A080610 -id:A080610 - cp's OEIS frontend

A094025 Expansion of (1+3x)/((1-x^2)(1-3x^2)).

1, 3, 4, 12, 13, 39, 40, 120, 121, 363, 364, 1092, 1093, 3279, 3280, 9840, 9841, 29523, 29524, 88572, 88573, 265719, 265720, 797160, 797161, 2391483, 2391484, 7174452, 7174453, 21523359, 21523360, 64570080, 64570081, 193710243, 193710244

Offset: 0

Paul Barry, Apr 22 2004

Add 1, triple, add 1, triple, ... (of course this is simply a restatement of one of Philippe Deléham's formulas). - Jon Perry, Aug 11 2014

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

Cf. A075427, A080610, A083416.

a(n)=4a(n-2)-3a(n-4); a(n)=3*3^(n/2)(1/4+sqrt(3)/4+(1/4-sqrt(3)/4)(-1)^n)+(-1)^n/2-1.
a(n) = a(n-1)*3 if n odd; a(n) = a(n-1)+1 if n even. - Philippe Deléham, Apr 22 2013
a(2n) = A003462(n+1); a(2n+1) = A123109(n+1) = A029858(n+1). - Philippe Deléham, Apr 22 2013

A094026 Expansion of x(1+10x)/((1-x^2)(1-10x^2)).

Original entry on oeis.org

0, 1, 10, 11, 110, 111, 1110, 1111, 11110, 11111, 111110, 111111, 1111110, 1111111, 11111110, 11111111, 111111110, 111111111, 1111111110, 1111111111, 11111111110, 11111111111, 111111111110, 111111111111, 1111111111110

Offset: 0

Paul Barry, Apr 22 2004

The expansion of x(1+kx)/((1-x^2)(1-kx^2)) has a(n)=k^((n+1)/2)/(2(sqrt(k)-1))-(-sqrt(k))^(n+1)/(2(sqrt(k)+1))-(-1)^n/2-(k+1)/(2(k-1)).

First 4 positive members are the divisors of 6 (the first perfect number), written in base 2 (see A135652, A135653, A135654, A135655). - Omar E. Pol, May 04 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (0,11,0,-10).

Cf. A075427, A094025, A080610.

I:=[0,1,10,11]; [n le 4 select I[n] else 11*Self(n-2)-10*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 25 2019

LinearRecurrence[{0, 11, 0, -10}, {0, 1, 10, 11}, 30] (* Vincenzo Librandi, Apr 25 2019 *)
CoefficientList[Series[x (1+10x)/((1-x^2)(1-10x^2)),{x,0,30}],x] (* Harvey P. Dale, Jul 07 2024 *)

a(n) = 10^(n/2)(5/9+sqrt(10)/18+(5/9-sqrt(10)/18)(-1)^n)-(-1)^n/2-11/18.

A094027 Expansion of x(1+100x)/((1-x^2)(1-100x^2)).

Original entry on oeis.org

0, 1, 100, 101, 10100, 10101, 1010100, 1010101, 101010100, 101010101, 10101010100, 10101010101, 1010101010100, 1010101010101, 101010101010100, 101010101010101, 10101010101010100, 10101010101010101

Offset: 0

Paul Barry, Apr 22 2004

The expansion of x(1+kx)/((1-x^2)(1-kx^2)) has a(n)=k^((n+1)/2)/(2(sqrt(k)-1))-(-sqrt(k))^(n+1)/(2(sqrt(k)+1))-(-1)^n/2-(k+1)/(2(k-1))

Index entries for linear recurrences with constant coefficients, signature (0,101,0,-100).

Cf. A075427, A094025, A080610 (interpreted as binary), A094026.

a(n)=2^n*5^(n+1)((-1)^n/11+1/9)-(-1)^n/2-101/198

cp's OEIS Frontend

A094025 Expansion of (1+3x)/((1-x^2)(1-3x^2)).

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A094026 Expansion of x(1+10x)/((1-x^2)(1-10x^2)).

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A094027 Expansion of x(1+100x)/((1-x^2)(1-100x^2)).

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