A123654 -id:A123654 - cp's OEIS frontend

A129857 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.

0, 235, 1696, 2571, 3796, 12075, 17140, 24255, 72468, 101983, 143448, 424447, 596472, 838147, 2475928, 3478563, 4887148, 14432835, 20276620, 28486455, 84122796, 118182871, 166033296, 490305655, 688822320, 967715035, 2857712848

Offset: 1

Mohamed Bouhamida, Jun 03 2007

Also values x of Pythagorean triples (x, x+857, y).
Corresponding values y of solutions (x, y) are in A160206.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 for n mod 3 = 0.

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. A160206, A001652, A123654, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat  Coefficients(R!(x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5)/((1-x)*(1-6*x^3+x^6))) );  // G. C. Greubel, May 03 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,235,1696,2571,3796,12075, 17140}, 30] (* or *) CoefficientList[Series[x (235+1461x+875x^2-185x^3- 487x^4- 185x^5)/((1-x)(1-6x^3+x^6)),{x,0,30}],x] (* Harvey P. Dale, Apr 24 2011 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1714*n+734449), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1714 for n > 6; a(1)=0, a(2)=235, a(3)=1696, a(4)=2571, a(5)=3796, a(6)=12075.
G.f.: x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 857*A001652(k) for k >= 0.

Edited and two terms added by Klaus Brockhaus, May 18 2009

A160204 Decimal expansion of (873+232*sqrt(2))/809.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 18 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A123654.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160203.

			(873+232*sqrt(2))/809 = 1.48466940231218547753...

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

Cf. A123654, A160203, A002193 (decimal expansion of sqrt(2)), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).

(873+232*Sqrt(2))/809; // G. C. Greubel, Apr 25 2018

RealDigits[(873+232Sqrt[2])/809,10,120][[1]] (* Harvey P. Dale, Mar 22 2018 *)

(873+232*sqrt(2))/809 \\ G. C. Greubel, Apr 25 2018

Equals (29+4*sqrt(2))/(29-4*sqrt(2)).

A160205 Decimal expansion of (989043+524338*sqrt(2))/809^2.

Original entry on oeis.org

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

Offset: 11

Klaus Brockhaus, May 18 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A123654.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160203.

			(989043+524338*sqrt(2))/809^2 = 2.64418510371971670942...

G. C. Greubel, Table of n, a(n) for n = 11..10000

Cf. A123654, A160203, A002193 (decimal expansion of sqrt(2)), A160204 (decimal expansion of (873+232*sqrt(2))/809).

(989043+524338*Sqrt(2))/809^2; // G. C. Greubel, Apr 25 2018

RealDigits[(989043+524338*Sqrt[2])/809^2, 10, 100][[1]] (* G. C. Greubel, Apr 25 2018 *)

(989043+524338*sqrt(2))/809^2 \\ G. C. Greubel, Apr 25 2018

Equals (1282 +409*sqrt(2))/(1282 -409*sqrt(2)).

Equals (3+2*sqrt(2))*(29-4*sqrt(2))^2/(29+4*sqrt(2))^2.

A160203 Positive numbers y such that y^2 is of the form x^2+(x+809)^2 with integer x.

Original entry on oeis.org

641, 809, 1105, 2741, 4045, 5989, 15805, 23461, 34829, 92089, 136721, 202985, 536729, 796865, 1183081, 3128285, 4644469, 6895501, 18232981, 27069949, 40189925, 106269601, 157775225, 234244049, 619384625, 919581401, 1365274369

Offset: 1

Klaus Brockhaus, May 18 2009

nonn

(-200, a(1)) and (A123654(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.
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 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (989043+524338*sqrt(2))/809^2 for n mod 3 = 1.

			(-200, a(1)) = (-200, 641) is a solution: (-200)^2+(-200+809)^2 = 40000+370881 = 410881 = 641^2.
(A123654(1), a(2)) = (0, 809) is a solution: 0^2+(0+809)^2 = 654481 = 809^2.
(A123654(3), a(4)) = (1491, 2741) is a solution: 1491^2+(1491+809)^2 = 2223081+5290000 = 7513081 = 2741^2.

Cf. A123654, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160204 (decimal expansion of (873+232*sqrt(2))/809), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).

{forstep(n=-200, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=641, a(2)=809, a(3)=1105, a(4)=2741, a(5)=4045, a(6)=5989.
G.f.: (1-x)*(641+1450*x+2555*x^2+1450*x^3+641*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 809*A001653(k) for k >= 1.

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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A129857 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.

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A160204 Decimal expansion of (873+232*sqrt(2))/809.

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A160205 Decimal expansion of (989043+524338*sqrt(2))/809^2.

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A160203 Positive numbers y such that y^2 is of the form x^2+(x+809)^2 with integer x.

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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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