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 1-10 of 13 results. Next

A118676 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+79)^2 = y^2.

Original entry on oeis.org

0, 20, 161, 237, 341, 1140, 1580, 2184, 6837, 9401, 12921, 40040, 54984, 75500, 233561, 320661, 440237, 1361484, 1869140, 2566080, 7935501, 10894337, 14956401, 46251680, 63497040, 87172484, 269574737, 370088061, 508078661, 1571196900, 2157031484, 2961299640
Offset: 1

Views

Author

Mohamed Bouhamida, May 19 2006

Keywords

Comments

Also values x of Pythagorean triples (x, x+79, y).
Corresponding values y of solutions (x, y) are in A159758.
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) = (83+18*sqrt(2))/79 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (10659+6110*sqrt(2))/79^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159758, A028871, A118337, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159759 (decimal expansion of (83+18*sqrt(2))/79), A159760 (decimal expansion of (10659+6110*sqrt(2))/79^2).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(20+141*x+76*x^2-16*x^3-47*x^4-16*x^5)/((1-x)*(1- 6*x^3+x^6)))); // G. C. Greubel, May 07 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,20,161,237,341,1140,1580},75] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)
  • PARI
    forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+158*n+6241), print1(n, ",")))

Formula

a(n) = 6*a(n-3) -a(n-6) +158 for n > 6; a(1)=0, a(2)=20, a(3)=161, a(4)=237, a(5)=341, a(6)=1140.
G.f.: x*(20+141*x+76*x^2-16*x^3-47*x^4-16*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 79*A001652(k) for k >= 0.

Extensions

Edited by Klaus Brockhaus, Apr 30 2009

A118675 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.

Original entry on oeis.org

0, 16, 85, 141, 225, 616, 940, 1428, 3705, 5593, 8437, 21708, 32712, 49288, 126637, 190773, 287385, 738208, 1112020, 1675116, 4302705, 6481441, 9763405, 25078116, 37776720, 56905408, 146166085, 220178973, 331669137, 851918488, 1283297212, 1933109508
Offset: 1

Views

Author

Mohamed Bouhamida, May 19 2006

Keywords

Comments

Also values x of Pythagorean triples (x, x+47, y).
Corresponding values y of solutions (x, y) are in A159750.
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) = (51+14*sqrt(2))/47 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159750, A028871, A118337, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3 +x^6)))); // G. C. Greubel, May 07 2018
  • Mathematica
    Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+47)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,16,85,141,225,616,940},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+94+2209), print1(n, ",")))}

Formula

a(n) = 6*a(n-3) -a(n-6) +94 for n > 6; a(1)=0, a(2)=16, a(3)=85, a(4)=141, a(5)=225, a(6)=616.
G.f.: x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 47*A001652(k) for k >= 0.

Extensions

Edited by Klaus Brockhaus, Apr 30 2009

A156567 Positive numbers y such that y^2 is of the form x^2+(x+23)^2 with integer x.

Original entry on oeis.org

17, 23, 37, 65, 115, 205, 373, 667, 1193, 2173, 3887, 6953, 12665, 22655, 40525, 73817, 132043, 236197, 430237, 769603, 1376657, 2507605, 4485575, 8023745, 14615393, 26143847, 46765813, 85184753, 152377507, 272571133, 496493125
Offset: 1

Views

Author

Klaus Brockhaus, Feb 11 2009 , Feb 16 2009

Keywords

Comments

(-8, a(1)) and(A118337(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+23)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (27+10*sqrt(2))/23 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (627+238*sqrt(2))/23^2 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>=2, 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)=3m^2-10m+8, a(4)=3p, a(5)=3m^2+10m+8, a(6)=20m^2-58m+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+2m+2, b(3)=5m^2-14m+10, b(4)=5p, b(5)=5m^2+14m+10, b(6)=29m^2-82m+58. [From Mohamed Bouhamida, Sep 09 2009]

Examples

			(-8, a(1)) = (-8, 17) is a solution: (-8)^2+(-8+23)^2 = 64+225 = 289 = 17^2.
(A118337(1), a(2)) = (0, 23) is a solution: 0^2+(0+23)^2 = 529 = 23^2.
(A118337(3), a(4)) = (33, 65) is a solution: 33^2+(33+23)^2 = 1089+3136 = 4225 = 65^2.

Crossrefs

Cf. A118337, A156035 (decimal expansion of 3+2*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23), A157472 (decimal expansion of (627+238*sqrt(2))/23^2).
A156570 (first trisection), A156568 (second trisection), A156569 (third trisection).

Programs

  • PARI
    {forstep(n=-8, 360000000, [1,3], if(issquare(2*n*(n+23)+529, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=17, a(2)=23, a(3)=37, a(4)=65, a(5)=115, a(6)=205.
G.f.: x*(1-x)*(17+40*x+77*x^2+40*x^3+17*x^4)/(1-6*x^3+x^6).

Extensions

G.f. corrected, third and fourth comment edited, cross-reference added by Klaus Brockhaus, Sep 18 2009

A130609 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.

Original entry on oeis.org

0, 32, 533, 669, 833, 3672, 4460, 5412, 21945, 26537, 32085, 128444, 155208, 187544, 749165, 905157, 1093625, 4366992, 5276180, 6374652, 25453233, 30752369, 37154733, 148352852, 179238480, 216554192, 864664325, 1044678957, 1262170865, 5039633544, 6088835708
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 17 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+223, y).
Corresponding values y of solutions (x, y) are in A159809.
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) = (227+30*sqrt(2))/223 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (105507+65798*sqrt(2))/223^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159809, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159810 (decimal expansion of (227+30*sqrt(2))/223), A159811 (decimal expansion of (105507+65798*sqrt(2))/223^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,32,533,669,833,3672,4460}, 70]  (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)
  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+446*n+49729), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+446 for n > 6; a(1)=0, a(2)=32, a(3)=533, a(4)=669, a(5)=833, a(6)=3672.
G.f.: x*(32+501*x+136*x^2-28*x^3-167*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 223*A001652(k) for k >= 0.

Extensions

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

A130610 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.

Original entry on oeis.org

0, 40, 901, 1077, 1281, 6160, 7180, 8364, 36777, 42721, 49621, 215220, 249864, 290080, 1255261, 1457181, 1691577, 7317064, 8493940, 9860100, 42647841, 49507177, 57469741, 248570700, 288549840, 334959064, 1448777077, 1681792581
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 17 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+359, y).
Corresponding values y of solutions (x, y) are in A159844.
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) = (363+38*sqrt(2))/359 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (293619+186550*sqrt(2))/359^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159844, A028871, A118337, A130609, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159845 (decimal expansion of (363+38*sqrt(2))/359), A159846 (decimal expansion of (293619+186550*sqrt(2))/359^2).

Programs

  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+718*n+128881), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+718 for n > 6; a(1)=0, a(2)=40, a(3)=901, a(4)=1077, a(5)=1281, a(6)=6160.
G.f.: x*(40+861*x+176*x^2-36*x^3-287*x^4-36*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 359*A001652(k) for k >= 0.

Extensions

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

A130645 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.

Original entry on oeis.org

0, 44, 1121, 1317, 1541, 7644, 8780, 10080, 45621, 52241, 59817, 266960, 305544, 349700, 1557017, 1781901, 2039261, 9076020, 10386740, 11886744, 52899981, 60539417, 69282081, 308324744, 352850640, 403806620, 1797049361, 2056565301, 2353558517, 10473972300
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 20 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+439, y).
Corresponding values y of solutions (x, y) are in A159890.
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) = (443+42*sqrt(2))/439 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (450483+287918*sqrt(2))/439^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159890, A028871, A118337, A118675, A118676, A001652, 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).

Programs

  • Mathematica
    LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 44, 1121, 1317, 1541, 7644, 8780}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)
  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+878*n+192721), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+878 for n > 6; a(1)=0, a(2)=44, a(3)=1121, a(4)=1317, a(5)=1541, a(6)=7644.
G.f.: x*(44+1077*x+196*x^2-40*x^3-359*x^4-40*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 439*A001652(k) for k >= 0.

Extensions

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

A130646 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+727)^2 = y^2.

Original entry on oeis.org

0, 56, 1925, 2181, 2465, 13056, 14540, 16188, 77865, 86513, 96117, 455588, 505992, 561968, 2657117, 2950893, 3277145, 15488568, 17200820, 19102356, 90275745, 100255481, 111338445, 526167356, 584333520, 648929768, 3066729845
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 20 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+727, y).
Corresponding values y of solutions (x, y) are in A159893.
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) = (731+54*sqrt(2))/727 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159893, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion of (1304787+843542*sqrt(2))/727^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,56,1925,2181,2465,13056,14540},40] (* or *) RecurrenceTable[{a[1]==0,a[2]==56,a[3]==1925,a[4]==2181,a[5] == 2465, a[6] == 13056, a[n] ==6a[n-3]-a[n-6]+1454},a,{n,40}] (* Harvey P. Dale, Jan 16 2013 *)
  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1454*n+528529), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1454 for n > 6; a(1)=0, a(2)=56, a(3)=1925, a(4)=2181, a(5)=2465, a(6)=13056.
G.f.: x*(56+1869*x+256*x^2-52*x^3-623*x^4-52*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 727*A001652(k) for k >= 0.

Extensions

Edited and one term added by Klaus Brockhaus, Apr 30 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

Views

Author

Mohamed Bouhamida, Jun 20 2007

Keywords

Comments

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.

Links

Crossrefs

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).

Programs

  • Magma
    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
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,60,2241,2517,2821,15180,16780},30] (* Harvey P. Dale, Jun 19 2014 *)
  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1678*n+703921), print1(n, ",")))}

Formula

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.

Extensions

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

A130608 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.

Original entry on oeis.org

0, 28, 385, 501, 645, 2668, 3340, 4176, 15957, 19873, 24745, 93408, 116232, 144628, 544825, 677853, 843357, 3175876, 3951220, 4915848, 18510765, 23029801, 28652065, 107889048, 134227920, 166996876, 628823857, 782338053, 973329525, 3665054428, 4559800732
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 17 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+167, y).
Corresponding values y of solutions (x, y) are in A159777.
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) = (171+26*sqrt(2))/167 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (56211+34510*sqrt(2))/167^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159777, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159778 (decimal expansion of (171+26*sqrt(2))/167), A159779 (decimal expansion of (56211+34510*sqrt(2))/167^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,28,385,501,645,2668,3340},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)
  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+334*n+27889), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+334 for n > 6; a(1)=0, a(2)=28, a(3)=385, a(4)=501, a(5)=645, a(6)=2668.
G.f.: x*(28+357*x+116*x^2-24*x^3-119*x^4-24*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 167*A001652(k) for k >= 0.

Extensions

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

A156571 Decimal expansion of (27 + 10*sqrt(2))/23.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Feb 10 2009

Keywords

Comments

Lim_{n -> infinity} a(n)/a(n-1) = (27+10*sqrt(2))/23 for n mod 3 = {1, 2}, b = A118337, A156567.
Lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((27+10*sqrt(2))/23)^2 for n mod 3 = 0, b = A118337, A156567.

Examples

			(27 + 10*sqrt(2))/23 = 1.78878850537960654295...

Links

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)).

Programs

  • Magma
    (27+10*Sqrt(2))/23; // G. C. Greubel, Jan 27 2018
  • Mathematica
    RealDigits[(27 + 10*Sqrt[2])/23, 10, 100][[1]] (* G. C. Greubel, Jan 28 2018 *)
  • PARI
    (27+10*sqrt(2))/23 \\ G. C. Greubel, Jan 27 2018
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