A129298 -id:A129298 - cp's OEIS frontend

A101152 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.

0, 111, 1260, 1707, 2280, 8791, 11380, 14707, 52624, 67711, 87100, 308091, 396024, 509031, 1797060, 2309571, 2968224, 10475407, 13462540, 17301451, 61056520, 78466807, 100841620, 355864851, 457339440, 587749407, 2074133724, 2665570971

Offset: 1

Mohamed Bouhamida, Jun 03 2007

Also values x of Pythagorean triples (x, x+569, y).
Corresponding values y of solutions (x, y) are in A160090.
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 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (617139+371510*sqrt(2))/569^2 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. A160090, A129298, A001652, 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:=[0,111,1260,1707,2280,8791,11380]; [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, Apr 21 2018

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,111,1260,1707,2280,8791,11380}, 50] (* G. C. Greubel, Apr 21 2018 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1138*n+323761), print1(n, ",")))}

x='x+O('x^30); concat([0], Vec(x*(111 +1149*x +447*x^2 -93*x^3 -383*x^4 -93*x^5)/((1-x)*(1-6*x^3 +x^6)))) \\ G. C. Greubel, Apr 21 2018

a(n) = 6*a(n-3) - a(n-6) + 1138 for n > 6; a(1)=0, a(2)=111, a(3)=1260, a(4)=1707, a(5)=2280, a(6)=8791.
G.f.: x*(111 +1149*x +447*x^2 -93*x^3 -383*x^4 -93*x^5)/((1-x)*(1-6*x^3 +x^6)).
a(3*k+1) = 569*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, May 04 2009

A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

Original entry on oeis.org

0, 115, 136, 411, 1036, 1155, 2740, 6375, 7068, 16303, 37488, 41527, 95352, 218827, 242368, 556083, 1275748, 1412955, 3241420, 7435935, 8235636, 18892711, 43340136, 48001135, 110115120, 252605155, 279771448, 641798283, 1472291068, 1630627827, 3740674852

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+137, y).
Corresponding values y of solutions (x, y) are in A157213.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 0.

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

Cf. A157213, A001652, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))), A129288, A129289, A129298.

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,115,136,411,1036,1155,2740},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 1500000000, [3, 1], if(issquare(2*n^2+274*n+18769), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+274 for n > 6; a(1)=0, a(2)=115, a(3)=136, a(4)=411, a(5)=1036, a(6)=1155.
G.f.: x*(115+21*x+275*x^2-65*x^3-7*x^4-65*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 137*A001652(k) for k >= 0.

Edited and extended by Klaus Brockhaus, Feb 25 2009

A129625 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

Original entry on oeis.org

0, 75, 432, 699, 1092, 3115, 4660, 6943, 18724, 27727, 41032, 109695, 162168, 239715, 639912, 945747, 1397724, 3730243, 5512780, 8147095, 21742012, 32131399, 47485312, 126722295, 187276080, 276765243, 738592224, 1091525547, 1613106612

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+233, y).
Corresponding values y of solutions (x, y) are in A157297.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 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. A157297, A001652, A129288, A129289, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).

I:=[0,75,432,699,1092,3115,4660]; [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, Mar 29 2018

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,75,432,699,1092,3115,4660}, 50] (* G. C. Greubel, Mar 29 2018 *)

{forstep(n=0, 1700000000, [3, 1], if(issquare(2*n^2+466*n+54289), print1(n, ",")))};

a(n) = 6*a(n-3) -a(n-6) +466 for n > 6; a(1)=0, a(2)=75, a(3)=432, a(4)=699, a(5)=1092, a(6)=3115.
G.f.: x*(75 +357*x +267*x^2 -57*x^3 -119*x^4 -57*x^5)/((1-x)*(1 -6*x^3 +x^6)).
a(3*k+1) = 233*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 11 2009

A129640 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+313)^2 = y^2.

Original entry on oeis.org

0, 155, 464, 939, 1764, 3515, 6260, 11055, 21252, 37247, 65192, 124623, 217848, 380723, 727112, 1270467, 2219772, 4238675, 7405580, 12938535, 24705564, 43163639, 75412064, 143995335, 251576880, 439534475, 839267072, 1466298267, 2561795412, 4891607723

Offset: 1

Mohamed Bouhamida, May 31 2007

Also values x of Pythagorean triples (x, x+313, y).
Corresponding values y of solutions (x, y) are in A160574.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (363+130*sqrt(2))/313 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (119187+47998*sqrt(2))/313^2 for n mod 3 = 0.

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

Cf. A160574, A001652, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A160575 (decimal expansion of (363+130*sqrt(2))/313), A160576 (decimal expansion of (119187+47998*sqrt(2))/313^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 155, 464, 939, 1764, 3515, 6260}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+626*n+97969), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+626 for n > 6; a(1)=0, a(2)=155, a(3)=464, a(4)=939, a(5)=1764, a(6)=3515.
G.f.: x*(155+309*x+475*x^2-105*x^3-103*x^4-105*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 313*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Jun 08 2009

A129626 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

Original entry on oeis.org

0, 76, 559, 843, 1239, 3976, 5620, 7920, 23859, 33439, 46843, 139740, 195576, 273700, 815143, 1140579, 1595919, 4751680, 6648460, 9302376, 27695499, 38750743, 54218899, 161421876, 225856560, 316011580, 940836319, 1316389179, 1841851143, 5483596600

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+281, y).
Corresponding values y of solutions (x, y) are in A157348.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 0.

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

Cf. A157348, A001652, A129288, A129289, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 76, 559, 843, 1239, 3976, 5620}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 1000000000, [3, 1], if(issquare(2*n^2+562*n+78961), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+562 for n > 6; a(1)=0, a(2)=76, a(3)=559, a(4)=843, a(5)=1239, a(6)=3976.
G.f.: x*(76+483*x+284*x^2-60*x^3-161*x^4-60*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 281*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Apr 12 2009

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.

A160056 Decimal expansion of (107+42*sqrt(2))/89.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 04 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129298.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160055.

			(107+42*sqrt(2))/89 = 1.86962887213112350617...

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

Cf. A129298, A160055, A002193 (decimal expansion of sqrt(2)), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).

(107+42*Sqrt(2))/89; // G. C. Greubel, Apr 15 2018

RealDigits[(107+42*Sqrt[2])/89, 10, 100][[1]] (* G. C. Greubel, Apr 15 2018 *)

(107+42*sqrt(2))/89 \\ G. C. Greubel, Apr 15 2018

Equals (14+3*sqrt(2))/(14-3*sqrt(2)).

A160057 Decimal expansion of (8979+2990*sqrt(2))/89^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 04 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129298.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160055.

			(8979+2990*sqrt(2))/89^2 = 1.66740292279959022799...

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

Cf. A129298, A160055, A002193 (decimal expansion of sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89).

(8979+2990*Sqrt(2))/89^2; // G. C. Greubel, Apr 15 2018

RealDigits[(8979+2990*Sqrt[2])/89^2, 10, 100][[1]] (* G. C. Greubel, Apr 15 2018 *)

(8979+2990*sqrt(2))/89^2 \\ G. C. Greubel, Apr 15 2018

Equals (130+23*sqrt(2))/(130-23*sqrt(2)).

Equals (3+2*sqrt(2))*(14- 3*sqrt(2) )^2/(14+3*sqrt(2))^2.

A201916 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2737)^2 = y^2.

Original entry on oeis.org

0, 75, 203, 323, 552, 708, 1020, 1127, 1311, 1428, 1608, 1820, 1955, 2336, 2675, 3128, 3311, 3627, 3927, 4140, 4508, 4743, 5535, 6003, 6800, 7280, 7848, 8211, 8588, 9240, 9860, 11063, 11895, 13583, 14168, 15180, 15827, 16827, 18011, 18768, 20915, 22836

Offset: 1

T. D. Noe, Feb 09 2012

Note that 2737 = 7 * 17 * 23, the product of the first three distinct primes in A058529 (and A001132) and hence the smallest such number. This sequence satisfies a linear difference equation of order 55 whose 55 initial terms can be found by running the Mathematica program.
There are many sequences like this one. What determines the order of the linear difference equation? All primes p have order 7. For those p, it appears that p^2 has order 11, p^3 order 15, and p^i order 3+4*i. It appears that for semiprimes p*q (with p > q), the order is 19. What is the next term of the sequence beginning 3, 7, 19, 55, 163? This could be sequence A052919, which is 1 + 2*3^f, where f is the number of primes.
The crossref list is thought to be complete up to Feb 14 2012.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A001652 (1), A076296 (7), A118120 (17), A118337 (23), A118674 (31).
Cf. A129288 (41), A118675 (47), A118554 (49), A118673 (71), A129289 (73).
Cf. A118676 (79), A129298 (89), A129836 (97), A157119 (103), A161478 (113).
Cf. A129837 (119), A129992 (127), A129544 (137), A161482 (151).
Cf. A206426 (161), A130608 (167), A161486 (191), A185394 (193).
Cf. A129993 (199), A198294 (217), A130609 (223), A129625 (233).
Cf. A204765 (239), A129991 (241), A207058 (263), A129626 (281).
Cf. A205644 (287), A207059 (289), A129640 (313), A205672 (329).
Cf. A129999 (337), A118611 (343), A130610 (359), A207060 (401).
Cf. A129641 (409), A207061 (433), A130645 (439), A130004 (449).
Cf. A129642 (457), A129725 (521), A101152 (569), A130005 (577).
Cf. A207075 (479), A207076 (487), A207077 (497), A207078 (511).
Cf. A111258 (601), A115135 (617), A130013 (647), A130646 (727).
Cf. A122694 (761), A123654 (809), A129010 (833), A130647 (839).
Cf. A129857 (857), A130014 (881), A129974 (937), A129975 (953).
Cf. A130017 (967), A118630 (2401), A118576 (16807).

d = 2737; terms = 100; t = Select[Range[0, 55000], IntegerQ[Sqrt[#^2 + (#+d)^2]] &]; Do[AppendTo[t, t[[-1]] + 6*t[[-27]] - 6*t[[-28]] - t[[-54]] + t[[-55]]], {terms-55}]; t

a(n) = a(n-1) + 6*a(n-27) - 6*a(n-28) - a(n-54) + a(n-55), where the 55 initial terms can be computed using the Mathematica program.

G.f.: x^2*(73*x^53 +116*x^52 +100*x^51 +171*x^50 +104*x^49 +184*x^48 +57*x^47 +92*x^46 +55*x^45 +80*x^44 +88*x^43 +53*x^42 +139*x^41 +113*x^40 +139*x^39 +53*x^38 +88*x^37 +80*x^36 +55*x^35 +92*x^34 +57*x^33 +184*x^32 +104*x^31 +171*x^30 +100*x^29 +116*x^28 +73*x^27 -363*x^26 -568*x^25 -480*x^24 -797*x^23 -468*x^22 -792*x^21 -235*x^20 -368*x^19 -213*x^18 -300*x^17 -316*x^16 -183*x^15 -453*x^14 -339*x^13 -381*x^12 -135*x^11 -212*x^10 -180*x^9 -117*x^8 -184*x^7 -107*x^6 -312*x^5 -156*x^4 -229*x^3 -120*x^2 -128*x -75) / ((x -1)*(x^54 -6*x^27 +1)). - Colin Barker, May 18 2015

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A101152 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.

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A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

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A129625 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

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A129640 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+313)^2 = y^2.

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A129626 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.

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

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A160056 Decimal expansion of (107+42*sqrt(2))/89.

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