A098158 -id:A098158 - cp's OEIS frontend

A152106 a(n) = (11^n + 7^n)/2.

1, 9, 85, 837, 8521, 88929, 944605, 10155357, 110061841, 1199150649, 13109949925, 143644498677, 1576134831961, 17309800577169, 190214028328045, 2090997865462797, 22991481397070881, 252839829506640489, 2780772863545070965, 30585244671799959717, 336414893599428810601

Offset: 0

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

			a(3) = (11^3 + 7^3)/2 = 837.

Harvey P. Dale, Table of n, a(n) for n = 0..960
Index entries for linear recurrences with constant coefficients, signature (18,-77).

Cf. A098158.

LinearRecurrence[{18,-77},{1,9},30] (* Harvey P. Dale, Jan 26 2025 *)

a(n)=(11^n+7^n)/2 \\ Charles R Greathouse IV, Aug 23 2024

a(n) = ((9 + sqrt(4))^n + (9 - sqrt(4))^n)/2.
From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 18*a(n-1) - 77*a(n-2), n > 1; a(0)=1, a(1)=9.
G.f.: (1-9*x)/(1-18*x+77*x^2).
a(n) = (1/9^n)*Sum_{k=0..n} A098158(n,k)*9^(2*k)*4^(n-k). (End)
E.g.f.: exp(9*x)*cosh(2*x). - Elmo R. Oliveira, Aug 23 2024

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

More terms by Elmo R. Oliveira, Aug 23 2024

A152261 a(n) = ((9 + sqrt(5))^n + (9 - sqrt(5))^n)/2.

Original entry on oeis.org

1, 9, 86, 864, 9016, 96624, 1054016, 11628864, 129214336, 1442064384, 16136869376, 180866755584, 2029199527936, 22779718078464, 255815761289216, 2873425129242624, 32279654468386816, 362653470608523264

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

nonn

Binomial transform of A152109. - Philippe Deléham, Dec 03 2008

G. C. Greubel, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (18,-76).

Cf. A152109.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((9+r5)^n+(9-r5)^n)/2: n in [0..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

[n le 2 select 9^(n-1) else 18*Self(n-1) -76*Self(n-2): n in [1..30]]; // G. C. Greubel, May 23 2023

LinearRecurrence[{18,-76}, {1,9}, 41] (* G. C. Greubel, May 23 2023 *)

@CachedFunction
def a(n): # a = A152261
    if (n<2): return 9^n
    else: return 18*a(n-1) -76*a(n-2)
[a(n) for n in range(41)] # G. C. Greubel, May 23 2023

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 18*a(n-1) - 76*a(n-2), n > 1; a(0)=1, a(1)=9.
G.f.: (1-9*x)/(1-18*x+76*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*9^(2k-n)*5^(n-k). (End)
a(n) = m^n*(ChebyshevU(n, 9/m) - (9/m)*ChebyshevU(n-1, 9/m)), where m = 2*sqrt(19). - G. C. Greubel, May 23 2023

Extended beyond a(6) by Klaus Brockhaus, Dec 03 2008

A152263 a(n) = ((8 + sqrt(6))^n + (8 - sqrt(6))^n)/2.

Original entry on oeis.org

1, 8, 70, 656, 6436, 64928, 665560, 6883136, 71527696, 745221248, 7774933600, 81176105216, 847871534656, 8857730451968, 92547138221440, 967005845328896, 10104359508418816, 105583413105625088, 1103281758201710080

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

nonn

Binomial transform of A152262. Inverse binomial transform of A152264. - Philippe Deléham, Dec 03 2008

Index entries for linear recurrences with constant coefficients, signature (16, -58).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((8+r6)^n+(8-r6)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

LinearRecurrence[{16,-58},{1,8},20] (* Harvey P. Dale, Jul 09 2021 *)

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 16*a(n-1) - 58*a(n-2), n > 1; a(0)=1, a(1)=8.
G.f.: (1-8*x)/(1-16*x+58*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*8^(2k-n)*6^(n-k). (End)

Extended beyond a(6) by Klaus Brockhaus, Dec 03 2008

A152264 a(n) = ((9+sqrt(6))^n + (9-sqrt(6))^n)/2.

Original entry on oeis.org

1, 9, 87, 891, 9513, 104409, 1165887, 13155291, 149353713, 1701720009, 19429431687, 222100769691, 2540606477913, 29073358875609, 332774973917487, 3809447614844091, 43611934023382113, 499306241307571209

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

nonn

Binomial transform of A152263. - Philippe Deléham, Dec 03 2008

Harvey P. Dale, Table of n, a(n) for n = 0..944
Index entries for linear recurrences with constant coefficients, signature (18, -75).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((9+r6)^n+(9-r6)^n)/2: n in [0..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

CoefficientList[Series[(1-9x)/(1-18x+75x^2),{x,0,20}],x] (* or *) LinearRecurrence[{18,-75},{1,9},20] (* Harvey P. Dale, Feb 07 2023 *)

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 18*a(n-1) - 75*a(n-2), n > 1; a(0)=1, a(1)=9.
G.f.: (1-9*x)/(1-18*x+75*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*9^(2k-n)*6^(n-k). (End)

Extended beyond a(6) by Klaus Brockhaus, Dec 03 2008

A152265 a(n) = ((8 + sqrt(7))^n + (8 - sqrt(7))^n)/2.

Original entry on oeis.org

1, 8, 71, 680, 6833, 70568, 739607, 7811336, 82823777, 879934280, 9357993191, 99571637096, 1059740581649, 11280265991912, 120079042716599, 1278289521926600, 13608126915979457, 144867527905855112

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

nonn

Binomial transform of A145302. Inverse binomial transform of A152266. - Philippe Deléham, Dec 03 2008

Index entries for linear recurrences with constant coefficients, signature (16, -57).

Cf. A056241, A098158, A145302, A152266.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((8+r7)^n+(8-r7)^n)/2: n in [0..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 16*a(n-1) - 57*a(n-2), n > 1; a(0)=1, a(1)=8.
G.f.: (1-8*x)/(1-16*x+57*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*8^(2k-n)*7^(n-k). (End)
a(n) = Sum_{k=1..n} A056241(n,k) * 7^(k-1). - J. Conrad, Nov 23 2022

Extended beyond a(6) by Klaus Brockhaus, Dec 03 2008

A152266 a(n) = ((9 + sqrt(7))^n + (9 - sqrt(7))^n)/2.

Original entry on oeis.org

1, 9, 88, 918, 10012, 112284, 1280224, 14735016, 170493712, 1978495632, 22996386688, 267526283616, 3113740490176, 36250383835584, 422090112767488, 4915093625981568, 57237016922874112, 666549376289097984

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

nonn

Binomial transform of A152265. - Philippe Deléham, Dec 03 2008

Index entries for linear recurrences with constant coefficients, signature (18, -74).

Cf. A098158, A152265.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((9+r7)^n+(9-r7)^n)/2: n in [0..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 18*a(n-1) - 74*a(n-2), n > 1; a(0)=1, a(1)=9.
G.f.: (1-9*x)/(1-18*x+74*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*9^(2k-n)*7^(n-k). (End)

Extended beyond a(6) by Klaus Brockhaus, Dec 03 2008

A152267 a(n) = ((9 + sqrt(8))^n + (9 - sqrt(8))^n)/2.

Original entry on oeis.org

1, 9, 89, 945, 10513, 120249, 1397033, 16368417, 192648097, 2272771305, 26846572409, 317325998097, 3752068179889, 44372429376921, 524802751652681, 6207262185233025, 73420118463548737, 868431992821866441

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008

nonn

Binomial transform of A145303. - Philippe Deléham, Dec 03 2008

Index entries for linear recurrences with constant coefficients, signature (18, -73).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-8); S:=[ ((9+r8)^n+(9-r8)^n)/2: n in [0..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008

LinearRecurrence[{18,-73},{1,9},30] (* Harvey P. Dale, May 14 2014 *)

From Philippe Deléham, Dec 03 2008: (Start)
a(n) = 18*a(n-1) - 73*a(n-2), n > 1; a(0)=1, a(1)=9.
G.f.: (1-9*x)/(1-18*x+73*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*9^(2k-n)*8^(n-k). (End)

Extended beyond a(6) by Klaus Brockhaus, Dec 03 2008

A143648 a(n) = ((4 + sqrt 6)^n + (4 - sqrt 6)^n)/2.

Original entry on oeis.org

1, 4, 22, 136, 868, 5584, 35992, 232096, 1496848, 9653824, 62262112, 401558656, 2589848128, 16703198464, 107727106432, 694784866816, 4481007870208, 28900214293504, 186391635645952, 1202130942232576, 7753131181401088

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Oct 27 2008

Binomial transform of A084120. Sequences defined by a(n) = ((A + sqrt(B))^n + (A - sqrt(B))^n)/2 have recurrences a(n) = 2*A*a(n-1) + (B - A^2)*a(n-2) and generating functions g.f.: (1-Ax)/(1-2Ax+(A^2-B)x^2). - R. J. Mathar, Nov 01 2008

			a(3) = 136.
a(4) = ((4 + sqrt(6))^4 + (4 - sqrt(6))^4)/2 = 4^4 + 6*sqrt(6)^2*4^2 + sqrt(6)^4 = 4^4 + 6*6*4^2 + 6^2 = 868. - _Klaus Brockhaus_, Nov 01 2008

Index entries for linear recurrences with constant coefficients, signature (8, -10).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((4+r6)^n+(4-r6)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 01 2008

Table[MatrixPower[{{4,2},{3,4}},n][[1]][[1]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)

From R. J. Mathar, Klaus Brockhaus and Philippe Deléham, Nov 01 2008: (Start)
a(n) = 8*a(n-1) - 10*a(n-2).
G.f.: (1-4x)/(1-8x+10x^2). (End)
a(n) = (Sum_{k=0..n} A098158(n,k)*4^(2*k)*6^(n-k))/4^n. - Philippe Deléham, Nov 06 2008

More terms from Klaus Brockhaus and R. J. Mathar, Nov 01 2008

A147962 a(n) = ((7+sqrt(3))^n + (7-sqrt(3))^n) / 2.

Original entry on oeis.org

1, 7, 52, 406, 3292, 27412, 232336, 1991752, 17197072, 149138416, 1296872512, 11295848032, 98485736896, 859191307072, 7498334401792, 65453881499776, 571430958514432, 4989154870212352, 43562344091309056, 380371693248558592, 3321335877279603712

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14, -46).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((7+r3)^n+(7-r3)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 19 2008

LinearRecurrence[{14,-46},{1,7},30] (* Harvey P. Dale, Aug 18 2012 *)

Vec((1-7*x) / (1-14*x+46*x^2) + O(x^30)) \\ Colin Barker, Sep 24 2017

From Zak Seidov, Nov 19 2008: (Start)
a(0) = 1; a(1) = 7; a(n) = 14 * a(n - 1) - 46 * a(n - 2);
a(n) = ((7+sqrt(3))^n + (7-sqrt(3))^n)/2. (End)
From Philippe Deléham, Nov 19 2008: (Start)
G.f.: (1-7*x)/(1-14*x+46*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2*k)*3^(n-k))/7^n. (End)

Extended beyond a(6) by Klaus Brockhaus and Zak Seidov, Nov 19 2008

A152055 a(n) = ((8 + sqrt(3))^n + (8 - sqrt(3))^n)/2.

Original entry on oeis.org

1, 8, 67, 584, 5257, 48488, 455131, 4324328, 41426257, 399036104, 3857575987, 37380013448, 362768079961, 3524108459048, 34256882467147, 333139503472424, 3240562225062817, 31527485889187208, 306765478498163491

Offset: 0

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

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16, -61).

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((8+r3)^n+(8-r3)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 26 2008

LinearRecurrence[{16,-61},{1,8},30] (* Harvey P. Dale, Sep 02 2018 *)

From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 16*a(n-1) - 61*a(n-2), n > 1; a(0)=1, a(1)=8.
G.f.: (1-8*x)/(1-16x+61*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*8^(2*k)*3^(n-k))/8^n. (End)

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

cp's OEIS Frontend

A152106 a(n) = (11^n + 7^n)/2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A152261 a(n) = ((9 + sqrt(5))^n + (9 - sqrt(5))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

SageMath

Formula

Extensions

A152263 a(n) = ((8 + sqrt(6))^n + (8 - sqrt(6))^n)/2.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

Formula

Extensions

A152264 a(n) = ((9+sqrt(6))^n + (9-sqrt(6))^n)/2.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica

Formula

Extensions

A152265 a(n) = ((8 + sqrt(7))^n + (8 - sqrt(7))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

Extensions

A152266 a(n) = ((9 + sqrt(7))^n + (9 - sqrt(7))^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

Extensions

A152267 a(n) = ((9 + sqrt(8))^n + (9 - sqrt(8))^n)/2.

Views

Author

Keywords

Comments

Links

Programs

Magma

Mathematica