A028871 -id:A028871 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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.

A235640 Primes p of the form n^2 + 1234567890 where 1234567890 is the first pandigital number with digits in order.

Original entry on oeis.org

1234567891, 1234568059, 1234569571, 1234574779, 1234576171, 1234579771, 1234592539, 1234595779, 1234609099, 1234625011, 1234625971, 1234634971, 1234647979, 1234669651, 1234692499, 1234743451, 1234753651, 1234769491, 1234780411, 1234900819, 1234948579

Offset: 1

K. D. Bajpai, Apr 20 2014

nonn

			1234567891 is a prime and appears in the sequence because 1234567891 = 1^2 + 1234567890.
1234568059 is a prime and appears in the sequence because 1234568059 = 13^2 + 1234567890.

K. D. Bajpai, Table of n, a(n) for n = 1..6694

Cf. A000040, A002496, A028871, A050289, A056899, A240587, A234812.

KD := proc() local a; a:=n^2+1234567890; if isprime(a) then RETURN (a); fi; end: seq(KD(), n=1..2000);

Select[Table[k^2+1234567890,{k,1,2000}],PrimeQ]
c=0; a=n^2+1234567890; Do[If[PrimeQ[a],c=c+1; Print[c," ",a]], {n,0,200000}]  (*b-file*)

A089623 Numbers n such that n^2 + 2n - 1 is prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 18, 20, 26, 28, 32, 34, 36, 42, 46, 48, 54, 60, 62, 68, 70, 74, 76, 88, 92, 102, 106, 116, 118, 120, 126, 130, 134, 138, 144, 154, 160, 168, 172, 176, 182, 190, 204, 210, 216, 222, 230, 232, 236, 238, 246, 252, 256, 258, 264, 266, 272

Offset: 1

Giovanni Teofilatto, Dec 31 2003

Equivalently, numbers n such that (n+1)^2 - 2 is prime.

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A088572. See A028871 for the actual primes.

[n: n in [0..400] | IsPrime(n^2+2*n-1)]; // Vincenzo Librandi, Dec 17 2010

Select[Range[1000], PrimeQ[#^2 + 2 # - 1] &] (* Vincenzo Librandi, May 24 2014 *)

is(n)=isprime(n^2+2*n-1) \\ Charles R Greathouse IV, Jun 05 2017

a(n) = 2*A088572(n).

Corrected (1 prepended and 96 replaced with 92) by Vincenzo Librandi, Dec 17 2010

cp's OEIS Frontend

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

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

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

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

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

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

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