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.

A276228 a(n+3) = -a(n+2) - 2*a(n+1) + a(n) with a(0)=3, a(1)=-1, a(2)=-3.

Original entry on oeis.org

3, -1, -3, 8, -3, -16, 30, -1, -75, 107, 42, -331, 354, 350, -1389, 1043, 2085, -5560, 2433, 10772, -21198, 2087, 51081, -76453, -23622, 227609, -256818, -222022, 963267, -776041, -1372515, 3887864, -1918875, -7229368, 14954982, -2415121, -34724211, 54509435, 12523866
Offset: 0

Views

Author

G. C. Greubel, Aug 24 2016

Keywords

Links

Crossrefs

Cf. A276225, A276229.

Programs

  • Magma
    I:=[3,-1,-3]; [n le 3 select I[n] else -Self(n-1)- 2*Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 25 2016
  • Mathematica
    LinearRecurrence[{-1, -2, 1}, {3, -1, -3}, 50]
CoefficientList[Series[(3 + 2 x + 2 x^2)/(1 + x + 2 x^2 - x^3), {x, 0, 38}], x] (* Michael De Vlieger, Aug 25 2016 *)
nxt[{a_,b_,c_}]:={b,c,a-2b-c}; NestList[nxt,{3,-1,-3},40][[All,1]] (* Harvey P. Dale, Dec 19 2022 *)
  • PARI
    Vec((3+2*x+2*x^2)/(1+x+2*x^2-x^3) + O(x^99)) \\ Altug Alkan, Aug 25 2016

Formula

G.f.: (3 + 2*x + 2*x^2)/(1 + x + 2*x^2 - x^3).
Let (b, c, d) be the three roots of x^3 = 2*x^2 + x + 1, then a(n) = b^(-n) + c^(-n) + d^(-n) = A276225(-n).
a(2*n) = -3*a(2*n-2) - 6*a(2*n-4) + a(2*n-6).
a(n) = 2*A276229(n) + 2*A276229(n+1) + 3*A276229(n+2).

Extensions

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A276226 a(n+3) = 2*a(n+2) + a(n+1) + a(n) with a(0)=0, a(1)=6, a(2)=8.

Original entry on oeis.org

0, 6, 8, 22, 58, 146, 372, 948, 2414, 6148, 15658, 39878, 101562, 258660, 658760, 1677742, 4272904, 10882310, 27715266, 70585746, 179769068, 457839148, 1166033110, 2969674436, 7563221130, 19262149806, 49057195178, 124939761292, 318198867568, 810394691606, 2063928012072, 5256449583318, 13387221870314, 34094821336018
Offset: 0

Views

Author

G. C. Greubel, Aug 24 2016

Keywords

Links

Crossrefs

Cf. A276225, A077939.

Programs

  • Magma
    I:=[0,6,8]; [n le 3 select I[n] else 2*Self(n-1)+ Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 25 2016
  • Mathematica
    LinearRecurrence[{2, 1, 1}, {0, 6, 8}, 50]
CoefficientList[Series[2 (3 x - 2 x^2)/(1 - 2 x - x^2 - x^3), {x, 0, 33}], x] (* Michael De Vlieger, Aug 25 2016 *)
  • PARI
    concat(0, Vec(2*(3*x-2*x^2)/(1-2*x-x^2-x^3) + O(x^99))) \\ Altug Alkan, Aug 25 2016

Formula

Let p = (4*(61 + 9*sqrt(29)))^(1/3), q = (4*(61 - 9*sqrt(29)))^(1/3), and x = (1/6)*(4 + p + q) then x^n = (1/6)*(2*A276225(n) + a(n)*(p + q) + A077939(n-1)*(p^2 + q^2)).G.f.: 2*(3*x - 2*x^2)/(1 - 2*x - x^2 - x^3).

A287327 Number of independent vertex sets (and vertex covers) in the 2n-crossed prism graph.

Original entry on oeis.org

2, 7, 35, 196, 1127, 6517, 37730, 218491, 1265327, 7327852, 42437675, 245768761, 1423317602, 8242841887, 47736669995, 276456796756, 1601040887327, 9272088633997, 53697334226690, 310976719148851, 1800955694455127, 10429852827143932, 60402279928821635
Offset: 0

Views

Author

Andrew Howroyd, Aug 31 2017

Keywords

Comments

Sequence extrapolated to n = 0 using recurrence.

Links

Crossrefs

Cf. A276225 (maximal independent vertex sets), A287062, A290708.

Programs

  • Mathematica
    CoefficientList[Series[(2 - 7 x)/(1 - 7 x + 7 x^2), {x, 0, 22}], x] (* Michael De Vlieger, Aug 31 2017 *)
Table[(1/2 (7 - Sqrt[21]))^n + (1/2 (7 + Sqrt[21]))^n, {n, 0, 20}] // Expand (* Eric W. Weisstein, Sep 21 2017 *)
LinearRecurrence[{7, -7}, {7, 35}, {0, 20}] (* Eric W. Weisstein, Sep 21 2017 *)
  • PARI
    Vec((2 - 7*x)/(1 - 7*x + 7*x^2) + O(x^30))

Formula

a(n) = 7*a(n-1) - 7*a(n-2) for n > 1.
G.f.: (2 - 7*x)/(1 - 7*x + 7*x^2).
Showing 1-3 of 3 results.

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

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