A118630 -id:A118630 - cp's OEIS frontend

A118120 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.

0, 7, 28, 51, 88, 207, 340, 555, 1248, 2023, 3276, 7315, 11832, 19135, 42676, 69003, 111568, 248775, 402220, 650307, 1450008, 2344351, 3790308, 8451307, 13663920, 22091575, 49257868, 79639203, 128759176, 287095935, 464171332, 750463515, 1673317776

Offset: 0

Mohamed Bouhamida, May 12 2006

Also values x of Pythagorean triples (x, x+17, y).
Corresponding values y of solutions (x, y) are in A155923.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a prime number in A066436, m >= 2 the associated value in A066049, the x values are given by the sequence defined by a(n) = 6*a(n-3)-a(n-6)+2p with a(0)=0, a(1)=2m+1, a(2)=6m^2-10m+4, a(3)=3p, a(4)=6m^2+10m+4, a(5)=40m^2-58m+21 (cf. A118673).
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(0)=p, b(1)=2*m^2+2m+1, b(2)=10m^2-14m+5, b(3)=5p, b(4)=10m^2+14m+5, b(5)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

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

Cf. A155923, A118673, A066436 (primes of the form 2*n^2-1), A066049 (2*n^2-1 is prime), A118554, A118611, A118630.

Cf. A155464 (first trisection), A155465 (second trisection), A155466 (third trisection).

[ n: n in [0..25000000] | IsSquare(2*n*(n+17)+289) ];

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+17)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,7,28,51,88,207,340},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

m=32; v=concat([0, 7, 28, 51, 88, 207], vector(m-6)); for(n=7, m, v[n]=6*v[n-3]-v[n-6]+34); v

a(n) = 6*a(n-3) -a(n-6) +34 for n > 5; a(0)=0, a(1)=7, a(2)=28, a(3)=51, a(4)=88, a(5)=207.

G. f.: x*(7 +21*x +23*x^2 -5*x^3 -7*x^4 -5*x^5)/((1-x)*(1-6*x^3+x^6)).

Edited and 248755 changed to 248775 by Klaus Brockhaus, Feb 01 2009

A129837 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+119)^2 = y^2.

Original entry on oeis.org

0, 24, 49, 57, 85, 136, 180, 196, 261, 357, 481, 616, 660, 816, 1105, 1357, 1449, 1824, 2380, 3100, 3885, 4141, 5049, 6732, 8200, 8736, 10921, 14161, 18357, 22932, 24424, 29716, 39525, 48081, 51205, 63940, 82824, 107280, 133945, 142641, 173485, 230656

Offset: 1

Mohamed Bouhamida, May 21 2007

nonn

Also values x of Pythagorean triples (x, x+119, y).
Corresponding values y of solutions (x, y) are in A156650.
lim_{n -> infinity} a(n)/a(n-9) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)/((19+6*sqrt(2))/17) for n mod 9 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = ((19+6*sqrt(2))/17)^2/((9+4*sqrt(2))/7) for n mod 9 = {0, 3}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 9 = {4, 8}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 9 = {5, 7}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((9+4*sqrt(2))/7)^2 for n mod 9 = 6.

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. A156650, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17), A118630.

LinearRecurrence[{1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,24,49,57,85,136,180,196,261,357,481,616,660,816,1105,1357,1449,1824,2380}, 140] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 240000, [1, 3], if(issquare(n^2+(n+119)^2), print1(n, ",")))}

a(n) = 6*a(n-9)-a(n-18)+238 for n > 18; a(1)=0, a(2)=24, a(3)=49, a(4)=57, a(5)=85, a(6)=136, a(7)=180, a(8)=196, a(9)=261, a(10)=357, a(11)=481, a(12)=616, a(13)=660, a(14)=816, a(15)=1105, a(16)=1357, a(17)=1449, a(18)=1824.

G.f.: x*(24+25*x+8*x^2+28*x^3+51*x^4+44*x^5+16*x^6+65*x^7+96*x^8-20*x^9-15*x^10-4*x^11-12*x^12-17*x^13-12*x^14-4*x^15-15*x^16-20*x^17 )/((1-x)*(1-6*x^9+x^18))

Edited and extended by Klaus Brockhaus, Feb 13 2009

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

A157247 Positive numbers y such that y^2 is of the form x^2+(x+2401)^2 with integer x.

Original entry on oeis.org

1715, 1781, 1855, 2009, 2401, 2989, 3451, 3821, 4459, 5831, 6865, 7679, 9065, 12005, 15925, 18851, 21145, 25039, 33271, 39409, 44219, 52381, 69629, 92561, 109655, 123049, 145775, 193795, 229589, 257635, 305221, 405769, 539441, 639079, 717149

Offset: 1

Klaus Brockhaus, Feb 25 2009

nonn

(-1029, a(1)), (-820, a(2)), (-672, a(3)), (-441, a(3)) and (A118630(n), a(n+4)) are solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-9) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 9 = {1, 5, 6}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 9 = {0, 2, 4, 7}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 9 = {3, 8}.

			(-1029, a(1)) = (-1029, 1715) is a solution: (-1029)^2+(-1029+2401)^2 = 1058841+1882384 = 2941225 = 1715^2.
(A118630(1), a(5)) = (0, 2401) is a solution: 0^2+(0+2401)^2 = 5764801 = 2401^2.
(A118630(3), a(7)) = (924, 3451) is a solution: 924^2+(924+2401)^2 = 853776+11055625 = 11909401 = 3451^2.

Cf. A118630, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

Sqrt[#]&/@Select[Table[2x^2+4802x+5764801,{x,-1200,510000}], IntegerQ[ Sqrt[ #]]&] (* Harvey P. Dale, Jul 21 2011 *)

{forstep(n=-1032, 540000, [3 ,1], if(issquare(n^2+(n+2401)^2, &k), print1(k, ",")))}

a(n)=6*a(n-9)-a(n-18) for n > 18; a(1)=1715, a(2)=1781, a(3)=1855, a(4)=2009, a(5)=2401, a(6)=2989, a(7)=3451, a(8)=3821, a(9)=4459, a(10)=5831, a(11)=6865, a(12)=7679, a(13)=9065, a(14)=12005, a(15)=15925, a(16)=18851, a(17)=21145, a(18)=25039.
G.f.: x * (1-x) * (1715 +3496*x +5351*x^2 +7360*x^3 +9761*x^4 +12750*x^5 +16201*x^6 +20022*x^7 +24481*x^8 +20022*x^9 +16201*x^10 +12750*x^11 +9761*x^12 +7360*x^13 +5351*x^14 +3496*x^15 +1715*x^16) / (1 -6*x^9 +x^18).
a(9*k-4) = 2401*A001653(k) for k >= 1.

G.f. adapted to the offset by Bruno Berselli, Apr 01 2011

cp's OEIS Frontend

A118120 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A129837 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+119)^2 = y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A157247 Positive numbers y such that y^2 is of the form x^2+(x+2401)^2 with integer x.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions