A156035 -id:A156035 - cp's OEIS frontend

A159890 Positive numbers y such that y^2 is of the form x^2+(x+439)^2 with integer x.

401, 439, 485, 1921, 2195, 2509, 11125, 12731, 14569, 64829, 74191, 84905, 377849, 432415, 494861, 2202265, 2520299, 2884261, 12835741, 14689379, 16810705, 74812181, 85615975, 97979969, 436037345, 499006471, 571069109, 2541411889

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-40, a(1)) and (A130645(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (443+42*sqrt(2))/439 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (450483+287918*sqrt(2))/439^2 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

			(-40, a(1)) = (-40, 401) is a solution: (-40)^2+(-40+439)^2 = 1600+159201 = 160801 = 401^2.
(A130645(1), a(2)) = (0, 439) is a solution: 0^2+(0+439)^2 = 192721 = 439^2.
(A130645(3), a(4)) = (1121, 1921) is a solution: 1121^2+(1121+439)^2 = 1256641+2433600 = 3690241 = 1921^2.

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. A130645, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159891 (decimal expansion of (443+42*sqrt(2))/439), A159892 (decimal expansion of (450483+287918*sqrt(2))/439^2).

I:=[401,439,485,1921,2195,2509]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 17 2018

LinearRecurrence[{0,0,6,0,0,-1}, {401,439,485,1921,2195,2509}, 50] (* G. C. Greubel, May 17 2018 *)

{forstep(n=-40, 10000000, [1, 3], if(issquare(2*n^2+878*n+192721, &k), print1(k, ",")))}

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=401, a(2)=439, a(3)=485, a(4)=1921, a(5)=2195, a(6)=2509.
G.f.: (1-x)*(401+840*x+1325*x^2+840*x^3+401*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 439*A001653(k) for k >= 1.

A159893 Positive numbers y such that y^2 is of the form x^2+(x+727)^2 with integer x.

Original entry on oeis.org

677, 727, 785, 3277, 3635, 4033, 18985, 21083, 23413, 110633, 122863, 136445, 644813, 716095, 795257, 3758245, 4173707, 4635097, 21904657, 24326147, 27015325, 127669697, 141783175, 157456853, 744113525, 826372903, 917725793

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-52, a(1)) and (A130646(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (731+54*sqrt(2))/727 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 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

			(-52, a(1)) = (-52, 677) is a solution: (-52)^2+(-52+727)^2 = 2704+455625 = 458329 = 677^2.
(A130646(1), a(2)) = (0, 727) is a solution: 0^2+(0+727)^2 = 528529 = 727^2.
(A130646(3), a(4)) = (1925, 3277) is a solution: 1925^2+(1925+727)^2 = 3705625+7033104 = 10738729 = 3277^2.

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. A130646, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).

I:=[677,727,785,3277,3635,4033]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 17 2018

LinearRecurrence[{0,0,6,0,0,-1}, {677,727,785,3277,3635,4033}, 50] (* G. C. Greubel, May 17 2018 *)

{forstep(n=-52, 10000000, [1, 3], if(issquare(2*n^2+1454*n+528529, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=677, a(2)=727, a(3)=785, a(4)=3277, a(5)=3635, a(6)=4033.
G.f.: (1-x)*(677+1404*x+2189*x^2+1404*x^3+677*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 727*A001653(k) for k >= 1.

A159896 Positive numbers y such that y^2 is of the form x^2+(x+839)^2 with integer x.

Original entry on oeis.org

785, 839, 901, 3809, 4195, 4621, 22069, 24331, 26825, 128605, 141791, 156329, 749561, 826415, 911149, 4368761, 4816699, 5310565, 25463005, 28073779, 30952241, 148409269, 163625975, 180402881, 864992609, 953682071, 1051465045

Offset: 1

Klaus Brockhaus, Apr 30 2009

(-56, a(1)) and (A130647(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (843+58*sqrt(2))/839 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1760979+1141390*sqrt(2))/839^2 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

			(-56, a(1)) = (-56, 785) is a solution: (-56)^2+(-56+839)^2 = 3136+613089 = 616225 = 785^2.
(A130647(1), a(2)) = (0, 839) is a solution: 0^2+(0+839)^2 = 703921 = 839^2.
(A130647(3), a(4)) = (2241, 3809) is a solution: 2241^2+(2241+839)^2 = 5022081+9486400 = 14508481 = 3809^2.

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. A130647, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159897 (decimal expansion of (843+58*sqrt(2))/839), A159898 (decimal expansion of (1760979+1141390*sqrt(2))/839^2).

I:=[785,839,901,3809,4195,4621]; [n le 6 select I[n] else 6*Self(n-3) -Self(n-6): n in [1..30]]; // G. C. Greubel, May 17 2018

LinearRecurrence[{0,0,6,0,0,-1}, {785,839,901,3809,4195,4621}, 30] (* Harvey P. Dale, Mar 03 2013 *)

{forstep(n=-56, 10000000, [1, 3], if(issquare(2*n^2+1678*n+703921, &k), print1(k, ",")))}

is(n,p=839)=for(m=sqrtint((max(n,984)^2-p^2)\2)-p\2,n,m^2+(m+p)^2A159896(n)=(matrix(6,6,i,j,if(i<6,i+1==j,j==4,6,j==1,-1))^n*[785,839,901,3809,4195,4621]~)[1] \\ M. F. Hasler, May 17 2018

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=785, a(2)=839, a(3)=901, a(4)=3809, a(5)=4195, a(6)=4621.
G.f.: (1-x)*(785+1624*x+2525*x^2+1624*x^3+785*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 839*A001653(k) for k >= 1.

A160041 Positive numbers y such that y^2 is of the form x^2+(x+73)^2 with integer x.

Original entry on oeis.org

53, 73, 125, 193, 365, 697, 1105, 2117, 4057, 6437, 12337, 23645, 37517, 71905, 137813, 218665, 419093, 803233, 1274473, 2442653, 4681585, 7428173, 14236825, 27286277, 43294565, 82978297, 159036077, 252339217, 483632957, 926930185

Offset: 1

Klaus Brockhaus, May 04 2009

nonn

(-28, a(1)) and (A129289(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+73)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (89+36*sqrt(2))/73 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (5907+1802*sqrt(2))/73^2 for n mod 3 = 1.

			(-28, a(1)) = (-28, 53) is a solution: (-28)^2+(-28+73)^2 = 784+2025 = 2809 = 53^2.
(A129289(1), a(2)) = (0, 73) is a solution: 0^2+(0+73)^2 = 5329 = 73^2.
(A129289(3), a(4)) = (95, 193) is a solution: 95^2+(95+73)^2 = 9025+28224 = 37249 = 193^2.

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. A129289, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160042 (decimal expansion of (89+36*sqrt(2))/73), A160043 (decimal expansion of (5907+1802*sqrt(2))/73^2).

I:=[53,73,125,193,365,697]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 21 2018

LinearRecurrence[{0,0,6,0,0,-1}, {53,73,125,193,365,697}, 50] (* G. C. Greubel, Apr 21 2018 *)

{forstep(n=-28, 10000000, [3, 1], if(issquare(2*n^2+146*n+5329, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(53 +126*x +251*x^2 +126*x^3 +53*x^4)/(1 -6*x^3+x^6)) \\ G. C. Greubel, Apr 21 2018

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=53, a(2)=73, a(3)=125, a(4)=193, a(5)=365, a(6)=697.
G.f.: (1-x)*(53 +126*x +251*x^2 +126*x^3 +53*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 73*A001653(k) for k >= 1.

A160055 Positive numbers y such that y^2 is of the form x^2+(x+89)^2 with integer x.

Original entry on oeis.org

65, 89, 149, 241, 445, 829, 1381, 2581, 4825, 8045, 15041, 28121, 46889, 87665, 163901, 273289, 510949, 955285, 1592845, 2978029, 5567809, 9283781, 17357225, 32451569, 54109841, 101165321, 189141605, 315375265, 589634701, 1102398061

Offset: 1

Klaus Brockhaus, May 04 2009

nonn

(-33, a(1)) and (A129298(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+89)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (107+42*sqrt(2))/89 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 1.

			(-33, a(1)) = (-33, 65) is a solution: (-33)^2+(-33+89)^2 = 1089+3136 = 4225 = 65^2.
(A129298(1), a(2)) = (0, 89) is a solution: 0^2+(0+89)^2 = 7921 = 89^2.
(A129298(3), a(4)) = (120, 241) is a solution: 120^2+(120+89)^2 = 14400+43681 = 58081 = 241^2.

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. A129298, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).

LinearRecurrence[{0,0,6,0,0,-1},{65,89,149,241,445,829},40] (* Harvey P. Dale, Feb 04 2015 *)

{forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=65, a(2)=89, a(3)=149, a(4)=241, a(5)=445, a(6)=829.
G.f.: (1-x)*(65+154*x+303*x^2+154*x^3+65*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 89*A001653(k) for k >= 1.

A160090 Positive numbers y such that y^2 is of the form x^2 + (x + 569)^2 with integer x.

Original entry on oeis.org

485, 569, 689, 2221, 2845, 3649, 12841, 16501, 21205, 74825, 96161, 123581, 436109, 560465, 720281, 2541829, 3266629, 4198105, 14814865, 19039309, 24468349, 86347361, 110969225, 142611989, 503269301, 646776041, 831203585, 2933268445

Offset: 1

Klaus Brockhaus, May 04 2009

nonn

(-93, a(1)) and (A101152(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (587+102*sqrt(2))/569 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (617139+371510*sqrt(2))/569^2 for n mod 3 = 1.

			(-93, a(1)) = (-93, 485) is a solution: (-93)^2+(-93+569)^2 = 8649+226576 = 235225 = 485^2.
(A101152(1), a(2)) = (0, 569) is a solution: 0^2+(0+569)^2 = 323761= 569^2.
(A101152(3), a(4)) = (1260, 2221) is a solution: 1260^2+(1260+569)^2 = 1587600+3345241 = 4932841 = 2221^2.

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. A101152, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160091 (decimal expansion of (587+102*sqrt(2))/569), A160092 (decimal expansion of (617139+371510*sqrt(2))/569^2).

I:=[485,569,689,2221,2845,3649]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 21 2018

LinearRecurrence[{0,0,6,0,0,-1}, {485,569,689,2221,2845,3649}, 50] (* G. C. Greubel, Apr 21 2018 *)

{forstep(n=-96, 10000000, [3, 1], if(issquare(2*n^2+1138*n+323761, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4)/(1-6*x^3+x^6)) \\ G. C. Greubel, Apr 21 2018

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=485, a(2)=569, a(3)=689, a(4)=2221, a(5)=2845, a(6)=3649.
G.f.: (1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 569*A001653(k) for k >= 1.

A160098 Positive numbers y such that y^2 is of the form x^2+(x+601)^2 with integer x.

Original entry on oeis.org

425, 601, 1261, 1289, 3005, 7141, 7309, 17429, 41585, 42565, 101569, 242369, 248081, 591985, 1412629, 1445921, 3450341, 8233405, 8427445, 20110061, 47987801, 49118749, 117210025, 279693401, 286285049, 683150089, 1630172605

Offset: 1

Klaus Brockhaus, May 18 2009

nonn

(-297, a(1)) and (A111258(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+601)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (843+418*sqrt(2))/601 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (361299+5950*sqrt(2))/601^2 for n mod 3 = 1.

			(-297, a(1)) = (-297, 425) is a solution: (-297)^2+(-297+601)^2 = 88209+92416 = 180625 = 425^2.
(A111258(1), a(2)) = (0, 601) is a solution: 0^2+(0+601)^2 = 361201 = 601^2.
(A111258(3), a(4)) = (560, 1289) is a solution: 560^2+(560+601)^2 = 313600+1347921 = 1661521 = 1289^2.

G. C. Greubel, Table of n, a(n) for n = 1..3875
Index entries for linear recurrences with constant coefficients, signature (0, 0, 6, 0, 0, -1).

Cf. A111258, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160099 (decimal expansion of (843+418*sqrt(2))/601), A160100 (decimal expansion of (361299+5950*sqrt(2))/601^2).

I:=[425,601,1261,1289,3005,7141]; [n le 6 select I[n] else 5*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 22 2018

LinearRecurrence[{0,0,6,0,0,-1}, {425,601,1261,1289,3005,7141}, 50] (* G. C. Greubel, Apr 22 2018 *)

{forstep(n=-300, 10000000, [3, 1], if(issquare(2*n^2+1202*n+361201, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4 )/(1-6*x^3+x^6)) \\ G. C. Greubel, Apr 22 2018

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=425, a(2)=601, a(3)=1261, a(4)=1289, a(5)=3005, a(6)=7141.
G.f.: (1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 601*A001653(k) for k >= 1.

A160176 Positive numbers y such that y^2 is of the form x^2+(x+617)^2 with integer x.

Original entry on oeis.org

533, 617, 733, 2465, 3085, 3865, 14257, 17893, 22457, 83077, 104273, 130877, 484205, 607745, 762805, 2822153, 3542197, 4445953, 16448713, 20645437, 25912913, 95870125, 120330425, 151031525, 558772037, 701337113, 880276237

Offset: 1

Klaus Brockhaus, May 18 2009

nonn

(-92, a(1)) and (A115135(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+617)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (633+100*sqrt(2))/617 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (755667+461578*sqrt(2))/617^2 for n mod 3 = 1.

			(-92, a(1)) = (-92, 533) is a solution: (-92)^2+(-92+617)^2 = 8464+275625 = 284089 = 533^2.
(A115135(1), a(2)) = (0, 617) is a solution: 0^2+(0+617)^2 = 380689 = 617^2.
(A115135(3), a(4)) = (1407, 2465) is a solution: 1407^2+(1407+617)^2 = 1979649+4096576 = 6076225 = 2465^2.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A115135, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160177 (decimal expansion of (633+100*sqrt(2))/617), A160178 (decimal expansion of (755667+461578*sqrt(2))/617^2).

I:=[533,617,733,2465,3085,3865]; [n le 6 select I[n] else 6*Self(n31) -Self(n-6): n in [1..30]]; // G. C. Greubel, May 04 2018

LinearRecurrence[{0,0,6,0,0,-1}, {533,617,733,2465,3085,3865}, 50] (* G. C. Greubel, May 04 2018 *)

{forstep(n=-92, 10000000, [3, 1], if(issquare(2*n^2+1234*n+380689, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6)) \\ G. C. Greubel, May 04 2018

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=533, a(2)=617, a(3)=733, a(4)=2465, a(5)=3085, a(6)=3865.
G.f.: (1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 617*A001653(k) for k >= 1.

A160206 Positive numbers y such that y^2 is of the form x^2+(x+857)^2 with integer x.

Original entry on oeis.org

697, 857, 1117, 3065, 4285, 6005, 17693, 24853, 34913, 103093, 144833, 203473, 600865, 844145, 1185925, 3502097, 4920037, 6912077, 20411717, 28676077, 40286537, 118968205, 167136425, 234807145, 693397513, 974142473, 1368556333

Offset: 1

Klaus Brockhaus, May 18 2009

nonn

(-185, a(1)) and (A129857(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 for n mod 3 = 1.

			(-185, a(1)) = (-185, 697) is a solution: (-185)^2+(-185+857)^2 = 34225+451584 = 485809 = 697^2.
(A129857(1), a(2)) = (0, 857) is a solution: 0^2+(0+857)^2 = 734449 = 857^2.
(A129857(3), a(4)) = (1696, 3065) is a solution: 1696^2+(1696+857)^2 = 2876416+6517809 = 9394225 = 3065^2.

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. A129857, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).

I:=[697,857,1117,3065,4285,6005]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..40]]; // G. C. Greubel, May 14 2018

LinearRecurrence[{0,0,6,0,0,-1}, {697,857,1117,3065,4285,6005}, 50] (* G. C. Greubel, May 14 2018 *)

{forstep(n=-188, 10000000, [3, 1], if(issquare(2*n^2 +1714*n +734449, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(697+1554*x+2671*x^2+1554*x^3 +697*x^4 )/(1-6*x^3+x^6)) \\ G. C. Greubel, May 14 2018

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=697, a(2)=857, a(3)=1117, a(4)=3065, a(5)=4285, a(6)=6005.
G.f.: (1-x)*(697+1554*x+2671*x^2+1554*x^3+697*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 857*A001653(k) for k >= 1.

A161478 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+113)^2 = y^2.

Original entry on oeis.org

0, 52, 175, 339, 615, 1312, 2260, 3864, 7923, 13447, 22795, 46452, 78648, 133132, 271015, 458667, 776223, 1579864, 2673580, 4524432, 9208395, 15583039, 26370595, 53670732, 90824880, 153699364, 312816223, 529366467, 895825815, 1823226832, 3085374148

Offset: 1

Klaus Brockhaus, Jun 13 2009

nonn

Corresponding values y of solutions (x, y) are in A161479.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (129+44*sqrt(2))/113 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (16131+6970*sqrt(2))/113^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 6, -6, 0, -1, 1).

Cf. A161479, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161480 (decimal expansion of (129+44*sqrt(2))/113), A161481 (decimal expansion of (16131+6970*sqrt(2))/113^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,52,175,339,615,1312,2260},72] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 100000000, [3, 1], if(issquare(2*n^2+226*n+12769), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+226 for n > 6; a(1)=0, a(2)=52, a(3)=175, a(4)=339, a(5)=615, a(6)=1312.
G.f.: x*(52+123*x+164*x^2-36*x^3-41*x^4-36*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 113*A001652(k) for k >= 0.

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A160090 Positive numbers y such that y^2 is of the form x^2 + (x + 569)^2 with integer x.

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