A098158 -id:A098158 - cp's OEIS frontend

A152056 a(n) = ((9+sqrt(3))^n + (9-sqrt(3))^n)/2.

1, 9, 84, 810, 8028, 81324, 837648, 8734392, 91882512, 972602640, 10340011584, 110257202592, 1178108743104, 12605895573696, 135013638364416, 1446985635811200, 15514677652177152, 166399318145915136

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008

nonn

Index entries for linear recurrences with constant coefficients, signature (18, -78).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((9+r3)^n+(9-r3)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 26 2008

From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 18*a(n-1) - 78*a(n-2), n > 1; a(0)=1, a(1)=9.
G.f.: (1-9*x)/(1-18*x+78*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*3^(3*k))/3^n. (End)

Extended beyond a(6) by Klaus Brockhaus, Nov 26 2008

A152107 a(n) = ((6+sqrt(5))^n+(6-sqrt(5))^n)/2.

Original entry on oeis.org

1, 6, 41, 306, 2401, 19326, 157481, 1290666, 10606081, 87262326, 718359401, 5915180706, 48713027041, 401185722606, 3304124833001, 27212740595226, 224125017319681, 1845905249384166, 15202987455699881, 125212786737489426

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008

nonn

			For n=3, (6+sqrt(5))^3 = 216 + 108*sqrt(5) + 18*5 + 5*sqrt(5) = 306 + 113*sqrt(5) and (6-sqrt(5))^3 = 306 - 113*sqrt(5), so a(3) = (306 + 113*sqrt(5) + 306 - 113*sqrt(5))/2 = 306. - _Michael B. Porter_, Aug 25 2016

Robert Israel, Table of n, a(n) for n = 0..1090
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (12,-31).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((6+r5)^n+(6-r5)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 26 2008

f:= gfun:-rectoproc({a(n)=12*a(n-1)-31*a(n-2),a(0)=1,a(1)=6},a(n),remember):
map(f, [$0..50]); # Robert Israel, Aug 25 2016

CoefficientList[Series[(1 - 6 x)/(1 - 12 x + 31 x^2), {x, 0, 19}], x] (* Michael De Vlieger, Aug 25 2016 *)
LinearRecurrence[{12,-31},{1,6},30] (* Harvey P. Dale, Aug 28 2024 *)

From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 12*a(n-1)-31*a(n-2), n>1 ; a(0)=1, a(1)=6 .
G.f.: (1-6*x)/(1-12*x+31*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2*k)*5^(n-k))/6^n. (End)
a(n) = Sum_{k=0..n} A027907(n,2k)*5^k . - J. Conrad, Aug 24 2016
E.g.f.: cosh(sqrt(5)*x)*exp(6*x). - Ilya Gutkovskiy, Aug 24 2016

Extended beyond a(6) by Klaus Brockhaus, Nov 26 2008

Typo in name corrected by J. Conrad, Aug 24 2016

A152108 a(n) = ((7+sqrt(5))^n + (7-sqrt(5))^n)/2.

Original entry on oeis.org

1, 7, 54, 448, 3896, 34832, 316224, 2894528, 26609536, 245174272, 2261620224, 20875015168, 192738922496, 1779844247552, 16437306875904, 151809149370368, 1402086588645376, 12949609668739072, 119602725461950464, 1104655331042787328, 10202654714273202176

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14, -44).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((7+r5)^n+(7-r5)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 26 2008

I:=[1,7]; [n le 2 select I[n] else 14*Self(n-1)-44*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 04 2018

With[{srt5=Sqrt[5]},Simplify/@Table[((7+srt5)^n+(7-srt5)^n)/2,{n,0,20}]] (* or *) LinearRecurrence[{14,-44},{1,7},20] (* Harvey P. Dale, Jan 16 2012 *)
CoefficientList[Series[(1 - 7 x) / (1 - 14 x + 44 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 04 2018 *)

From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 14*a(n-1) - 44*a(n-2), n > 1; a(0)=1, a(1)=7.
G.f.: (1-7*x)/(1-14*x+44*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2*k)*5^(n-k))/7^n. (End)

Extended beyond a(6) by Klaus Brockhaus, Nov 26 2008

A152109 a(n) = ((8+sqrt(5))^n + (8-sqrt(5))^n)/2.

Original entry on oeis.org

1, 8, 69, 632, 6041, 59368, 593469, 5992792, 60870001, 620345288, 6334194549, 64746740792, 662230374281, 6775628281768, 69338460425709, 709653298187032, 7263483605875681, 74346193100976008, 760993556868950949

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008

nonn

Index entries for linear recurrences with constant coefficients, signature (16, -59).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((8+r5)^n+(8-r5)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 26 2008

LinearRecurrence[{16,-59},{1,8},30] (* Harvey P. Dale, Dec 18 2011 *)

From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 16*a(n-1) - 59*a(n-2), n > 1; a(0)=1, a(1)=8.
G.f.: (1-8*x)/(1-16*x+59*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*8^(2*k)*5^(n-k))/8^n. (End)

Extended beyond a(6) by Klaus Brockhaus, Nov 26 2008

A152429 a(n) = (11^n + 5^n)/2.

Original entry on oeis.org

1, 8, 73, 728, 7633, 82088, 893593, 9782648, 107374753, 1179950408, 12973595113, 142680249368, 1569336258673, 17261966423528, 189877968549433, 2088639343496888, 22974941225731393, 252723895719373448

Offset: 0

Philippe Deléham, Dec 03 2008

Binomial transform of A081343.

Inverse binomial transform of A143079.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (16,-55).

Cf. A162516.

List([0..20], n-> (11^n+5^n)/2); # G. C. Greubel, Jan 08 2020

[(11^n+5^n)/2: n in [0..20]]; // Vincenzo Librandi, Jun 01 2011

seq( (11^n+5^n)/2, n=0..20); # G. C. Greubel, Jan 08 2020

LinearRecurrence[{16,-55}, {1,8}, 20] (* G. C. Greubel, Jan 08 2020 *)

vector(21, n, (11^(n-1) + 5^(n-1))/2 ) \\ G. C. Greubel, Jan 08 2020

[(11^n+5^n)/2 for n in (0..20)] # G. C. Greubel, Jan 08 2020

a(n) = 16*a(n-1) - 55*a(n-2), with a(0)=1, a(1)=8.
G.f.: (1-8*x)/(1 - 16*x + 55*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*8^(2k-n)*9^(n-k).
a(n) = ((8 + sqrt(9))^n + (8 - sqrt(9))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Dec 08 2008
E.g.f.: (exp(11*x) + exp(5*x))/2. - G. C. Greubel, Jan 08 2020

A147688 a(n) = ((6 + sqrt(8))^n + (6 - sqrt(8))^n)/2.

Original entry on oeis.org

1, 6, 44, 360, 3088, 26976, 237248, 2091648, 18456832, 162915840, 1438198784, 12696741888, 112091336704, 989587267584, 8736489783296, 77129433907200, 680931492954112, 6011553766047744, 53072563389857792, 468547255228956672

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 10 2008

Index entries for linear recurrences with constant coefficients, signature (12, -28).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((6+r8)^n+(6-r8)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 13 2008

LinearRecurrence[{12,-28},{1,6},30]  (* Harvey P. Dale, Apr 23 2011 *)

From Philippe Deléham, Nov 13 2008: (Start)
a(n) = 12*a(n-1) - 28*a(n-2), a(0)=1, a(1)=6.
G.f.: (1-6x)/(1-12x+28x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2k)*8^(n-k))/6^n. (End)
a(n) = 2^n*A083878(n). - R. J. Mathar, Feb 04 2021

Extended beyond a(6) by Klaus Brockhaus, Nov 13 2008

A152105 a(n) = (10^n + 6^n)/2.

Original entry on oeis.org

1, 8, 68, 608, 5648, 53888, 523328, 5139968, 50839808, 505038848, 5030233088, 50181398528, 501088391168, 5006530347008, 50039182082048, 500235092492288, 5001410554953728, 50008463329722368, 500050779978334208, 5000304679870005248, 50001828079220031488, 500010968475320188928

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008

Index entries for linear recurrences with constant coefficients, signature (16,-60).

Cf. A081186, A098158.

LinearRecurrence[{16,-60},{1,8},30] (* Harvey P. Dale, Jan 27 2015 *)

a(n) = (10^n + 6^n)/2. - Klaus Brockhaus, Nov 26 2008
From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 16*a(n-1) - 60*a(n-2), n > 1; a(0)=1, a(1)=8.
G.f.: (1-8*x)/((1-6*x)*(1-10*x)).
a(n) = (Sum_{k=0..n} A098158(n,k)*2^(4*k))/2^n. (End)
a(n) = 2^n*A081186(n). - R. J. Mathar, Feb 04 2021
E.g.f.: exp(8*x)*cosh(2*x). - Elmo R. Oliveira, Aug 23 2024

Extended beyond a(6) by Klaus Brockhaus, Nov 26 2008

a(19)-a(21) from Elmo R. Oliveira, Aug 23 2024

cp's OEIS Frontend

A152056 a(n) = ((9+sqrt(3))^n + (9-sqrt(3))^n)/2.

Views

Author

Keywords

Links

Programs

Magma

Formula

Extensions

A152107 a(n) = ((6+sqrt(5))^n+(6-sqrt(5))^n)/2.

Views

Author

Keywords

Examples

Links

Programs

Magma

Maple

Mathematica

Formula

Extensions

A152108 a(n) = ((7+sqrt(5))^n + (7-sqrt(5))^n)/2.

Views

Author

Keywords

Links

Programs

Magma

Magma

Mathematica

Formula

Extensions

A152109 a(n) = ((8+sqrt(5))^n + (8-sqrt(5))^n)/2.

Views

Author

Keywords

Links

Programs

Magma

Mathematica

Formula

Extensions

A152429 a(n) = (11^n + 5^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A147688 a(n) = ((6 + sqrt(8))^n + (6 - sqrt(8))^n)/2.

Views

Author

Keywords

Links

Programs

Magma

Mathematica

Formula

Extensions

A152105 a(n) = (10^n + 6^n)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions