A118576 -id:A118576 - cp's OEIS frontend

A118611 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2.

0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860, 9177, 10904, 14553, 19404, 25853, 30660, 40817, 54320, 64385, 85652, 113925, 151512, 179529, 238728, 317429, 376092, 500045, 664832, 883905, 1047200, 1392237, 1850940, 2192853

Offset: 1

Mohamed Bouhamida, May 08 2006

Also values x of Pythagorean triples (x, x+343, y); 343=7^3.
Corresponding values y of solutions (x, y) are in A157246.
Limit_{n -> oo} a(n)/a(n-7) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 7 = {1, 2, 4, 5, 6}.
Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 7 = {0, 3}.

			132^2+(132+343)^2 = 17424+225625 = 243049 = 493^2.

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

Cf. A157246, A001652, A118576, A118554, A118611, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

LinearRecurrence[{1, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0, -1, 1}, {0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

{forstep(n=0, 1400000, [1, 3], if(issquare(n^2+(n+343)^2), print1(n, ",")))}

a(n) = 6*a(n-7)-a(n-14)+686 for n > 14; a(1)=0, a(2)=77, a(3)=132, a(4)=245, a(5)=392, a(6)=585, a(7)=728, a(8)=1029, a(9)=1428, a(10)=1725, a(11)=2352, a(12)=3185, a(13)=4292, a(14)=5117.
G.f.: x*(77+55*x+113*x^2+147*x^3+193*x^4+143*x^5+301*x^6-63*x^7 -33*x^8-51*x^9-49*x^10-51*x^11-33*x^12-63*x^13)/((1-x)*(1-6*x^7+x^14)).
a(7*k+1) = 343*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Feb 25 2009

A118630 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^2.

Original entry on oeis.org

0, 539, 924, 1220, 1715, 2744, 3503, 4095, 5096, 7203, 9996, 12075, 13703, 16464, 22295, 26640, 30044, 35819, 48020, 64239, 76328, 85800, 101871, 135828, 161139, 180971, 214620, 285719, 380240, 450695, 505899, 599564, 797475, 944996, 1060584

Offset: 1

Mohamed Bouhamida, May 09 2006

Also values x of Pythagorean triples (x, x+2401, y); 2401=7^4.
Corresponding values y of solutions (x, y) are in A157247.
Limit_{n -> oo} a(n)/a(n-9) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 9 = {1, 2, 6}.
Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 9 = {0, 3, 5, 7}.
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 9 = {4, 8}.

			924^2+(924+2401)^2 = 853776+11055625 = 11909401 = 3451^2.

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. A157247, A001652, A118576, A118554, A118611, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

{forstep(n=0, 1100000, [3 ,1], if(issquare(n^2+(n+2401)^2), print1(n, ",")))}

a(n) = 6*a(n-9)-a(n-18)+4802 for n > 18; a(1)=0, a(2)=539, a(3)=924, a(4)=1220, a(5)=1715, a(6)=2744, a(7)=3503, a(8)=4095, a(9)=5096, a(10)=7203, a(11)=9996, a(12)=12075, a(13)=13703, a(14)=16464,a (15)=22295, a(16)=26640, a(17)=30044, a(18)=35819.
G.f.: x*(539+385*x+296*x^2+495*x^3+1029*x^4+759*x^5+592*x^6 +1001*x^7+2107*x^8-441*x^9-231*x^10-148*x^11-209*x^12-343*x^13 -209*x^14-148*x^15-231*x^16-441*x^17) / ((1-x)*(1-6*x^9+x^18)).
a(9*k+1) = 2401*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Feb 25 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

A156713 Positive numbers y such that y^2 is of the form x^2+(x+16807)^2 with integer x.

Original entry on oeis.org

12005, 12467, 12985, 14063, 15025, 16807, 19073, 20923, 24157, 26747, 31213, 40817, 48055, 53753, 63455, 71077, 84035, 99413, 111475, 131957, 148015, 175273, 232897, 275863, 309533, 366667, 411437, 487403, 577405, 647927, 767585, 861343

Offset: 1

Klaus Brockhaus, Feb 17 2009

nonn

(-7203, a(1)), (-5740, a(2)), (-4704, a(3)), (-3087, a(4)), (-1903, a(5)), and (A118576(n), a(n+5)) are solutions (x, y) to the Diophantine equation x^2+(x+16807)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-11) = 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 11 = 1.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 11 = {0, 2, 4, 6, 7, 9}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 11 = {3, 5, 8, 10}.

			(-7203, a(1)) = (-7203, 12005) is a solution: (-7203)^2+(-7203+16807)^2 = 51883209+92236816 = 144120025 = 12005^2.
(A118576(1), a(6)) = (0, 16807) is a solution: 0^2+(0+16807)^2 = 258791569 = 16807^2.
(A118576(3), a(8)) = (3773, 20923) is a solution: 3773^2+(3773+16807)^2 = 14235529+423536400 = 437771929 = 20923^2.

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

Cf. A118576, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

CoefficientList[Series[(1-x)(12005+24472x+37457x^2+51520x^3+66545x^4+83352x^5+ 102425x^6+123348x^7+147505x^8+ 174252x^9+205465x^10+ 174252x^11+ 147505x^12+ 123348x^13+ 102425x^14+83352x^15+66545x^16+51520x^17+ 37457x^18+ 24472x^19+ 12005x^20)/(1-6x^11+x^22),{x,0,40}],x] (* or *) LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,-1},{12005,12467,12985,14063,15025,16807,19073,20923,24157,26747,31213,40817,48055,53753,63455,71077,84035,99413,111475,131957,148015,175273},40] (* Harvey P. Dale, Oct 02 2021 *)

{forstep(n=-7220, 700000, [1, 3], if(issquare(2*n^2+33614*n+282475249, &k),print1(k, ",")))}

a(n) = 6*a(n-11)-a(n-22) for n > 22; a(1) = 12005, a(2) = 12467, a(3) = 12985, a(4) = 14063, a(5) = 15025, a(6) = 16807, a(7) = 19073, a(8) = 20923, a(9) = 24157, a(10) = 26747, a(11) = 31213, a(12) = 40817, a(13) = 48055, a(14) = 53753, a(15) = 63455, a(16) = 71077, a(17) = 84035, a(18) = 99413, a(19) = 111475, a(20) = 131957, a(21) = 148015, a(22) = 175273.

G.f.: (1-x)*(12005 +24472*x+37457*x^2+51520*x^3+66545*x^4+83352*x^5+102425*x^6+123348*x^7+147505*x^8+174252*x^9+205465*x^10+174252*x^11+147505*x^12+123348*x^13+102425*x^14+83352*x^15+66545*x^16+51520*x^17 +37457*x^18+24472*x^19+12005*x^20)/(1-6*x^11+x^22).

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A118611 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2.

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A118630 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^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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A156713 Positive numbers y such that y^2 is of the form x^2+(x+16807)^2 with integer x.

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