A207059 -id:A207059 - cp's OEIS frontend

A207060 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+401)^2 = y^2.

0, 259, 496, 1203, 2596, 3939, 8020, 16119, 23940, 47719, 94920, 140503, 279096, 554203, 819880, 1627659, 3231100, 4779579, 9487660, 18833199, 27858396, 55299103, 109768896, 162371599, 322307760, 639780979, 946372000, 1878548259, 3728917780, 5515861203

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

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

Cf. A204765, A205644, A205672, A207058, A207059.

LinearRecurrence[ {1, 0, 6, -6, 0, -1, 1}, {0, 259, 496, 1203, 2596, 3939, 8020}, 50]

G.f.: x^2*(161*x^5+79*x^4+161*x^3-707*x^2-237*x-259)/((x-1)*(x^6-6*x^3+1)). - Colin Barker, Aug 05 2012

A207061 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+433)^2 = y^2.

Original entry on oeis.org

0, 92, 935, 1299, 1775, 6552, 8660, 11424, 39243, 51527, 67635, 229772, 301368, 395252, 1340255, 1757547, 2304743, 7812624, 10244780, 13434072, 45536355, 59711999, 78300555, 265406372, 348028080, 456370124, 1546902743, 2028457347, 2659921055, 9016010952

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

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

Cf. A205644, A205672, A207058, A207059, A207060.

LinearRecurrence[ {1, 0, 6, -6, 0, -1, 1}, {0, 92, 935, 1299, 1775, 6552, 8660}, 50]

G.f.: x^2*(76*x^5+281*x^4+76*x^3-364*x^2-843*x-92)/((x-1)*(x^6-6*x^3+1)). - Colin Barker, Aug 05 2012

A207075 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+479)^2 = y^2.

Original entry on oeis.org

0, 280, 637, 1437, 2937, 4960, 9580, 18300, 30081, 57001, 107821, 176484, 333384, 629584, 1029781, 1944261, 3670641, 6003160, 11333140, 21395220, 34990137, 66055537, 124701637, 203938620, 385001040, 726815560, 1188642541, 2243951661, 4236192681, 6927917584

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

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

Cf. A205672, A207058, A207059, A207060, A207061

LinearRecurrence[ {1, 0, 6, -6, 0, -1, 1}, {0, 280, 637, 1437, 2937, 4960, 9580}, 50]

G.f.: x^2*(180*x^5 + 119*x^4 + 180*x^3 - 800*x^2 - 357*x - 280)/((x-1)*(x^6-6*x^3+1)). [Colin Barker, Aug 05 2012]

A207076 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+487)^2 = y^2.

Original entry on oeis.org

0, 185, 836, 1461, 2436, 6125, 9740, 15405, 36888, 57953, 90968, 216177, 338952, 531377, 1261148, 1976733, 3098268, 7351685, 11522420, 18059205, 42849936, 67158761, 105257936, 249748905, 391431120, 613489385, 1455644468, 2281428933, 3575679348, 8484118877

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

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

Cf. A207058, A207059, A207060, A207061, A207075

LinearRecurrence[ {1, 0, 6, -6, 0, -1, 1}, {0, 185, 836, 1461, 2436, 6125, 9740}, 50]

G.f.: x^2*(135*x^5+217*x^4+135*x^3-625*x^2-651*x-185)/((x-1)*(x^6-6*x^3+1)). [Colin Barker, Aug 05 2012]

A207077 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+497)^2 = y^2.

Original entry on oeis.org

0, 91, 216, 355, 387, 528, 568, 795, 1120, 1491, 1960, 2635, 3408, 3588, 4387, 4615, 5916, 7791, 9940, 12663, 16588, 21087, 22135, 26788, 28116, 35695, 46620, 59143, 75012, 97887, 124108, 130216, 157335, 165075, 209248, 272923, 345912, 438403, 571728, 724555

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 14 2012

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

Cf. A207059, A207060, A207061, A207075, A207076.

LinearRecurrence[ {1, 0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0, 0, 0, -1, 1}, {0, 91, 216, 355, 387, 528, 568, 795, 1120, 1491, 1960, 2635, 3408, 3588, 4387, 4615, 5916, 7791, 9940}, 50]

G.f.: x^2*(77*x^17 +75*x^16 +61*x^15 +12*x^14 +47*x^13 +12*x^12 +61*x^11 +75*x^10 +77*x^9 -371*x^8 -325*x^7 -227*x^6 -40*x^5 -141*x^4 -32*x^3 -139*x^2 -125*x -91)/((x -1)*(x^18 -6*x^9 +1)). - Colin Barker, Aug 05 2012

A309998 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 529)^2 = y^2.

Original entry on oeis.org

0, 276, 287, 740, 759, 1587, 3059, 3120, 5687, 5796, 10580, 19136, 19491, 34440, 35075, 62951, 112815, 114884, 202011, 205712, 368184, 658812, 670871, 1178684, 1200255, 2147211, 3841115, 3911400, 6871151, 6996876, 12516140, 22388936, 22798587, 40049280, 40782059, 72950687

Offset: 1

Mohamed Bouhamida, Aug 26 2019

For the generic case x^2 + (x + p^2)^2 = y^2 with p = m^2 - 2 a prime number in A028871, m>=5, (0; p^2) and (2*m^3 + 2*m^2 - 4*m - 4; m^4 + 2*m^3 - 4*m - 4) are solutions.

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

Cf. A207059.

Rest@ CoefficientList[Series[x^2*(276 + 11 x + 453 x^2 + 19 x^3 + 828 x^4 - 184 x^5 - 5 x^6 - 151 x^7 - 5 x^8 - 184 x^9)/((1 - x) (1 - 6 x^5 + x^10)), {x, 0, 36}], x] (* Michael De Vlieger, Sep 29 2019 *)

concat(0, Vec(x^2*(276 + 11*x + 453*x^2 + 19*x^3 + 828*x^4 - 184*x^5 - 5*x^6 - 151*x^7 - 5*x^8 - 184*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)) + O(x^40))) \\ Colin Barker, Aug 27 2019

a(n) = 6*a(n-5) - a(n-10) + 1058 with a(0) = 0, a(1) = 276, a(2) = 287, a(3) = 740, a(4) = 759, a(5) = 1587, a(6) = 3059, a(7) = 3120, a(8) = 5687, a(9) = 5796.
From Colin Barker, Aug 27 2019: (Start)
G.f.: x^2*(276 + 11*x + 453*x^2 + 19*x^3 + 828*x^4 - 184*x^5 - 5*x^6 - 151*x^7 - 5*x^8 - 184*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)).
a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - a(n-10) + a(n-11) for n>11.
(End)

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

A331265 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31^2)^2 = y^2.

Original entry on oeis.org

0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299, 19220, 27683, 39780, 55719, 79856, 114359, 163680, 234183, 327080, 467759, 668856, 956319, 1367240, 1908683, 2728620, 3900699, 5576156, 7971179, 11126940, 15905883, 22737260, 32502539, 46461756, 64854879, 92708600, 132524783

Offset: 1

Mohamed Bouhamida, Feb 12 2020

For the generic case x^2 + (x + p^2)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436, m >= 4 (means p >= 31), the first five consecutive solutions are (0, p^2), (4*m^3+2*m^2-2*m-1, 4*m^4+4*m^3-2*m-1), (8*m^3+8*m^2+4*m, 4*m^4+8*m^3+12*m^2+4*m+1), (12*m^4-40*m^3+44*m^2-20*m+3, 20*m^4-56*m^3+60*m^2-28*m+5), (12*m^4-20*m^3+2*m^2+10*m-4, 20*m^4-28*m^3+14*m-5) and the other solutions are defined by (X(n), Y(n)) = (3*X(n-5) + 2*Y(n-5) + p^2, 4*X(n-5) + 3*Y(n-5) + 2*p^2).

X(n) = 6*X(n-5) - X(n-10) + 2*p^2, and Y(n) = 6*Y(n-5) - Y(n-10) (can be easily proved using X(n) = 3*X(n-5) + 2*Y(n-5) + p^2, and Y(n) = 4*X(n-5) + 3*Y(n-5) + 2*p^2).

			For p=31 (m=4) the first five (5) consecutive solutions are (0, 961), (279, 1271), (656, 1745), (1139, 2389), (1860, 3379).

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

Cf. A066436 (Primes of the form 2*m^2 - 1).

Solutions x to x^2+(x+p^2)^2=y^2: A118554 (p=7), A207059 (p=17), A309998 (p=23), this sequence (p=31), A332000 (p=47).

I:=[0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299]; [n le 10 select I[n] else 6*Self(n-5) - Self(n-10)+1922: n in [1..100]];

LinearRecurrence[{1, 0, 0, 0, 6, -6, 0, 0, 0, -1, 1}, {0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299, 19220}, 36] (* Jean-François Alcover, Feb 12 2020 *)

concat(0, Vec(x^2*(279 + 377*x + 483*x^2 + 721*x^3 + 1023*x^4 - 217*x^5 - 183*x^6 - 161*x^7 - 183*x^8 - 217*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)) + O(x^30))) \\ Colin Barker, Feb 12 2020

a(n) = 6*a(n-5) - a(n-10) + 1922 for n >= 11; a(1)=0, a(2)=279, a(3)=656, a(4)=1139, a(5)=1860, a(6)=2883, a(7)=4340, a(8)=6419, a(9)=9156, a(10)=13299.
From Colin Barker, Feb 12 2020: (Start)
G.f.: x^2*(279 + 377*x + 483*x^2 + 721*x^3 + 1023*x^4 - 217*x^5 - 183*x^6 - 161*x^7 - 183*x^8 - 217*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)).
a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - a(n-10) + a(n-11) for n>11.
(End)

A332000 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 47^2)^2 = y^2.

Original entry on oeis.org

0, 752, 1820, 2231, 3995, 6627, 10575, 16511, 18840, 28952, 44180, 67116, 101664, 115227, 174135, 262871, 396539, 597891, 676940, 1020276, 1537464, 2316536, 3490100, 3950831, 5951939, 8966331, 13507095, 20347127, 23032464, 34695776, 52264940, 78730452, 118597080, 134248371

Offset: 1

Mohamed Bouhamida, Feb 04 2020

For the generic case x^2 + (x + p^2)^2 = y^2 with p = m^2 - 2 a (prime) number in A028871, m>=7 (means p>=47), the first five consecutive solutions are: (0; p^2), (2*m^3+2*m^2-4*m-4; m^4+2*m^3-4*m-4), (4*m^3+8*m^2+8*m; m^4+4*m^3+12*m^2+8*m+4), (3*m^4-20*m^3+44*m^2-40*m+12; 5*m^4-28*m^3+60*m^2-56*m+20), (3*m^4-10*m^3+2*m^2+20*m-16; 5*m^4-14*m^3+28*m-20) and the other solutions are defined by: (X(n); Y(n))= (3*X(n-5)+2*Y(n-5)+p^2; 4*X(n-5)+3*Y(n-5)+2*p^2).

X(n) = 6*X(n-5) - X(n-10) + 2*p^2, and Y(n) = 6*Y(n-5) - Y(n-10) (can be easily proved using X(n) = 3*X(n-5) + 2*Y(n-5) + p^2, and Y(n) = 4*X(n-5) + 3*Y(n-5) + 2*p^2).

			For p=47 (m=7) the first five (5) consecutive solutions are (0, 2209), (752, 3055), (1820, 4421), (2231, 4969), (3995, 7379).

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

Cf. A028871 (Primes of the form m^2 - 2).

Solutions x to x^2+(x+p^2)^2=y^2: A118554 (p=7), A207059 (p=17), A309998 (p=23), A331265 (p=31), this sequence (p=47).

I:=[0, 752, 1820, 2231, 3995, 6627, 10575, 16511, 18840, 28952]; [n le 10 select I[n] else 6*Self(n-5) - Self(n-10)+4418: n in [1..100]];

LinearRecurrence[{1, 0, 0, 0, 6, -6, 0, 0, 0, -1, 1}, {0, 752, 1820, 2231, 3995, 6627, 10575, 16511, 18840, 28952, 44180}, 40] (* Jean-François Alcover, Feb 08 2020 *)

concat(0, Vec(x^2*(752 + 1068*x + 411*x^2 + 1764*x^3 + 2632*x^4 - 564*x^5 - 472*x^6 - 137*x^7 - 472*x^8 - 564*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)) + O(x^40))) \\ Colin Barker, Feb 04 2020

a(n) = 6*a(n-5) - a(n-10) + 4418 for n >= 11; a(1)=0, a(2)=752, a(3)=1820, a(4)=2231, a(5)=3995, a(6)=6627, a(7)=10575, a(8)=16511, a(9)=18840, a(10)=28952.
From Colin Barker, Feb 04 2020: (Start)
G.f.: x^2*(752 + 1068*x + 411*x^2 + 1764*x^3 + 2632*x^4 - 564*x^5 - 472*x^6 - 137*x^7 - 472*x^8 - 564*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)).
a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - a(n-10) + a(n-11) for n>11.
(End)

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A207060 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+401)^2 = y^2.

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A207061 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+433)^2 = y^2.

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A207075 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+479)^2 = y^2.

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A207076 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+487)^2 = y^2.

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A207077 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+497)^2 = y^2.

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A309998 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 529)^2 = y^2.

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

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A331265 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31^2)^2 = y^2.

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