A080873 -id:A080873 - cp's OEIS frontend

A080871 a(n)a(n+3) - a(n+1)a(n+2) = 3, given a(0)=a(1)=1, a(2)=4.

1, 1, 4, 7, 31, 55, 244, 433, 1921, 3409, 15124, 26839, 119071, 211303, 937444, 1663585, 7380481, 13097377, 58106404, 103115431, 457470751, 811826071, 3601659604, 6391493137, 28355806081, 50320119025, 223244789044

Offset: 0

Paul D. Hanna, Feb 22 2003

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0, 8, 0, -1).

Cf. A001519, A079496, A080872, A080873, A080874, A080875.

Bisections are A001091 and A070997.

RecurrenceTable[{a[0]==a[1]==1,a[2]==4,a[n]==(3+a[n+1]a[n+2])/a[n+3]},a,{n,30}] (* Harvey P. Dale, Jun 08 2017 *)

a(n) = (3 + a(n-1)*a(n-2))/a(n-3) for n>2.
G.f.: (-x^3 - 4*x^2 + x + 1)/(x^4 - 8*x^2 + 1)
a(n+4) = 8*a(n+2)-a(n). [Richard Choulet, Dec 04 2008]
a(n) = (0.25 + sqrt(10)/20)*(sqrt(4 + sqrt(15)))^n + (0.25 + sqrt(10)/20)*(sqrt(4 - sqrt(15)))^n + ( - 1/20*10^(1/2) + 1/4)*( - sqrt(4 + sqrt(15)))^n + ( - 1/20*10^(1/2) + 1/4)*( - (sqrt(4 - sqrt(15))))^n. [Richard Choulet, Dec 06 2008]

A080872 a(n)a(n+3) - a(n+1)a(n+2) = 4, given a(0)=a(1)=1, a(2)=5.

Original entry on oeis.org

1, 1, 5, 9, 49, 89, 485, 881, 4801, 8721, 47525, 86329, 470449, 854569, 4656965, 8459361, 46099201, 83739041, 456335045, 828931049, 4517251249, 8205571449, 44716177445, 81226783441, 442644523201, 804062262961, 4381729054565, 7959395846169, 43374646022449, 78789896198729, 429364731169925

Offset: 0

Paul D. Hanna, Feb 22 2003

Seiichi Manyama, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0, 10, 0, -1).

Cf. A080871, A080873, A080874, A080875.

Bisections are A001079 and A072256.

CoefficientList[Series[(-x^3-5 x^2+x+1)/(x^4-10 x^2+1),{x,0,30}],x] (* or *) LinearRecurrence[{0,10,0,-1},{1,1,5,9},30] (* Harvey P. Dale, May 06 2012 *)

Vec( (-x^3 - 5*x^2 + x + 1)/(x^4 - 10*x^2 + 1) + O(x^66) ) \\ Joerg Arndt, Jan 29 2016

G.f.: (-x^3 - 5*x^2 + x + 1)/(x^4 - 10*x^2 + 1).
a(n) = (3+sqrt(3))/12*(sqrt(3)-sqrt(2))^n+(3-sqrt(3))/12*(-sqrt(3)+sqrt(2))^n+(3+sqrt(3))/12*(sqrt(3)+sqrt(2))^n+(3-sqrt(3))/12*(-sqrt(3)-sqrt(2))^n. [Richard Choulet, Dec 03 2008]
a(n+4) = 10*a(n+2)-a(n). [Richard Choulet, Dec 04 2008]

A080874 a(n)a(n+3) - a(n+1)a(n+2) = 5, given a(0)=a(1)=1, a(2)=3.

Original entry on oeis.org

1, 1, 3, 8, 29, 79, 287, 782, 2841, 7741, 28123, 76628, 278389, 758539, 2755767, 7508762, 27279281, 74329081, 270037043, 735782048, 2673091149, 7283491399, 26460874447, 72099131942, 261935653321, 713707828021, 2592895658763

Offset: 0

Paul D. Hanna, Feb 22 2003

nonn

Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A080871, A080872, A080873, A080875.

LinearRecurrence[{0,10,0,-1},{1,1,3,8},30] (* Harvey P. Dale, Sep 18 2016 *)

G.f.: (1+x-7x^2-2x^3)/(1-10x^2+x^4). a(n)=10a(n-2)-a(n-4). - Michael Somos, Mar 04 2003.

a(n) = ( - 1/24*3^(1/2)*2^(1/2) + 1/4 - 1/16*2^(1/2) + 1/8*3^(1/2))*(sqrt(3) + sqrt(2))^n + (1/24*3^(1/2)*2^(1/2) + 1/4 + 1/16*2^(1/2) + 1/8*3^(1/2))*(sqrt(3) - sqrt(2))^n + ( - 1/24*3^(1/2)*2^(1/2) - 1/8*3^(1/2) + 1/16*2^(1/2) + 1/4)*( - sqrt(3) - sqrt(2))^n + ( - 1/16*2^(1/2) + 1/4 + 1/24*3^(1/2)*2^(1/2) - 1/8*3^(1/2))*( - sqrt(3) + sqrt(2))^n [From Richard Choulet, Dec 04 2008]

A080875 a(n)a(n+3) - a(n+1)a(n+2) = 5, given a(0)=a(1)=1, a(2)=6.

Original entry on oeis.org

1, 1, 6, 11, 71, 131, 846, 1561, 10081, 18601, 120126, 221651, 1431431, 2641211, 17057046, 31472881, 203253121, 375033361, 2421980406, 4468927451, 28860511751, 53252096051, 343904160606, 634556225161, 4097989415521

Offset: 0

Paul D. Hanna, Feb 22 2003

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1500
Index entries for linear recurrences with constant coefficients, signature (0, 12, 0, -1).

Cf. A080871, A080872, A080873, A080874.

Bisections are A023038 and A077417.

LinearRecurrence[{0,12,0,-1},{1,1,6,11},30] (* Harvey P. Dale, Jul 14 2024 *)

G.f.: (-x^3 - 6*x^2 + x + 1)/(x^4 - 12*x^2 + 1).
a(n+4) = 12*a(n+2)-a(n). [Richard Choulet, Dec 04 2008]
a(n) = (1/4 + ((sqrt(6 + sqrt(35)) - sqrt(6 - sqrt(35)))/(4*sqrt(35))))*(sqrt(6 + sqrt(35)))^n + (1/4 + ((sqrt(6 + sqrt(35)) - sqrt(6 - sqrt(35)))/(4*sqrt(35))))*(sqrt(6 - sqrt(35)))^n + (1/4 - ((sqrt(6 + sqrt(35)) - sqrt(6 - sqrt(35)))/(4*sqrt(35))))*( - sqrt(6 + sqrt(35)))^n + (1/4 - ((sqrt(6 + sqrt(35)) - sqrt(6 - sqrt(35)))/(4*sqrt(35))))*( - (sqrt(6 - sqrt(35))))^n. [Richard Choulet, Dec 06 2008]

cp's OEIS Frontend

A080871 a(n)a(n+3) - a(n+1)a(n+2) = 3, given a(0)=a(1)=1, a(2)=4.

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A080872 a(n)a(n+3) - a(n+1)a(n+2) = 4, given a(0)=a(1)=1, a(2)=5.

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A080874 a(n)a(n+3) - a(n+1)a(n+2) = 5, given a(0)=a(1)=1, a(2)=3.

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A080875 a(n)a(n+3) - a(n+1)a(n+2) = 5, given a(0)=a(1)=1, a(2)=6.

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cp's OEIS Frontend

A080871 a(n)*a(n+3) - a(n+1)*a(n+2) = 3, given a(0)=a(1)=1, a(2)=4.

Views

Author

Keywords

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Programs

Mathematica

Formula

A080872 a(n)*a(n+3) - a(n+1)*a(n+2) = 4, given a(0)=a(1)=1, a(2)=5.

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Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula

A080874 a(n)*a(n+3) - a(n+1)*a(n+2) = 5, given a(0)=a(1)=1, a(2)=3.

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A080875 a(n)*a(n+3) - a(n+1)*a(n+2) = 5, given a(0)=a(1)=1, a(2)=6.

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A080871 a(n)a(n+3) - a(n+1)a(n+2) = 3, given a(0)=a(1)=1, a(2)=4.

A080872 a(n)a(n+3) - a(n+1)a(n+2) = 4, given a(0)=a(1)=1, a(2)=5.

A080874 a(n)a(n+3) - a(n+1)a(n+2) = 5, given a(0)=a(1)=1, a(2)=3.

A080875 a(n)a(n+3) - a(n+1)a(n+2) = 5, given a(0)=a(1)=1, a(2)=6.