A130645 -id:A130645 - 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

A130647 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.

Original entry on oeis.org

0, 60, 2241, 2517, 2821, 15180, 16780, 18544, 90517, 99841, 110121, 529600, 583944, 643860, 3088761, 3405501, 3754717, 18004644, 19850740, 21886120, 104940781, 115700617, 127563681, 611641720, 674354640, 743497644, 3564911217

Offset: 1

Mohamed Bouhamida, Jun 20 2007

Also values x of Pythagorean triples (x, x+839, y).
Corresponding values y of solutions (x, y) are in A159896.
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) = (843+58*sqrt(2))/839 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1760979+1141390*sqrt(2))/839^2 for n mod 3 = 0.

G. C. Greubel, Table of n, a(n) for n = 1..3895
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159896, A028871, A118337, A130645, A130646, A001652, 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:=[0,60,2241,2517,2821,15180,16780]; [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 17 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,60,2241,2517,2821,15180,16780},30] (* Harvey P. Dale, Jun 19 2014 *)

{forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1678*n+703921), print1(n, ",")))}

a(n) = 6*a(n-3) -a(n-6) +1678 for n > 6; a(1)=0, a(2)=60, a(3)=2241, a(4)=2517, a(5)=2821, a(6)=15180.
G.f.: x*(60+2181*x+276*x^2-56*x^3-727*x^4-56*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 839*A001652(k) for k >= 0.

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

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.

A159891 Decimal expansion of (443+42*sqrt(2))/439.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2009

lim_{n -> infinity} b(n)/b(n-1) = (443+42*sqrt(2))/439 for n mod 3 = {1, 2}, b = A130645.

lim_{n -> infinity} b(n)/b(n-1) = (443+42*sqrt(2))/439 for n mod 3 = {0, 2}, b = A159890.

			(443+42*sqrt(2))/439 = 1.14441223147988608667...

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

Cf. A130645, A159890, A002193 (decimal expansion of sqrt(2)), A159892 (decimal expansion of (450483+287918*sqrt(2))/439^2).

(443 +42*Sqrt(2))/439; // G. C. Greubel, May 17 2018

RealDigits[N[(443+42*Sqrt[2])/439,300]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)

(443 +42*sqrt(2))/439 \\ G. C. Greubel, May 17 2018

Equals (21 +sqrt(2))/(21 -sqrt(2)).

A159892 Decimal expansion of (450483+287918*sqrt(2))/439^2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2009

lim_{n -> infinity} b(n)/b(n-1) = (450483+287918*sqrt(2))/439^2 for n mod 3 = 0, b = A130645.

lim_{n -> infinity} b(n)/b(n-1) = (450483+287918*sqrt(2))/439^2 for n mod 3 = 1, b = A159890.

			(450483+287918*sqrt(2))/439^2 = 4.45027028944088490751...

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

Cf. A130645, A159890, A002193 (decimal expansion of sqrt(2)), A159891 (decimal expansion of (443+42*sqrt(2))/439).

(450483 +287918*Sqrt(2))/439^2; // G. C. Greubel, May 17 2018

RealDigits[N[(450483+287918*Sqrt[2])/439^2,300]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)

(450483+287918*sqrt(2))/439^2 \\ G. C. Greubel, May 17 2018

Equals (802 +359*sqrt(2))/(802 -359*sqrt(2)).

Equals (3 +2*sqrt(2))*(21 -sqrt(2))^2/(21 +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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A130647 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.

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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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A159891 Decimal expansion of (443+42*sqrt(2))/439.

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A159892 Decimal expansion of (450483+287918*sqrt(2))/439^2.

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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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