A167708 -id:A167708 - cp's OEIS frontend

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

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

Offset: 0

Richard Choulet, Nov 11 2009

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

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

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

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

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);

LinearRecurrence[{340, -1}, {9, 1530}, 50] (* G. C. Greubel, Jun 23 2016 *)

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

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

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

Original entry on oeis.org

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, ...

cp's OEIS Frontend

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

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

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