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 11 results. Next

A167822 Subsequence of A167709 whose indices are congruent to 1 mod 5, i.e., a(n) = A167709(5*n+1).

Original entry on oeis.org

1, 560, 190399, 64735100, 22009743601, 7483248089240, 2544282340597999, 865048512555230420, 294113949986437744801, 99997877946876278001920, 33998984387987948082907999, 11559554694037955471910717740, 3930214596988516872501561123601
Offset: 0

Views

Author

Richard Choulet, Nov 13 2009

Keywords

Comments

Solutions to the Pell equation b(n)^2 - 19*a(n)^2 = 81. The corresponding b(n) are in A167755. - Klaus Purath, Aug 31 2025

Examples

			a(0) = A167709(1) = 1, a(1) = A167709(6) = 560.

Links

Crossrefs

Cf. A167709, A167755

Programs

  • Magma
    I:=[1,560]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 28 2016
  • Maple
    w(0):=1:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((10*sqrt(19)+19)/38*(170+39*sqrt(19))^(n)+(-10*sqrt(19)+19)/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((1+560*z-1*340*z)/(1-340*z+z^2)),z=0,21);
  • Mathematica
    LinearRecurrence[{340,-1},{1,560},50] (* G. C. Greubel, Jun 27 2016 *)
RecurrenceTable[{a[1] == 1, a[2] == 560, a[n] == 340 a[n-1] - a[n-2]}, a, {n, 15}] (* Vincenzo Librandi, Jun 28 2016 *)

Formula

a(n+2) = 340*a(n+1) - a(n).
a(n+1) = 170*a(n) + 39*sqrt(19*(a(n))^2 + 81).
G.f.: (1 + 220*z)/(1 - 340*z + z^2).
a(n) = (10*sqrt(19) + 19)/38*(170 + 39*sqrt(19))^n + (-10*sqrt(19) + 19)/38*(170 - 39*sqrt(19))^n.

A167820 Subsequence of A167709 whose indices are congruent to 0 mod 5, i.e., a(n) = A167709(5*n).

Original entry on oeis.org

0, 351, 119340, 40575249, 13795465320, 4690417633551, 1594728199942020, 542202897562653249, 184347390443102162640, 62677570547757172644351, 21310189638846995596916700, 7245401799637430745779033649, 2463415301687087606569274523960
Offset: 0

Views

Author

Richard Choulet, Nov 13 2009

Keywords

Examples

			a(0)=0 because a(0) = A167709(0) = 0, a(1)=351 because a(1) = A167709(5) = 351.

Links

Crossrefs

Cf. A167709.

Programs

  • Magma
    I:=[0,351]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 28 2016
  • Maple
    w(0):=0:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((9*sqrt(19))/38*(170+39*sqrt(19))^(n)+(-9*sqrt(19))/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((351*z)/(1-340*z+z^2)),z=0,21);A167709
  • Mathematica
    RecurrenceTable[{a[1] == 0, a[2] == 351, a[n] == 340 a[n-1] - a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Jun 28 2016 *)
LinearRecurrence[{340,-1},{0,351},20] (* Harvey P. Dale, Dec 25 2018 *)

Formula

a(n+2) = 340*a(n+1) - a(n).
a(n+1) = 170*a(n) + 39*sqrt(19*(a(n))^2 + 81).
G.f.: 351*z/(1 - 340*z + z^2).
a(n) = (9*sqrt(19))/38*(170 + 39*sqrt(19))^n + (-9*sqrt(19))/38*(170 - 39*sqrt(19))^n.

A167823 Subsequence of A167709 whose indices are congruent to 2 mod 5, i.e., a(n) = A167709(5*n+2).

Original entry on oeis.org

15, 5124, 1742145, 592324176, 201388477695, 68471490092124, 23280105242844465, 7915167311077025976, 2691133605660945987375, 914977510757410558681524, 311089662523913929005730785, 105769570280619978451389785376, 35961342805748268759543521297055
Offset: 0

Views

Author

Richard Choulet, Nov 13 2009

Keywords

Examples

			a(0) = A167709(2) = 15, a(1) = A167709(7) = 5124.

Links

Programs

  • Magma
    I:=[15,5124]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 28 2016
  • Maple
    w(0):=15:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((66*sqrt(19)+285)/38*(170+39*sqrt(19))^(n)+(-66*sqrt(19)+285)/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((15+5124*z-15*340*z)/(1-340*z+z^2)),z=0,21);
  • Mathematica
    LinearRecurrence[{340,-1},{15, 5124}, 50] (* G. C. Greubel, Jun 27 2016 *)
RecurrenceTable[{a[1] == 15, a[2] == 5124, a[n] == 340 a[n-1] - a[n-2]}, a, {n, 15}] (* Vincenzo Librandi, Jun 28 2016 *)

Formula

a(n+2) = 340*a(n+1) - a(n).
a(n+1) = 170*a(n) + 39*sqrt(19*(a(n))^2 + 81).
G.f.: (15 + 24*z)/(1 - 340*z + z^2).
a(n) = (66*sqrt(19) + 285)/38*(170 + 39*sqrt(19))^n + (-66*sqrt(19) + 285)/38*(170 - 39*sqrt(19))^n.

A167824 Subsequence of A167709 whose indices are congruent to 3 mod 5, i.e., a(n) = A167709(5*n+3).

Original entry on oeis.org

24, 8175, 2779476, 945013665, 321301866624, 109241689638495, 37141853175221676, 12628120837885731345, 4293523943027973435624, 1459785512508673082380815, 496322780729005820036041476, 168748285662349470139171721025
Offset: 0

Views

Author

Richard Choulet, Nov 13 2009

Keywords

Examples

			a(0) = A167709(3) = 24, a(1) = A167709(8) = 8175.

Links

Programs

  • Magma
    I:=[24,8175]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 28 2016
  • Maple
    w(0):=24:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((105*sqrt(19)+456)/38*(170+39*sqrt(19))^(n)+(-105*sqrt(19)+456)/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((24+8175*z-24*340*z)/(1-340*z+z^2)),z=0,21);
  • Mathematica
    LinearRecurrence[{340, -1}, {24, 8175}, 50] (* G. C. Greubel, Jun 27 2016 *)
RecurrenceTable[{a[1] == 24, a[2] == 8175, a[n] == 340*a[n-1] -a[n-2]}, a, {n, 15}] (* Vincenzo Librandi, Jun 28 2016 *)

Formula

a(n+2) = 340*a(n+1) - a(n).
a(n+1) = 170*a(n) + 39*sqrt(19*(a(n))^2 + 81).
G.f.: (24 + 15*x)/(1 - 340*x + x^2).
a(n) = ((105*sqrt(19) + 456)/38)*(170 + 39*sqrt(19))^n + ((-105*sqrt(19) + 456)/38)*(170 - 39*sqrt(19))^n.

A167825 Subsequence of A167709 whose indices are congruent to 4 mod 5, i.e., a(n) = A167709(5*n+4).

Original entry on oeis.org

220, 74801, 25432120, 8646845999, 2939902207540, 999558103717601, 339846815361776800, 115546917664900394399, 39285612159250772318860, 13356992587227597688018001, 4541338194045223963153801480
Offset: 0

Views

Author

Richard Choulet, Nov 13 2009

Keywords

Examples

			a(0) = A167709(4) = 220, a(1) = A167709(9) = 74801.

Links

Programs

  • Magma
    I:=[220,74801]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 28 2016
  • Maple
    w(0):=220:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((959*sqrt(19)+4180)/38*(170+39*sqrt(19))^(n)+(-959*sqrt(19)+4180)/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((220+74801*z-220*340*z)/(1-340*z+z^2)),z=0,21);
  • Mathematica
    LinearRecurrence[{340, -1}, {220, 74801}, 50] (* G. C. Greubel, Jun 27 2016 *)
RecurrenceTable[{a[1] == 220, a[2] == 74801, a[n] == 340 a[n-1] - a[n-2]}, a, {n, 15}] (* Vincenzo Librandi, Jun 28 2016 *)

Formula

a(n+2) = 340*a(n+1) - a(n).
a(n+1) = 170*a(n) + 39*sqrt(19*(w(n))^2 + 81).
G.f.: (220 + x)/(1 - 340*x + x^2).
a(n) = ((959*sqrt(19) + 4180)/38)*(170 + 39*sqrt(19))^n + ((-959*sqrt(19) + 4180)/38)*(170 - 39*sqrt(19))^n.

A167774 Subsequence of A167708 whose indices are congruent to 1 mod 5, i.e., a(n) = A167708(5*n+1).

Original entry on oeis.org

9, 1530, 520191, 176863410, 60133039209, 20445056467650, 6951259065961791, 2363407637370541290, 803551645446918076809, 273205196044314775573770, 92888963103421576777004991, 31581974249967291789406123170, 10737778356025775786821304872809
Offset: 0

Views

Author

Richard Choulet, Nov 11 2009

Keywords

Examples

			a(0)=A167708(1)=9, a(1)=A167708(6)=1530.

Links

Crossrefs

Cf. A114048, A167708, A167709, A167775, A167778, A167779, A167780.

Programs

  • Magma
    I:=[9, 1530]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 24 2016
  • Maple
    u(0):=9:for n from 0 to 20 do u(n+1):=170*u(n)+39*sqrt(19*u(n)^2-1539):od:seq(u(n),n=0..20); taylor(((9+1530*z-9*z*340)/(1-340*z+z^2)),z=0,20);
  • Mathematica
    LinearRecurrence[{340, -1}, {9, 1530}, 50] (* G. C. Greubel, Jun 23 2016 *)

Formula

Recurrence formulas: a(n+2) = 340*a(n+1) - a(n) or a(n+1) = 170*a(n) + 39*sqrt(19*a(n)^2 - 1539).
G.f.: (9 - 1530*z)/(1 - 340*z + z^2).
a(n) = (9/2)*(170 + 39*sqrt(19))^(n) + (9/2)*(170 - 39*sqrt(19))^(n).
a(n) = 9*A114048(n+1). - R. J. Mathar, Feb 19 2016

A167708 Numbers k such that (k^2 - 81)/19 is a square.

Original entry on oeis.org

9, 10, 66, 105, 959, 1530, 2441, 22335, 35634, 326050, 520191, 829930, 7593834, 12115455, 110856041, 176863410, 282173759, 2581881225, 4119219066, 37690727890, 60133039209, 95938248130, 877832022666, 1400522366985
Offset: 1

Views

Author

Richard Choulet, Nov 10 2009, corrected Nov 12 2009

Keywords

Examples

			a(0)=9 because (9^2 - 81)/19=0 in N; a(1)=10 because (10^2 - 81)/19=1 in N.

References

  • A. H. Beiler, "Recreations in the theory of numbers": Ex. 38, p. 298 (Dover Publications, Inc., New York, 1966).

Links

Crossrefs

Cf. A167709, A167774, A167775, A167778, A167779, A167780.

Programs

  • Maple
    a(0):=9:a(1):=10:a(2):=66:a(3):=105:a(4):=959:a(5):=1530:a(6):=2441:a(7):=22335:a(8):=35634:a(9):=326050:for n from 0 to 40 do a(n+10):=340*a(n+5)-a(n):od:seq(a(n),n=0..40);
  • Mathematica
    Table[n /. {ToRules[Reduce[n > 0 && k >= 0 && n^2-81 == 19*k^2, n, Integers] /. C[1] -> c]} // Simplify, {c, 0, 5}] // Flatten // Union (* Jean-François Alcover, Dec 19 2013 *)
LinearRecurrence[{0, 0, 0, 0, 340, 0, 0, 0, 0, -1}, {9, 10, 66, 105, 959, 1530, 2441, 22335, 35634, 326050}, 100] (* G. C. Greubel, Jun 20 2016 *)

Formula

G.f.: (9 + 10*z + 66*z^2 + 105*z^3 + 959*z^4 + 1530*z^5 + 2441*z^6 + 22335*z^7 + 35634*z^8 + 326050*z^9 - 340*z^5*(9 + 10*z + 66*z^2 + 105*z^3 +959*z^4))/ (1 - 340*z^5 + z^10).
a(n+10) = 340*a(n+5) - a(n).
On every subsequence mod 5: a(n+2) = 340*a(n+1) - a(n).
On every subsequence mod 5: a(n+2) = 170*a(n) + 39*sqrt(19*a(n)^2 - 1539).
for n == 0 (mod 5): a(n) = (9/2)*(170 + 39*sqrt(19))^(n) + (9/2)*(170 - 39*sqrt(19))^(n); the subsequence is 9, 1530, 520191, 176863410, 60133039209, 20445056467650, 6951259065961791, 2363407637370541290, 803551645446918076809, ...
for n == 1 (mod 5): a(n) = (sqrt(19) + 10)/2*(170 + 39*sqrt(19))^(n) + (-sqrt(19) + 10)/2*(170 - 39*sqrt(19))^(n); the subsequence is 10, 2441, 829930, 282173759, 95938248130, 32618722190441, 11090269606501810, 3770659047488424959, 1282012985876457984250, ...
for n == 2 (mod 5): a(n) = (15*sqrt(19) + 66)/2*(170 + 39*sqrt(19))^(n) + (-15*sqrt(19) + 66)/2*(170 - 39*sqrt(19))^(n); the subsequence is 66, 22335, 7593834, 2581881225, 877832022666, 298460305825215, 101475626148550434, 34501414430201322345, 11730379430642301046866, ...
for n == 3 (mod 5): a(n) = (24*sqrt(19) + 105)/2*(170 + 39*sqrt(19))^(n) + (-24*sqrt(19) + 105)/2*(170 - 39*sqrt(19))^(n); the subsequence is 105, 35634, 12115455, 4119219066, 1400522366985, 476173485555834, 161897584566616575, 55044702579164079666, 18715036979331220469865, ...
for n == 4 (mod 5): a(n) = (220*sqrt(19) + 959)/2*(170 + 39*sqrt(19))^(n) + (-220*sqrt(19) + 959)/2*(170 - 39*sqrt(19))^(n); the subsequence is 959, 326050, 110856041, 37690727890, 12814736626559, 4356972762302170, 1481357924446111241, 503657337338915519770, 171242013337306830610559, ...

A167775 Subsequence of A167708 whose indices are congruent to 1 mod 5, i.e., a(n) = A167708(5n+1).

Original entry on oeis.org

10, 2441, 829930, 282173759, 95938248130, 32618722190441, 11090269606501810, 3770659047488424959, 1282012985876457984250, 435880644538948226220041, 148198137130256520456829690, 50386930743642678007095874559, 17131408254701380265892140520370
Offset: 0

Views

Author

Richard Choulet, Nov 11 2009

Keywords

Comments

Solutions to the Pell equation a(n)^2 - 19*b(n)^2 = 81. The corresponding b(n) are in A167822. - Klaus Purath, Aug 31 2025

Examples

			a(0) = A167708(1) = 10, a(1) = A167708(6) = 2441, ...

Links

Crossrefs

Cf. A167708, A167709, A167774, A167778, A167779, A167780, A167822.

Programs

  • Magma
    I:=[10,2441]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 24 2016
  • Maple
    u(0):=10:u(1):=2441:for n from 0 to 20 do u(n+2):=340*u(n+1)-u(n):od:seq(u(n),n=0..20); taylor(((10+2441*z-10*z*340)/(1-340*z+z^2)),z=0,20);
  • Mathematica
    LinearRecurrence[{340, -1}, {10, 2441}, 50] (* G. C. Greubel, Jun 23 2016 *)

Formula

a(n+2) = 340*a(n+1) - a(n).
a(n+1) = 170*a(n) + 39*sqrt(19*a(n)^2 - 1539).
G.f.: (10 + 2441*z - 10*z*340)/(1 - 340*z + z^2).
a(n) = ((10 + sqrt(19))/2)*(170 + 39*sqrt(19))^(n) + ((10 - sqrt(19))/2)* (170 - 39*sqrt(19))^(n).

A167778 Subsequence of A167708 whose indices are 2 mod 5.

Original entry on oeis.org

66, 22335, 7593834, 2581881225, 877832022666, 298460305825215, 101475626148550434, 34501414430201322345, 11730379430642301046866, 3988294505003952154612095, 1356008401321913090267065434, 461038868154945446738647635465, 156751859164280129978049928992666
Offset: 0

Views

Author

Richard Choulet, Nov 11 2009

Keywords

Links

Crossrefs

Cf. A167708, A167709, A167774, A167775, A167779, A167780. - Richard Choulet, Nov 11 2009

Programs

  • Magma
    I:=[66,22335]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 17 2015
  • Maple
    u(0):=66:u(1):=22335:for n from 0 to 20 do u(n+2):=340*u(n+1)-u(n):od:seq(u(n),n=0..20); taylor(((66+22335*z-66*z*340)/(1-340*z+z^2)),z=0,20); for n from 0 to 20 do u(n):=simplify((15*sqrt(19)+66)/2*(170+39*sqrt(19))^(n)+(-15*sqrt(19)+66)/2*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);
  • Mathematica
    LinearRecurrence[{340, -1}, {66, 22335}, 20] (* Bruno Berselli, Nov 17 2015 *)
  • PARI
    Vec(-3*(35*x-22)/(x^2-340*x+1) + O(x^20)) \\ Colin Barker, Nov 16 2015

Formula

a(n) = A167708(5n+2).
a(n+2) = 340*a(n+1) - a(n).
a(n+1) = 170*a(n) + 39*sqrt(19*a(n)^2-1539).
G.f.: (66 + 22335*x - 66*x*340)/(1 - 340*x + x^2).
a(n) = ((66 + 15*sqrt(19))/2)*(170 + 39*sqrt(19))^n + ((66 - 15*sqrt(19)) /2)*(170 - 39*sqrt(19))^n. - Richard Choulet, Nov 13 2009
G.f.: -3*(35*x - 22) / (x^2 - 340*x + 1). - Colin Barker, Nov 16 2015

Extensions

Definition corrected by Richard Choulet, Nov 15 2009
Typo in title fixed by Colin Barker, Nov 16 2015

A167779 Subsequence of A167708 whose indices are congruent to 4 mod 5, i.e., a(n) = A167708(5n+4).

Original entry on oeis.org

105, 35634, 12115455, 4119219066, 1400522366985, 476173485555834, 161897584566616575, 55044702579164079666, 18715036979331220469865, 6363057528270035795674434, 2163420844574832839308837695, 735556724097914895329209141866, 250087122772446489579091799396745
Offset: 0

Views

Author

Richard Choulet, Nov 11 2009

Keywords

Examples

			a(0) = A167708(4) = 105, a(1) = A167708(9) = 35634, ...

Links

Crossrefs

Cf. A167708, A167709, A167774, A167775, A167778, A167780.

Programs

  • Magma
    I:=[105,35634]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 25 2016
  • Maple
    u(0):=105:u(1):=35634:for n from 0 to 20 do u(n+2):=340*u(n+1)-u(n):od:seq(u(n),n=0..20); taylor(((105+35634*z-105*z*340)/(1-340*z+z^2)),z=0,20); for n from 0 to 20 do u(n):=simplify((24*sqrt(19)+105)/2*(170+39*sqrt(19))^(n)+(-24*sqrt(19)+105)/2*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);
  • Mathematica
    LinearRecurrence[{340, -1}, {105, 35634}, 50] (* G. C. Greubel, Jun 23 2016 *)

Formula

a(n+2) = 340*a(n+1) - a(n).
a(n+1) = 170*a(n) + 39*sqrt(19*a(n)^2 - 1539).
G.f.: (105 - 66*z)/(1 - 340*z + z^2).
a(n) = ((105 + 24*sqrt(19))/2)*(170 + 39*sqrt(19))^(n) + ((105 - 24*sqrt(19) )/2)*(170 - 39*sqrt(19))^(n).
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