A167774 -id:A167774 - cp's OEIS frontend

A167709 Numbers y such that 19*y^2 + 81 is a square.

0, 1, 15, 24, 220, 351, 560, 5124, 8175, 74801, 119340, 190399, 1742145, 2779476, 25432120, 40575249, 64735100, 592324176, 945013665, 8646845999, 13795465320, 22009743601, 201388477695, 321301866624, 2939902207540, 4690417633551

Offset: 0

Richard Choulet, Nov 10 2009

			a(0) = 0 because 19*0 + 81 = 9^2, a(1)=1 because 19*1 + 81 = 10^2.

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,340,0,0,0,0,-1).

Cf. A167708, A167774, A167775, A167778, A167779, A167780.

a(0):=0:a(1):=1:a(2):=15:a(3):=24:a(4):=220:a(5):=351:a(6):=560: a(7):=5124: a(8):=8175:a(9):=74801:for n from 0 to 40 do a(n+10):=340*a(n+5)-a(n):od:seq(a(n),n=0..40);

a[0]=0; a[1]=a[-1]=1; a[2]=a[-2]=15; a[n_] := a[n] = 170*a[n-5]+39*Sqrt[19*a[n-5]^2+81]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 19 2013 *)
LinearRecurrence[{0, 0, 0, 0, 340, 0, 0, 0, 0, -1}, {0, 1, 15, 24,
  220, 351, 560, 5124, 8175, 74801}, 100] (* G. C. Greubel, Jun 20 2016 *)

G.f.: (z + 15*z^2 + 24*z^3 + 220*z^4 + 351*z^5 + 560*z^6 + 5124*z^7 + 8175*z^8 + 74801*z^9 - 340*z^5*(z + 15*z^2 + 24*z^3 + 220*z^4) ) / (1 - 340*z^5 + z^10).
a(n+10) = 340*a(n+5) - a(n).
a(n+5) = 170*a(n) + 39*sqrt(19*a(n)^2 + 81).
For n == 0 (mod 5): a(n) = ( 9*sqrt(19) )/38*(170 + 39*sqrt(19))^(n) + (-9*sqrt(19))/38*(170 - 39*sqrt(19))^(n); the subsequence is 0, 351, 119340, 40575249, 13795465320, 4690417633551, 1594728199942020, 542202897562653249, 184347390443102162640, ...
For n == 1 (mod 5): a(n) = (10*sqrt(19) + 19)/38*(170 + 39*sqrt(19))^(n) + (-10*sqrt(19) + 19)/38*(170 - 39*sqrt(19))^(n); the subsequence is 1, 560, 190399, 64735100, 22009743601, 7483248089240, 2544282340597999, 865048512555230420, 294113949986437744801, ...
For n == 2 (mod 5): a(n) = (66*sqrt(19) + 285)/38*(170 + 39*sqrt(19))^(n) + (-66*sqrt(19) + 285)/38*(170 - 39*sqrt(19))^(n); the subsequence is 15, 5124, 1742145, 592324176, 201388477695, 68471490092124, 23280105242844465, 7915167311077025976, 2691133605660945987375, ...
For n == 3 (mod 5): a(n) = (105*sqrt(19) + 456)/38*(170 + 39*sqrt(19))^(n) + (-105*sqrt(19) + 456)/38*(170 - 39*sqrt(19))^(n); the subsequence is 24, 8175, 2779476, 945013665, 321301866624, 109241689638495, 37141853175221676, 12628120837885731345, 4293523943027973435624, ...
For n == 4 (mod 5): a(n) = (959*sqrt(19) + 4180)/38*(170 + 39*sqrt(19))^(n) + (-959*sqrt(19) + 4180)/38*(170 - 39*sqrt(19))^(n); the subsequence is 220, 74801, 25432120, 8646845999, 2939902207540, 999558103717601, 339846815361776800, 115546917664900394399, 39285612159250772318860, ...

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

Richard Choulet, Nov 10 2009, corrected Nov 12 2009

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

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,340,0,0,0,0,-1).

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

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);

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 *)

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

Richard Choulet, Nov 11 2009

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

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

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (340,-1).

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

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

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);

LinearRecurrence[{340, -1}, {10, 2441}, 50] (* G. C. Greubel, Jun 23 2016 *)

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

Richard Choulet, Nov 11 2009

Colin Barker, Table of n, a(n) for n = 0..394
Index entries for linear recurrences with constant coefficients, signature (340,-1).

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

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

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);

LinearRecurrence[{340, -1}, {66, 22335}, 20] (* Bruno Berselli, Nov 17 2015 *)

Vec(-3*(35*x-22)/(x^2-340*x+1) + O(x^20)) \\ Colin Barker, Nov 16 2015

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

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

Richard Choulet, Nov 11 2009

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

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (340,-1).

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

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

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);

LinearRecurrence[{340, -1}, {105, 35634}, 50] (* G. C. Greubel, Jun 23 2016 *)

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).

A167780 Subsequence of A167708 whose indices are 0 mod 5, that is, a(n) = A167708(5n+5).

Original entry on oeis.org

959, 326050, 110856041, 37690727890, 12814736626559, 4356972762302170, 1481357924446111241, 503657337338915519770, 171242013337306830610559, 58221780877346983492070290, 19795234256284637080473288041

Offset: 0

Richard Choulet, Nov 11 2009

			a(0) = A167708(5) = 959, a(1) = A167708(10) = 326050,...

Index entries for linear recurrences with constant coefficients, signature (340, -1).

Cf. A167708, A167709, A167774, A167775, A167778, A167779.

u(0):=959:u(1):=326050:for n from 0 to 20 do u(n+2):=340*u(n+1)-u(n):od:seq(u(n),n=0..20); taylor(((959+326050*z-959*z*340)/(1-340*z+z^2)),z=0,20); for n from 0 to 20 do u(n):=simplify((220*sqrt(19)+959)/2*(170+39*sqrt(19))^(n)+(-220*sqrt(19)+959)/2*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);

LinearRecurrence[{340,-1}, {959,326050}, 20] (* Harvey P. Dale, Aug 06 2013 *)

a(n+2) = 340*a(n+1) - a(n).
a(n+1) = 170*a(n) + 39*sqrt(19*a(n)^2 - 1539).
G.f.: (959 - 10*z)/(1 - 340*z + z^2).
a(n) = (220*sqrt(19) + 959)/2*(170 + 39*sqrt(19))^n + (-220*sqrt(19) + 959)/2*(170 - 39*sqrt(19))^n. - Richard Choulet, Nov 13 2009
a(n) = 10*cosh(x*log(170 + 39*Sqrt[19])) - Sqrt[19]*sinh(x*log(170 + 39*Sqrt[19])). - Harvey P. Dale, Aug 06 2013

A114048 x-values in the solution to x^2 - 19*y^2 = 1.

Original entry on oeis.org

1, 170, 57799, 19651490, 6681448801, 2271672940850, 772362118440199, 262600848596726810, 89283516160768675201, 30356132893812752841530, 10320995900380175197444999, 3509108249996365754378458130

Offset: 1

Cino Hilliard, Feb 01 2006

This sequence is computed with g(1e9,19) in the PARI program.
A Pellian equation (Pell's equation). - Benoit Cloitre, Feb 03 2006
The corresponding values of y of this Pell equation are in A174765. - Vincenzo Librandi, Dec 21 2011
Also numbers k such that 19*(k-1)*(k+1) is a square. - Bruno Berselli, May 31 2025

			(170^2 - 1)/19 = 39^2.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Tanya Khovanova, Recursive Sequences
John Robertson, Home page.
Index entries for linear recurrences with constant coefficients, signature (340,-1).

Cf. A167774, A174765.

I:=[1,170]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..40]]; // Vincenzo Librandi, Dec 21 2011

LinearRecurrence[{340,-1},{1,170},30] (* Vincenzo Librandi, Dec 21 2011 *)

g(n,k) = for(y=0,n,x=k*y^2+1;if(issquare(x),print1(floor(sqrt(x))",")))

a(n)=real((170+39*quadgen(4*19))^n) /* Michael Somos, Feb 15 2006 */

a=vector(12); a[1]=1; a[2]=170; for(i=3, #a, a[i]=340*a[i-1]-a[i-2]); a \\ Benoit Cloitre

a(n) = 340*a(n-1) - a(n-2) for n >= 3; a(1)=1, a(2)=170. - Benoit Cloitre, Feb 03 2006
G.f.: x*(1-170x)/(1-340x+x^2). - Philippe Deléham, Nov 18 2008
a(n) = A167774(n-1)/9. - Hugo Pfoertner, Feb 11 2024

More terms from Benoit Cloitre, Feb 03 2006

Offset changed from 0 to 1 and g.f. adapted by Vincenzo Librandi, Dec 21 2011

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A167709 Numbers y such that 19*y^2 + 81 is a square.

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

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

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A167778 Subsequence of A167708 whose indices are 2 mod 5.

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

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A167780 Subsequence of A167708 whose indices are 0 mod 5, that is, a(n) = A167708(5n+5).

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A114048 x-values in the solution to x^2 - 19*y^2 = 1.

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