A094025 -id:A094025 - cp's OEIS frontend

A080610 Partial sums of Jacobsthal gap sequence.

0, 1, 4, 5, 20, 21, 84, 85, 340, 341, 1364, 1365, 5460, 5461, 21844, 21845, 87380, 87381, 349524, 349525, 1398100, 1398101, 5592404, 5592405, 22369620, 22369621, 89478484, 89478485, 357913940, 357913941, 1431655764, 1431655765, 5726623060

Offset: 0

Paul Barry, Feb 26 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Cf. A075427, A080924, A094025.

[2^n+(-2)^n/3-(-1)^n/2-5/6: n in [0..30]]; // Vincenzo Librandi, Aug 05 2013

CoefficientList[Series[x (1 + 4 x) / ((1 - x^2) (1 - 4 x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{0,5,0,-4},{0,1,4,5},40] (* Harvey P. Dale, Nov 11 2021 *)

a(2n-1) = A001045(2n) = A002450(n); a(2n) = A001045(2n) - 1 = A002450(n) - 1.
G.f.: x*(1+4*x)/((1-x^2)*(1-4x^2)). - Ralf Stephan, Sep 16 2003
a(n) = 2^n+(-2)^n/3-(-1)^n/2-5/6. - Paul Barry, Apr 22 2004
a(n) = a(n-1)*4 if n even; a(n) = a(n-1)+1 if n odd. - 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

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A080610 Partial sums of Jacobsthal gap sequence.

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

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Author

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Crossrefs

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Magma

Mathematica

Formula

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

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Author

Keywords

Comments

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Crossrefs

Formula