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.

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A203570 Bisection of A201207 (half-convolution of the Lucas sequence A000032 with itself); even part.

Original entry on oeis.org

4, 7, 27, 84, 270, 826, 2488, 7353, 21461, 61960, 177344, 503892, 1422892, 3996619, 11173935, 31114236, 86328978, 238764238, 658478176, 1811322045, 4970928809, 13613135152, 37208048132, 101518052904, 276527670100, 752102592271
Offset: 0

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Author

Wolfdieter Lang, Jan 03 2012

Keywords

Comments

The odd part of the bisection of A201207 is given in A203574.
See a comment on A201204 for the definition of the half-convolution of a sequence with itself, and the rule for the o.g.f.s of the bisection. Here the o.g.f. is (Lconve(x) + L2(x))/2, with the o.g.f. Lconve(x) = (4-11*x+11*x^2+x^3)/
(1-3*x+x^2)^2 of A203573 and the o.g.f. L2(x)= (4-7*x-x^2)/ ((1+x)*(1-3*x+x^2)) of A001254. This leads to the o.g.f. given in the formula section.

Crossrefs

Cf. A201207, A203574.

Formula

a(n) = A201207(2*n), n>=0.
a(n) = (2*(4*n+6)*F(2*n+1)-4*(n+1)*F(2*n))/4 + (-1)^n, with the Fibonacci numbers F(n)=A000045(n).
O.g.f.: (4-13*x+4*x^3+12*x^2)/((1-3*x+x^2)^2*(1+x)). See a comment above.

A203574 Bisection of A099924 (Lucas convolution); one half of the terms with odd arguments.

Original entry on oeis.org

2, 11, 41, 137, 435, 1338, 4024, 11899, 34723, 100255, 286947, 815316, 2302286, 6466667, 18079805, 50343893, 139683219, 386328654, 1065440068, 2930780635, 8043131767, 22026515371, 60203886531, 164259660072, 447431169050, 1216927557323
Offset: 0

Views

Author

Wolfdieter Lang, Jan 03 2012

Keywords

Comments

The even part of this bisection of A099924 is found in A203573.
This is also the odd part of the bisection of A201207 (half-convolution of the Lucas sequence with itself). See a comment on A201204 for the definition of half-convolution of a sequence with itself. There the rule for the o.g.f. is given.

Links

Crossrefs

Cf. A000032, A000045, A099924, A203573, A201207.

Programs

  • Magma
    I:=[2,11,41,137]; [n le 4 select I[n] else 6*Self(n-1) - 11*Self(n-2) + 6*Self(n-3) - Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 22 2017
  • Mathematica
    CoefficientList[Series[(2-x-3x^2)/(1-3x+x^2)^2,{x,0,30}],x] (* or *) LinearRecurrence[{6,-11,6,-1},{2,11,41,137},30] (* Harvey P. Dale, Oct 12 2015 *)
  • PARI
    x='x+O('x^30); Vec((2-x-3x^2)/(1-3x+x^2)^2) \\ G. C. Greubel, Dec 22 2017

Formula

a(n) = A099924(2*n+1)/2, n>=0.
O.g.f.: (2-x-3*x^2)/(1-3*x+x^2)^2.
a(n) = (3+2*n)*F(2*n) + (2+n)*F(2*n+1), with the Fibonacci numbers F(n)=A000045(n). From the partial fraction decomposition of the o.g.f. and the Fibonacci recurrence.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4); a(0)=2, a(1)=11, a(2)=41, a(3)=137. - Harvey P. Dale, Oct 12 2015
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