1, 3], if(issquare(2*n^2+446*n+49729, &k), print1(k, ",")))};

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=197, a(2)=223, a(3)=257, a(4)=925, a(5)=1115, a(6)=1345.
G.f.: (1-x)*(197+420*x+677*x^2+420*x^3+197*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 223*A001653(k) for k >= 1.

A159844 Positive numbers y such that y^2 is of the form x^2+(x+359)^2 with integer x.

Original entry on oeis.org

325, 359, 401, 1549, 1795, 2081, 8969, 10411, 12085, 52265, 60671, 70429, 304621, 353615, 410489, 1775461, 2061019, 2392505, 10348145, 12012499, 13944541, 60313409, 70013975, 81274741, 351532309, 408071351, 473703905, 2048880445

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-36, a(1)) and (A130610(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (363+38*sqrt(2))/359 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (293619+186550*sqrt(2))/359^2 for n mod 3 = 1.
For the generic case x^2+(x+p)^2=y^2 with p=m^2-2 a prime number in A028871, m>=5, 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+2, a(3)=3m^2-10m+8, a(4)=3p, a(5)=3m^2+10m+8, a(6)=20m^2-58m+42.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=m^2+2m+2, b(3)=5m^2-14m+10, b(4)=5p, b(5)=5m^2+14m+10, b(6)=29m^2-82m+58. [Mohamed Bouhamida, Sep 09 2009]

			(-36, a(1)) = (-36, 325) is a solution: (-36)^2+(-36+359)^2 = 1296+104329 = 105625 = 325^2.
(A130610(1), a(2)) = (0, 359) is a solution: 0^2+(0+359)^2 = 128881 = 359^2.
(A130610(3), a(4)) = (901, 1549) is a solution: 901^2+(901+359)^2 = 811801+1587600 = 2399401 = 1549^2.

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

Cf. A130610, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159845 (decimal expansion of (363+38*sqrt(2))/359), A159846 (decimal expansion of (293619+186550*sqrt(2))/359^2).

I:=[325, 359, 401, 1549, 1795, 2081]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 19 2018

t={325,359,401,1549,1795,2081}; Do[AppendTo[t, 6*t[[-3]]-t[[-6]]], {25}]; t
CoefficientList[Series[(325+359 x+401 x^2-401 x^3-359 x^4-325 x^5)/(1-6 x^3+x^6),{x,0,30}],x]  (* Harvey P. Dale, Feb 16 2011 *)
LinearRecurrence[{0,0,6,0,0,-1}, {325, 359, 401, 1549, 1795, 2081}, 50] (* G. C. Greubel, May 19 2018 *)

{forstep(n=-36, 10000000, [1, 3], if(issquare(2*n^2+718*n+128881, &k), print1(k, ",")))}

V=[]; v=[[-323,-325], [-323,325], [0,-359], [-359,359], [-399,-401], [399,401]]; for(n=1,100,u=[]; for(i=1,#v,if(v[i][2]>0, u=concat(u,v[i][2])); t=3*v[i][1]+2*v[i][2]+359; v[i][2]=4*v[i][1]+3*v[i][2]+718; v[i][1]=t); V=concat(V,u)); vecsort(V,,8) \\ Charles R Greathouse IV, Feb 14 2011

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=325, a(2)=359, a(3)=401, a(4)=1549, a(5)=1795, a(6)=2081.
G.f.: (1-x)*(325+684*x+1085*x^2+684*x^3+325*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 359*A001653(k) for k >= 1.

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