cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A166754 a(n) = 4*A061547(n+1) - 3*A166753(n).

Original entry on oeis.org

1, 2, 9, 22, 53, 114, 241, 494, 1005, 2026, 4073, 8166, 16357, 32738, 65505, 131038, 262109, 524250, 1048537, 2097110, 4194261, 8388562, 16777169, 33554382, 67108813, 134217674, 268435401, 536870854, 1073741765, 2147483586
Offset: 0

Views

Author

Paul Barry, Oct 21 2009

Keywords

Links

Crossrefs

Cf. A061547, A166753.
Cf. A000975, A011377.

Programs

  • GAP
    List([0..40], n-> (2^(n+3) + (-1)^n - (4*n+7))/2) # G. C. Greubel, Jun 04 2019
  • Magma
    [(2^(n+3) +(-1)^n -(4*n+7))/2: n in [0..40]]; // G. C. Greubel, Oct 10 2017
  • Mathematica
    LinearRecurrence[{3,-1,-3,2}, {1,2,9,22}, 40] (* G. C. Greubel, May 24 2016 *)
  • PARI
    my(x='x+O('x^40)); Vec((1-x+4*x^2)/((1+x)*(1-x)^2*(1-2*x))) \\ G. C. Greubel, Oct 10 2017
  • Sage
    [(2^(n+3) + (-1)^n - (4*n+7))/2 for n in (0..40)] # G. C. Greubel, Jun 04 2019

Formula

G.f.: (1-x+4*x^2)/((1+x)*(1-x)^2*(1-2*x)).
a(n) = (2^(n+3) + (-1)^n - (4*n+7))/2.
a(n) = A000975(n) - 4*A011377(n-2).
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).
E.g.f.: (8*exp(2*x) + exp(-x) - (4*x+7)*exp(x))/2. - G. C. Greubel, Jun 04 2019

A166752 Interleave A007583 and A000012.

Original entry on oeis.org

1, 1, 3, 1, 11, 1, 43, 1, 171, 1, 683, 1, 2731, 1, 10923, 1, 43691, 1, 174763, 1, 699051, 1, 2796203, 1, 11184811, 1, 44739243, 1, 178956971, 1, 715827883, 1, 2863311531, 1, 11453246123, 1, 45812984491, 1, 183251937963, 1, 733007751851, 1
Offset: 0

Views

Author

Paul Barry, Oct 21 2009

Keywords

Comments

Partial sums are A166753.

Links

Programs

  • Magma
    [(4*4^Floor(n/2)-1)/3 - 2*Floor(2^n/3): n in [0..25]]; // G. C. Greubel, Oct 10 2017
  • Mathematica
    LinearRecurrence[{0, 5, 0, -4}, {1, 1, 3, 1}, 100] (* G. C. Greubel, May 24 2016 *)
  • PARI
    x='x+O('x^50); Vec((1+x-2*x^2-4*x^3)/(1-5*x^2+4*x^4)) \\ G. C. Greubel, Oct 10 2017

Formula

G.f.: (1+x-2*x^2-4*x^3)/(1-5*x^2+4*x^4).
G.f.: (1+x)/(1-5*x^2+4*x^4) - 2*x^2*(1+2*x)/(1-5*x^2+4*x^4).
a(n) = (4*4^floor(n/2)-1)/3 - 2*floor(2^n/3).
a(n) = 4*4^floor(n/2)/3 - 2*2^n/3 - (-1)^n/3 + 2/3.
a(n) = A002450(floor(n/2)+1) - 2*A000975(n-1).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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