cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A118120 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.

Original entry on oeis.org

0, 7, 28, 51, 88, 207, 340, 555, 1248, 2023, 3276, 7315, 11832, 19135, 42676, 69003, 111568, 248775, 402220, 650307, 1450008, 2344351, 3790308, 8451307, 13663920, 22091575, 49257868, 79639203, 128759176, 287095935, 464171332, 750463515, 1673317776
Offset: 0

Views

Author

Mohamed Bouhamida, May 12 2006

Keywords

Comments

Also values x of Pythagorean triples (x, x+17, y).
Corresponding values y of solutions (x, y) are in A155923.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a prime number in A066436, m >= 2 the associated value in A066049, the x values are given by the sequence defined by a(n) = 6*a(n-3)-a(n-6)+2p with a(0)=0, a(1)=2m+1, a(2)=6m^2-10m+4, a(3)=3p, a(4)=6m^2+10m+4, a(5)=40m^2-58m+21 (cf. A118673).
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(0)=p, b(1)=2*m^2+2m+1, b(2)=10m^2-14m+5, b(3)=5p, b(4)=10m^2+14m+5, b(5)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

Links

Crossrefs

Cf. A155923, A118673, A066436 (primes of the form 2*n^2-1), A066049 (2*n^2-1 is prime), A118554, A118611, A118630.
Cf. A155464 (first trisection), A155465 (second trisection), A155466 (third trisection).

Programs

  • Magma
    [ n: n in [0..25000000] | IsSquare(2*n*(n+17)+289) ];
  • Mathematica
    Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+17)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,7,28,51,88,207,340},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
  • PARI
    m=32; v=concat([0, 7, 28, 51, 88, 207], vector(m-6)); for(n=7, m, v[n]=6*v[n-3]-v[n-6]+34); v

Formula

a(n) = 6*a(n-3) -a(n-6) +34 for n > 5; a(0)=0, a(1)=7, a(2)=28, a(3)=51, a(4)=88, a(5)=207.
G. f.: x*(7 +21*x +23*x^2 -5*x^3 -7*x^4 -5*x^5)/((1-x)*(1-6*x^3+x^6)).

Extensions

Edited and 248755 changed to 248775 by Klaus Brockhaus, Feb 01 2009

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

Original entry on oeis.org

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

Views

Author

Mohamed Bouhamida, May 08 2006

Keywords

Comments

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}.

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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.

Extensions

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

Views

Author

Mohamed Bouhamida, May 09 2006

Keywords

Comments

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}.

Examples

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

Links

Crossrefs

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

Programs

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

Formula

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.

Extensions

Edited by Klaus Brockhaus, Feb 25 2009

A129010 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+833)^2 = y^2.

Original entry on oeis.org

0, 124, 168, 187, 343, 399, 595, 624, 915, 952, 1260, 1372, 1768, 1827, 1975, 2499, 3135, 3367, 3468, 4312, 4620, 5712, 5875, 7524, 7735, 9499, 10143, 12427, 12768, 13624, 16660, 20352, 21700, 22287, 27195, 28987, 35343, 36292, 45895, 47124
Offset: 1

Views

Author

Mohamed Bouhamida, May 27 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+833, y); 833=7^2*17.
Corresponding values y of solutions (x, y) are in A156835.
lim_{n -> infinity} a(n)/a(n-15) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 15 = {1, 2, 5, 7, 11, 13}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 15 = {0, 3, 6, 12}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*((19+6*sqrt(2))/17)/((9+4*sqrt(2))/7)^3 for n mod 15 = {4, 8, 10, 14}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^4/((3+2*sqrt(2))*((19+6*sqrt(2))/17)^2) for n mod 15 = 9.

Examples

			124^2+(124+833)^2 = 15376+915849 = 931225 = 965^2.

Crossrefs

Cf. A156835, A076296, A118120, A118554, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17).

Programs

  • PARI
    {forstep(n=0, 50000, [3, 1], if(issquare(2*n^2+1666*n+693889), print1(n, ",")))}

Formula

a(n) = 6*a(n-15)-a(n-30)+1666 for n > 30; a(1) = 0, a(2) = 124, a(3) = 168, a(4) = 187, a(5) = 343, a(6) = 399, a(7) = 595, a(8) = 624, a(9) = 915, a(10) = 952, a(11) = 1260, a(12) = 1372, a(13) = 1768, a(14) = 1827, a(15) = 1975, a(16) = 2499, a(17) = 3135, a(18) = 3367, a(19) = 3468, a(20) = 4312, a(21) = 4620, a(22) = 5712, a(23) = 5875, a(24) = 7524, a(25) = 7735, a(26) = 9499, a(27) = 10143, a(28) = 12427, a(29) = 12768, a(30) = 13624.
G.f.: x*(124+44*x+19*x^2+156*x^3+56*x^4+196*x^5+29*x^6+291*x^7+37*x^8+308*x^9+112*x^10+396*x^11+59*x^12+148*x^13+524*x^14-108*x^15-32*x^16-13*x^17-92*x^18-28*x^19-84*x^20-11*x^21-97*x^22-11*x^23-84*x^24-28*x^25-92*x^26-13*x^27-32*x^28-108*x^29)/((1-x)*(1-6*x^15+x^30)).

Extensions

Edited by Klaus Brockhaus, Feb 16 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

Views

Author

T. D. Noe, Feb 09 2012

Keywords

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.

Links

Crossrefs

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).

Programs

  • 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

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

A331265 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31^2)^2 = y^2.

Original entry on oeis.org

0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299, 19220, 27683, 39780, 55719, 79856, 114359, 163680, 234183, 327080, 467759, 668856, 956319, 1367240, 1908683, 2728620, 3900699, 5576156, 7971179, 11126940, 15905883, 22737260, 32502539, 46461756, 64854879, 92708600, 132524783
Offset: 1

Views

Author

Mohamed Bouhamida, Feb 12 2020

Keywords

Comments

For the generic case x^2 + (x + p^2)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436, m >= 4 (means p >= 31), the first five consecutive solutions are (0, p^2), (4*m^3+2*m^2-2*m-1, 4*m^4+4*m^3-2*m-1), (8*m^3+8*m^2+4*m, 4*m^4+8*m^3+12*m^2+4*m+1), (12*m^4-40*m^3+44*m^2-20*m+3, 20*m^4-56*m^3+60*m^2-28*m+5), (12*m^4-20*m^3+2*m^2+10*m-4, 20*m^4-28*m^3+14*m-5) and the other solutions are defined by (X(n), Y(n)) = (3*X(n-5) + 2*Y(n-5) + p^2, 4*X(n-5) + 3*Y(n-5) + 2*p^2).
X(n) = 6*X(n-5) - X(n-10) + 2*p^2, and Y(n) = 6*Y(n-5) - Y(n-10) (can be easily proved using X(n) = 3*X(n-5) + 2*Y(n-5) + p^2, and Y(n) = 4*X(n-5) + 3*Y(n-5) + 2*p^2).

Examples

			For p=31 (m=4) the first five (5) consecutive solutions are (0, 961), (279, 1271), (656, 1745), (1139, 2389), (1860, 3379).

Links

Crossrefs

Cf. A066436 (Primes of the form 2*m^2 - 1).
Solutions x to x^2+(x+p^2)^2=y^2: A118554 (p=7), A207059 (p=17), A309998 (p=23), this sequence (p=31), A332000 (p=47).

Programs

  • Magma
    I:=[0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299]; [n le 10 select I[n] else 6*Self(n-5) - Self(n-10)+1922: n in [1..100]];
  • Mathematica
    LinearRecurrence[{1, 0, 0, 0, 6, -6, 0, 0, 0, -1, 1}, {0, 279, 656, 1139, 1860, 2883, 4340, 6419, 9156, 13299, 19220}, 36] (* Jean-François Alcover, Feb 12 2020 *)
  • PARI
    concat(0, Vec(x^2*(279 + 377*x + 483*x^2 + 721*x^3 + 1023*x^4 - 217*x^5 - 183*x^6 - 161*x^7 - 183*x^8 - 217*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)) + O(x^30))) \\ Colin Barker, Feb 12 2020

Formula

a(n) = 6*a(n-5) - a(n-10) + 1922 for n >= 11; a(1)=0, a(2)=279, a(3)=656, a(4)=1139, a(5)=1860, a(6)=2883, a(7)=4340, a(8)=6419, a(9)=9156, a(10)=13299.
From Colin Barker, Feb 12 2020: (Start)
G.f.: x^2*(279 + 377*x + 483*x^2 + 721*x^3 + 1023*x^4 - 217*x^5 - 183*x^6 - 161*x^7 - 183*x^8 - 217*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)).
a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - a(n-10) + a(n-11) for n>11.
(End)

A332000 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 47^2)^2 = y^2.

Original entry on oeis.org

0, 752, 1820, 2231, 3995, 6627, 10575, 16511, 18840, 28952, 44180, 67116, 101664, 115227, 174135, 262871, 396539, 597891, 676940, 1020276, 1537464, 2316536, 3490100, 3950831, 5951939, 8966331, 13507095, 20347127, 23032464, 34695776, 52264940, 78730452, 118597080, 134248371
Offset: 1

Views

Author

Mohamed Bouhamida, Feb 04 2020

Keywords

Comments

For the generic case x^2 + (x + p^2)^2 = y^2 with p = m^2 - 2 a (prime) number in A028871, m>=7 (means p>=47), the first five consecutive solutions are: (0; p^2), (2*m^3+2*m^2-4*m-4; m^4+2*m^3-4*m-4), (4*m^3+8*m^2+8*m; m^4+4*m^3+12*m^2+8*m+4), (3*m^4-20*m^3+44*m^2-40*m+12; 5*m^4-28*m^3+60*m^2-56*m+20), (3*m^4-10*m^3+2*m^2+20*m-16; 5*m^4-14*m^3+28*m-20) and the other solutions are defined by: (X(n); Y(n))= (3*X(n-5)+2*Y(n-5)+p^2; 4*X(n-5)+3*Y(n-5)+2*p^2).
X(n) = 6*X(n-5) - X(n-10) + 2*p^2, and Y(n) = 6*Y(n-5) - Y(n-10) (can be easily proved using X(n) = 3*X(n-5) + 2*Y(n-5) + p^2, and Y(n) = 4*X(n-5) + 3*Y(n-5) + 2*p^2).

Examples

			For p=47 (m=7) the first five (5) consecutive solutions are (0, 2209), (752, 3055), (1820, 4421), (2231, 4969), (3995, 7379).

Links

Crossrefs

Cf. A028871 (Primes of the form m^2 - 2).
Solutions x to x^2+(x+p^2)^2=y^2: A118554 (p=7), A207059 (p=17), A309998 (p=23), A331265 (p=31), this sequence (p=47).

Programs

  • Magma
    I:=[0, 752, 1820, 2231, 3995, 6627, 10575, 16511, 18840, 28952]; [n le 10 select I[n] else 6*Self(n-5) - Self(n-10)+4418: n in [1..100]];
  • Mathematica
    LinearRecurrence[{1, 0, 0, 0, 6, -6, 0, 0, 0, -1, 1}, {0, 752, 1820, 2231, 3995, 6627, 10575, 16511, 18840, 28952, 44180}, 40] (* Jean-François Alcover, Feb 08 2020 *)
  • PARI
    concat(0, Vec(x^2*(752 + 1068*x + 411*x^2 + 1764*x^3 + 2632*x^4 - 564*x^5 - 472*x^6 - 137*x^7 - 472*x^8 - 564*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)) + O(x^40))) \\ Colin Barker, Feb 04 2020

Formula

a(n) = 6*a(n-5) - a(n-10) + 4418 for n >= 11; a(1)=0, a(2)=752, a(3)=1820, a(4)=2231, a(5)=3995, a(6)=6627, a(7)=10575, a(8)=16511, a(9)=18840, a(10)=28952.
From Colin Barker, Feb 04 2020: (Start)
G.f.: x^2*(752 + 1068*x + 411*x^2 + 1764*x^3 + 2632*x^4 - 564*x^5 - 472*x^6 - 137*x^7 - 472*x^8 - 564*x^9) / ((1 - x)*(1 - 6*x^5 + x^10)).
a(n) = a(n-1) + 6*a(n-5) - 6*a(n-6) - a(n-10) + a(n-11) for n>11.
(End)
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