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Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.
+10
6
0, 264, 1491, 2427, 3811, 10764, 16180, 24220, 64711, 96271, 143127, 379120, 563064, 836160, 2211627, 3283731, 4875451, 12892260, 19140940, 28418164, 75143551, 111563527, 165635151, 437970664, 650241840, 965394360, 2552682051
COMMENTS
Also values x of Pythagorean triples (x, x+809, y).
Corresponding values y of solutions (x, y) are in A160203. lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (873+232*sqrt(2))/809 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (989043+524338*sqrt(2))/809^2 for n mod 3 = 0.
FORMULA
a(n) = 6*a(n-3)-a(n-6)+1618 for n > 6; a(1)=0, a(2)=264, a(3)=1491, a(4)=2427, a(5)=3811, a(6)=10764.
G.f.: x*(264+1227*x+936*x^2-200*x^3-409*x^4-200*x^5) / ((1-x)*(1-6*x^3 +x^6)).
a(3*k+1) = 809* A001652(k) for k >= 0.
MATHEMATICA
LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 264, 1491, 2427, 3811, 10764, 16180}, 50] (* G. C. Greubel, May 04 2018 *)
PROG
(PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481), print1(n, ", ")))}
(PARI) x='x+O('x^30); concat([0], Vec(x*(264+1227*x+936*x^2-200*x^3 -409*x^4 -200*x^5)/((1-x)*(1-6*x^3 +x^6)))) \\ G. C. Greubel, May 04 2018 (Magma) I:=[0, 264, 1491, 2427, 3811, 10764, 16180]; [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
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+761)^2 = y^2.
+10
5
0, 583, 820, 2283, 5440, 6783, 15220, 33579, 41400, 90559, 197556, 243139, 529656, 1153279, 1418956, 3088899, 6723640, 8272119, 18005260, 39190083, 48215280, 104944183, 228418380, 281021083, 611661360, 1331321719, 1637912740
COMMENTS
Also values x of Pythagorean triples (x, x+761, y).
Corresponding values y of solutions (x, y) are in A160200. lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2) = A156035. lim_{n -> infinity} a(n)/a(n-1) = (1003+462*sqrt(2))/761 = A160201 for n mod 3 = {1, 2}. lim_{n -> infinity} a(n)/a(n-1) = (591603+85478*sqrt(2))/761^2 = A160202 for n mod 3 = 0.
FORMULA
a(n) = 6*a(n-3) -a(n-6) +1522 for n > 6; a(1)=0, a(2)=583, a(3)=820, a(4)=2283, a(5)=5440, a(6)=6783.
G.f.: x*(583 +237*x +1463*x^2 -341*x^3 -79*x^4 -341*x^5)/((1-x)*(1 -6*x^3 +x^6)).
a(3*k+1) = 761* A001652(k) for k >= 0.
MATHEMATICA
LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 583, 820, 2283, 5440, 6783, 15220}, 27] (* Jean-François Alcover, Nov 13 2017 *)
PROG
(PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1522*n+579121), print1(n, ", ")))}
(PARI) x='x+O('x^30); concat([0], Vec(x*(583 +237*x +1463*x^2 -341*x^3 -79*x^4 -341*x^5)/((1-x)*(1 -6*x^3 +x^6)))) \\ G. C. Greubel, May 04 2018 (Magma) I:=[0, 583, 820, 2283, 5440, 6783, 15220]; [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
Positive numbers y such that y^2 is of the form x^2+(x+617)^2 with integer x.
+10
4
533, 617, 733, 2465, 3085, 3865, 14257, 17893, 22457, 83077, 104273, 130877, 484205, 607745, 762805, 2822153, 3542197, 4445953, 16448713, 20645437, 25912913, 95870125, 120330425, 151031525, 558772037, 701337113, 880276237
COMMENTS
(-92, a(1)) and ( A115135(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+617)^2 = y^2. lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (633+100*sqrt(2))/617 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (755667+461578*sqrt(2))/617^2 for n mod 3 = 1.
FORMULA
a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=533, a(2)=617, a(3)=733, a(4)=2465, a(5)=3085, a(6)=3865.
G.f.: (1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 617* A001653(k) for k >= 1.
EXAMPLE
(-92, a(1)) = (-92, 533) is a solution: (-92)^2+(-92+617)^2 = 8464+275625 = 284089 = 533^2.
( A115135(1), a(2)) = (0, 617) is a solution: 0^2+(0+617)^2 = 380689 = 617^2. ( A115135(3), a(4)) = (1407, 2465) is a solution: 1407^2+(1407+617)^2 = 1979649+4096576 = 6076225 = 2465^2.
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {533, 617, 733, 2465, 3085, 3865}, 50] (* G. C. Greubel, May 04 2018 *)
PROG
(PARI) {forstep(n=-92, 10000000, [3, 1], if(issquare(2*n^2+1234*n+380689, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6)) \\ G. C. Greubel, May 04 2018 (Magma) I:=[533, 617, 733, 2465, 3085, 3865]; [n le 6 select I[n] else 6*Self(n31) -Self(n-6): n in [1..30]]; // G. C. Greubel, May 04 2018
CROSSREFS
Cf. A115135, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160177 (decimal expansion of (633+100*sqrt(2))/617), A160178 (decimal expansion of (755667+461578*sqrt(2))/617^2).
Decimal expansion of (633+100*sqrt(2))/617.
+10
4
1, 2, 5, 5, 1, 3, 9, 9, 6, 1, 4, 8, 6, 7, 2, 5, 2, 9, 1, 5, 3, 9, 9, 8, 1, 9, 6, 5, 0, 2, 5, 8, 8, 2, 9, 9, 4, 9, 0, 5, 9, 4, 3, 5, 7, 7, 8, 5, 6, 8, 7, 9, 7, 5, 2, 5, 4, 0, 7, 9, 0, 8, 8, 1, 3, 5, 9, 9, 2, 4, 2, 2, 6, 6, 5, 5, 1, 2, 2, 6, 9, 6, 7, 3, 4, 2, 6, 0, 5, 4, 0, 4, 0, 9, 6, 6, 6, 3, 8, 1, 3, 1, 8, 4, 7
COMMENTS
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A115135. Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160176.
FORMULA
Equals (25 +2*sqrt(2))/(25 -2*sqrt(2)).
EXAMPLE
(633+100*sqrt(2))/617 = 1.25513996148672529153...
MATHEMATICA
RealDigits[(633+100Sqrt[2])/617, 10, 120][[1]] (* Harvey P. Dale, Nov 25 2013 *)
Decimal expansion of (755667+461578*sqrt(2))/617^2.
+10
4
3, 6, 9, 9, 7, 0, 4, 6, 6, 1, 0, 0, 4, 2, 5, 4, 0, 4, 0, 5, 3, 1, 2, 1, 0, 1, 9, 5, 0, 4, 9, 6, 7, 0, 5, 1, 7, 6, 9, 3, 0, 0, 4, 0, 1, 3, 4, 9, 3, 2, 7, 3, 8, 7, 3, 7, 4, 1, 4, 1, 4, 8, 5, 5, 9, 6, 4, 5, 4, 4, 0, 6, 6, 0, 0, 5, 5, 3, 8, 4, 9, 1, 4, 6, 8, 0, 5, 9, 6, 9, 5, 2, 3, 6, 4, 9, 0, 0, 0, 0, 6, 5, 6, 3, 9
COMMENTS
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A115135. Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160176.
FORMULA
Equals (1066 +433*sqrt(2))/(1066 -433*sqrt(2)).
Equals (3 +2*sqrt(2))*(25 -2*sqrt(2))^2/(25 +2*sqrt(2))^2.
EXAMPLE
(755667+461578*sqrt(2))/617^2 = 3.69970466100425404053...
MATHEMATICA
RealDigits[(755667+461578*Sqrt[2])/617^2, 10, 120][[1]] (* Harvey P. Dale, Oct 14 2017 *)
PROG
(PARI) (755667+461578*sqrt(2))/617^2 \\ G. C. Greubel, Apr 25 2018 (Magma) (755667+461578*Sqrt(2))/617^2; // G. C. Greubel, Apr 25 2018
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2737)^2 = y^2.
+10
2
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
COMMENTS
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.
FORMULA
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
MATHEMATICA
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
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