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 1-3 of 3 results.

A098600 a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n.

Original entry on oeis.org

1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350, 15126, 24477, 39602, 64080, 103681, 167762, 271442, 439205, 710646, 1149852, 1860497, 3010350, 4870846, 7881197, 12752042, 20633240, 33385281, 54018522
Offset: 0

Views

Author

Paul Barry, Sep 17 2004

Keywords

Comments

Row sums of A098599.

Links

Crossrefs

Cf. A000032, A000035, A000045, A001350, A008346.
Cf. A020878, A098599, A099925, A181716, A182143.
First differences of A014217 and A062724.

Programs

  • Magma
    [Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n: n in [0..50]]; // Vincenzo Librandi, Aug 31 2014
  • Mathematica
    Table[-(-1)^n + LucasL[n], {n, 0, 39}] (* Alonso del Arte, Aug 30 2014 *)
Table[Fibonacci[n - 1] + Fibonacci[n + 1] - (-1)^n, {n, 0, 40}] (* Vincenzo Librandi, Aug 31 2014 *)
CoefficientList[ Series[-(1 + 2x)/(-1 + 2x^2 + x^3), {x, 0, 40}], x] (* or *)
LinearRecurrence[{0, 2, 1}, {1, 2, 2}, 40] (* Robert G. Wilson v, Mar 09 2018 *)
  • PARI
    a(n)=fibonacci(n-1) + fibonacci(n+1) - (-1)^n; \\ Joerg Arndt, Oct 18 2014
  • PARI
    Vec((1+2*x)/((1+x)*(1-x-x^2)) + O(x^30)) \\ Colin Barker, Jun 03 2016
  • SageMath
    [lucas_number2(n,1,-1) -(-1)^n for n in range(51)] # G. C. Greubel, Mar 26 2024

Formula

G.f.: (1+2*x) / ((1+x)*(1-x-x^2)).
a(n) = Sum_{k = 0..n} binomial(k, n-k) + binomial(k-1, n-k-1).
a(n) = A020878(n) - 1 = A001350(n) + 1.
a(n) = Lucas(n) - (-1)^n. - Paul Barry, Dec 01 2004
a(n) = A181716(n+1). - Richard R. Forberg, Aug 30 2014
a(n) = [x^n] ( (1 + x + sqrt(1 + 6*x + 5*x^2))/2 )^n. exp( Sum_{n >= 1} a(n)*x^n/n ) = Sum_{n >= 0} Fibonacci(n+2)*x^n. Cf. A182143. - Peter Bala, Jun 29 2015
From Colin Barker, Jun 03 2016: (Start)
a(n) = (-(-1)^n + ((1/2)*(1-sqrt(5)))^n + ((1/2)*(1+sqrt(5)))^n).
a(n) = 2*a(n-2) + a(n-3) for n > 2. (End)
E.g.f.: (2*exp(3*x/2)*cosh(sqrt(5)*x/2) - 1)*exp(-x). - Ilya Gutkovskiy, Jun 03 2016
a(n) = A014217(n) + A000035(n). - Paul Curtz, Jul 27 2023

A284663 Number of dominating sets in the Moebius ladder M_n.

Original entry on oeis.org

3, 15, 51, 179, 663, 2439, 8935, 32771, 120219, 440975, 1617531, 5933267, 21763823, 79831879, 292831311, 1074134531, 3940032883, 14452434639, 53012975555, 194456895859, 713287340551, 2616409296967, 9597250953527, 35203676264195, 129130605057163
Offset: 1

Views

Author

Eric W. Weisstein, Mar 31 2017

Keywords

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, May 10 2017

Links

Crossrefs

Cf. A182143, A284702, A218348 (ladder).

Programs

  • Mathematica
    LinearRecurrence[{3, 1, 5, 1, 1, -1, -1}, {3, 15, 51, 179, 663, 2439,
  8935}, 20] (* Eric W. Weisstein, May 17 2017 *)
Rest[CoefficientList[Series[x*(1 - x)*(1 + x)*(3*x^4 + 2*x^3 + 6*x^2 + 6*x + 3)/((x^2 + 1)*(x^5 + x^4 - 2*x^3 - 2*x^2 - 3*x + 1)), {x,0,50}], x]] (* G. C. Greubel, May 17 2017 *)
Table[RootSum[1 + # - 2 #^2 - 2 #^3 - 3 #^4 + #^5 &, #^n &] - 2 Cos[n Pi/2], {n, 20}] (* Eric W. Weisstein, Jun 14 2017 *)
  • PARI
    Vec((1-x)*(1+x)*(3*x^4+2*x^3+6*x^2+6*x+3)/((x^2+1)*(x^5+x^4-2*x^3-2*x^2-3*x+1))+O(x^50)) \\ Andrew Howroyd, May 10 2017

Formula

From Andrew Howroyd, May 10 2017 (Start)
a(n) = 3*a(n-1)+a(n-2)+5*a(n-3)+a(n-4)+a(n-5)-a(n-6)-a(n-7) for n>7.
G.f.: x*(1-x)*(1+x)*(3*x^4+2*x^3+6*x^2+6*x+3)/((x^2+1)*(x^5+x^4-2*x^3 -2*x^2-3*x+1)). (End)

Extensions

a(1)-(2) and a(16)-a(25) from Andrew Howroyd, May 10 2017

A286910 Number of independent vertex sets and vertex covers in the n-antiprism graph.

Original entry on oeis.org

3, 1, 5, 10, 21, 46, 98, 211, 453, 973, 2090, 4489, 9642, 20710, 44483, 95545, 205221, 440794, 946781, 2033590, 4367946, 9381907, 20151389, 43283149, 92967834, 199685521, 428904338, 921243214, 1978737411, 4250128177, 9128846213, 19607839978, 42115660581
Offset: 0

Views

Author

Andrew Howroyd, May 15 2017

Keywords

Comments

Sequence extrapolated to n=0 using recurrence.

Links

Crossrefs

Cf. A051927, A182143, A192742, A284700.

Programs

  • Magma
    I:=[3,1,5]; [n le 3 select I[n] else Self(n-1)+2*Self(n-2)+Self(n-3): n in [1..33]]; // Vincenzo Librandi, May 16 2017
  • Mathematica
    CoefficientList[Series[(- 2 x^2 - 2 x + 3) / (- x^3 - 2 x^2 - x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, May 16 2017 *)
LinearRecurrence[{1, 2, 1}, {3, 1, 5}, 40] (* Vincenzo Librandi, May 16 2017 *)
Table[RootSum[-1 - 2 # - #^2 + #^3 &, #^n &], {n, 20}] (* Eric W. Weisstein, Aug 16 2017 *)
RootSum[-1 - 2 # - #^2 + #^3 &, #^Range[20] &] (* Eric W. Weisstein, Aug 16 2017 *)
  • PARI
    Vec((-2*x^2 - 2*x + 3)/(-x^3 - 2*x^2 - x + 1)+O(x^30))

Formula

a(n) = a(n-1) + 2*a(n-2) + a(n-3) for n>=3.
G.f.: (2*x^2 + 2*x - 3)/(x^3 + 2*x^2 + x - 1).
a(n) = n*Sum_{k=1..n} C(2*k,n-k)/k, a(0)=3. - Vladimir Kruchinin, Jun 13 2020
Showing 1-3 of 3 results.

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