A130609 -id:A130609 - cp's OEIS frontend

A118337 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 23)^2 = y^2.

0, 12, 33, 69, 133, 252, 460, 832, 1525, 2737, 4905, 8944, 16008, 28644, 52185, 93357, 167005, 304212, 544180, 973432, 1773133, 3171769, 5673633, 10334632, 18486480, 33068412, 60234705, 107747157, 192736885, 351073644, 627996508, 1123352944, 2046207205

Offset: 1

Mohamed Bouhamida, May 14 2006

Also values x of Pythagorean triples (x, x+23, y).
Corresponding values y of solutions (x, y) are in A156567.
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) = 3*m^2 - 10m + 8, a(4)=3p, a(5) = 3*m^2 + 10m + 8, a(6) = 20*m^2 - 58m + 42. Pairs (p, m) are (23, 5), (47, 7), (79, 9), (167, 13), (223, 15), (359, 19), (439, 21), (727, 27), (839, 29), ...
Limit_{n -> oo} a(n)/a(n-3) = 3 + 2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (27 + 10*sqrt(2))/23 for n mod 3 = {1, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (3 + 2*sqrt(2))/((27 + 10*sqrt(2))/23)^2 for n mod 3 = 0.
For the generic case x^2 + (x + p)^2=y^2 with p = m^2 - 2 a prime number in A028871, m>=5, 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
For the generic case x^2 + (x + p)^2 = y^2 with p = m^2 - 2 a prime number, m>=5, the first three consecutive solutions are: (0;p), (2m+2; m^2+2m+2), (3*m^2-10m+8; 5*m^2-14m+10) and the other solutions are defined by: (X(n); Y(n))= (3*X(n-3)+2*Y(n-3)+p; 4*X(n-3)+3*Y(n-3)+2p). - Mohamed Bouhamida, Aug 19 2019
X(n) = 6*X(n-3) - X(n-6) + 2*p, and Y(n) = 6*Y(n-3) - Y(n-6) (can be easily proved using X(n) = 3*X(n-3) + 2*Y(n-3) + p, and Y(n) = 4*X(n-3) + 3*Y(n-3) + 2*p). - Mohamed Bouhamida, Aug 20 2019

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. A156567, A028871 (primes of form n^2 - 2), A156035 (decimal expansion of 3 + 2*sqrt(2)), A156571 (decimal expansion of (27 + 10*sqrt(2))/23).

Cf. A118675 (p=47), A118676 (p=79), A130608 (p=167), A130609 (p=223), A130610 (p=359), A130645 (p=439), A130646 (p=727), A130647 (p=839).

I:=[0,12,33,69,133,252,460]; [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, May 04 2018

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+23)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,12,33,69,133,252,460},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

forstep(n=0, 1124000000, [1, 3], if(issquare(2*n*(n+23)+529), print1(n, ",")))

x='x+O('x^30); concat([0], Vec(x*(12+21*x+36*x^2-8*x^3-7*x^4-8*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, May 04 2018

a(n) = 6*a(n-3) - a(n-6) + 46 for n > 6; a(1)=0, a(2)=12, a(3)=33, a(4)=69, a(5)=133, a(6)=252.

G.f.: x*(12 + 21*x + 36*x^2 - 8*x^3 - 7*x^4 - 8*x^5)/((1-x)*(1 - 6*x^3 + x^6)).

Edited by Klaus Brockhaus, Feb 10 2009

A130610 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.

Original entry on oeis.org

0, 40, 901, 1077, 1281, 6160, 7180, 8364, 36777, 42721, 49621, 215220, 249864, 290080, 1255261, 1457181, 1691577, 7317064, 8493940, 9860100, 42647841, 49507177, 57469741, 248570700, 288549840, 334959064, 1448777077, 1681792581

Offset: 1

Mohamed Bouhamida, Jun 17 2007

Also values x of Pythagorean triples (x, x+359, y).
Corresponding values y of solutions (x, y) are in A159844.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
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 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (293619+186550*sqrt(2))/359^2 for n mod 3 = 0.

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

Cf. A159844, A028871, A118337, A130609, A001652, 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).

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+718*n+128881), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+718 for n > 6; a(1)=0, a(2)=40, a(3)=901, a(4)=1077, a(5)=1281, a(6)=6160.
G.f.: x*(40+861*x+176*x^2-36*x^3-287*x^4-36*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 359*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

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.

A159810 Decimal expansion of (227+30*sqrt(2))/223.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A130609.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A159809.

			(227+30*sqrt(2))/223 = 1.20819016534167197965...

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

Cf. A130609, A159809, A002193 (decimal expansion of sqrt(2)), A159811 (decimal expansion of (105507+65798*sqrt(2))/223^2).

(227 +30*Sqrt(2))/223; // G. C. Greubel, May 19 2018

RealDigits[(227+30*Sqrt[2])/223,10,120][[1]] (* Harvey P. Dale, Sep 10 2017 *)

(227 +30*sqrt(2))/223 \\ G. C. Greubel, May 19 2018

Equals (15 +sqrt(2))/(15 -sqrt(2)).

A159811 Decimal expansion of (105507 + 65798*sqrt(2))/223^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A130609.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A159809.

			(105507+65798*sqrt(2))/223^2 = 3.99282961605954087194...

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

Cf. A130609, A159809, A002193 (decimal expansion of sqrt(2)), A159810 (decimal expansion of (227+30*sqrt(2))/223).

(105507+65798*Sqrt(2))/223^2; // G. C. Greubel, May 19 2018

RealDigits[(105507 + 65798*Sqrt[2])/223^2, 10, 100][[1]] (* G. C. Greubel, May 19 2018 *)

(105507+65798*sqrt(2))/223^2 \\ G. C. Greubel, May 19 2018

Equals (394 + 167*sqrt(2))/(394 - 167*sqrt(2)).

Equals (3 + 2*sqrt(2))*(15 - sqrt(2))^2/(15 + 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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A118337 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 23)^2 = y^2.

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A130610 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.

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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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A159810 Decimal expansion of (227+30*sqrt(2))/223.

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A159811 Decimal expansion of (105507 + 65798*sqrt(2))/223^2.

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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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