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.

A247560 a(n) = 3*a(n-1) - 4*a(n-2) with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, -1, -7, -17, -23, -1, 89, 271, 457, 287, -967, -4049, -8279, -8641, 7193, 56143, 139657, 194399, 24569, -703889, -2209943, -3814273, -2603047, 7447951, 32756041, 68476319, 74404793, -50690897, -449691863, -1146312001, -1640168551, -335257649, 5554901257
Offset: 0

Views

Author

Michael Somos, Sep 19 2014

Keywords

Examples

			G.f. = 1 + x - x^2 - 7*x^3 - 17*x^4 - 23*x^5 - x^6 + 89*x^7 + 271*x^8 + ...

Links

Crossrefs

Cf. A087168, A247487, A247564, A247565.

Programs

  • Haskell
    a247560 n = a247560_list !! n
a247560_list = 1 : 1 : zipWith (-) (map (* 3) $ tail a247560_list)
                                   (map (* 4) a247560_list)
-- Reinhard Zumkeller, Sep 20 2014
  • Magma
    I:=[1, 1]; [n le 2 select I[n] else 3*Self(n-1) - 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 04 2018
  • Maple
    A247560:=n->simplify((1/14*I)*sqrt(7)*((3/2+(1/2*I)*sqrt(7))^n-(3/2-(1/2*I)*sqrt(7))^n)+1/2*((3/2+(1/2*I)*sqrt(7))^n+(3/2-(1/2*I)*sqrt(7))^n)): seq(A247560(n), n=0..40); # Wesley Ivan Hurt, Oct 02 2014
  • Mathematica
    a[ n_] := Re[ (1 - 1/Sqrt[-7]) (3 + Sqrt[-7])^n / 2^n];
LinearRecurrence[{3,-4},{1,1},40] (* Harvey P. Dale, Jun 13 2017 *)
  • PARI
    {a(n) = real( (1 + quadgen(-7))^n )};
  • Sage
    [((1-1/sqrt(-7))*(3+sqrt(-7))^n/2^n).real() for n in range(34)] # Peter Luschny, Oct 02 2014 (after Somos)

Formula

G.f.: (1 - 2*x) / (1 - 3*x + 4*x^2).
a(n) = 3*a(n-1) - 4*a(n-2) for all n in Z.
a(n) = a(-1-n) * 2^(2*n + 1) for all n in Z.
a(n) = (-1)^n * A087168(n) for all n in Z.
A247565(n) = 2^n + a(n) for all n in Z.
a(n) = A247487(2*n + 1) = A247564(2*n + 1) for all n in Z.

A247564 a(n) = 3*a(n-2) - 4*a(n-4) with a(0) = 2, a(1) = 1, a(2) = 3, a(3) = 1.

Original entry on oeis.org

2, 1, 3, 1, 1, -1, -9, -7, -31, -17, -57, -23, -47, -1, 87, 89, 449, 271, 999, 457, 1201, 287, -393, -967, -5983, -4049, -16377, -8279, -25199, -8641, -10089, 7193, 70529, 56143, 251943, 139657, 473713, 194399, 413367, 24569, -654751, -703889, -3617721
Offset: 0

Views

Author

Michael Somos, Sep 20 2014

Keywords

Examples

			G.f. = 2 + x + 3*x^2 + x^3 + x^4 - x^5 - 9*x^6 - 7*x^7 - 31*x^8 - 17*x^9 + ...

Links

Crossrefs

Cf. A247487, A247560, A247563.

Programs

  • Haskell
    a247564 n = a247564_list !! n
a247564_list = [2,1,3,1] ++ zipWith (-) (map (* 3) $ drop 2 a247564_list)
                                        (map (* 4) $ a247564_list)
-- Reinhard Zumkeller, Sep 20 2014
  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2+x-3*x^2-2*x^3)/(1-3*x^2+4*x^4)));  // G. C. Greubel, Aug 04 2018
  • Maple
    H := (n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], 8):
a := n -> `if`(n < 3, [2, 1, 3][n+1], (-1)^iquo(n, 2)*H(n, irem(n, 2), 1/2)):
seq(simplify(a(n)), n=0..42); # Peter Luschny, Sep 03 2019
# second Maple program:
a:= n-> (<<0|1>, <-4|3>>^iquo(n, 2, 'r').<[<2, 3>, <1, 1>][1+r]>)[1,1]:
seq(a(n), n=0..42);  # Alois P. Heinz, Sep 03 2019
  • Mathematica
    CoefficientList[Series[(2+x-3*x^2-2*x^3)/(1-3*x^2+4*x^4), {x,0,60}], x] (* G. C. Greubel, Aug 04 2018 *)
  • PARI
    {a(n) = if( n<0, n=-n; 2^-n, 1) * polcoeff( (2 + x - 3*x^2 - 2*x^3) / (1 - 3*x^2 + 4*x^4) + x * O(x^n), n)};

Formula

G.f.: (2 + x - 3*x^2 - 2*x^3) / (1 - 3*x^2 + 4*x^4).
a(n) = A247487(n) * 3^( n == 1 (mod 4) ) for all n in Z.
a(2*n) = A247563(n). a(2*n + 1) = A247560(n).
0 = a(n)*(+2*a(n+2)) + a(n+1)*(+2*a(n+1) - 8*a(n+2) + a(n+3)) + a(n+2)*(+a(n+2)) for all n in Z.
a(n) = (-1)^floor(n/2)*H(n, n mod 2, 1/2) for n >= 3 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 8). - Peter Luschny, Sep 03 2019

A247518 a(n) = a(n-1) * (11*a(n-1) - 16*a(n-2)) / (a(n-1) + 10*a(n-2)) with a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 2, 1, -1, 3, -21, 651, 11067, 70091, 230299, 349163, 20539, -31349, 121307, -1158869, 314053499, 3606200523, 18517231899, 49530502251, 52454812347, -20635376629, 43663915099, -217519736917, 3068771480059, 127881095493451, 1094857900300187, 4611286015320811
Offset: 1

Views

Author

Michael Somos, Sep 18 2014

Keywords

Links

Crossrefs

Cf. A247487.

Programs

  • GAP
    a:=[1,2];; for n in [3..30] do a[n]:=a[n-1]*(11*a[n-1]-16*a[n-2])/(a[n-1]+10*a[n-2]); od; a; # Muniru A Asiru, Aug 05 2018
  • Magma
    I:=[1, 2]; [n le 2 select I[n] else Self(n-1)*(11*Self(n-1) - 16*Self(n-2))/(Self(n-1) + 10*Self(n-2)): n in [1..30]]; // G. C. Greubel, Aug 05 2018
  • Mathematica
    RecurrenceTable[{a[n] == a[n - 1]*(11*a[n - 1] - 16*a[n - 2])/(a[n - 1] + 10*a[n - 2]), a[1] == 1, a[2] == 2}, a, {n, 1, 50}] (* G. C. Greubel, Aug 05 2018 *)
nxt[{a_,b_}]:={b, b (11b-16a)/(b+10a)}; NestList[nxt,{1,2},30][[;;,1]] (* Harvey P. Dale, May 13 2023 *)
  • PARI
    {a(n) = my(A, t=1); if( n<1, t = 4^(2*n - 1); n = 1-n); t * if( n<3, n, A = vector(n, k, k); for(k=3, n, A[k] = A[k-1] * (11*A[k-1] - 16*A[k-2]) / (A[k-1] + 10*A[k-2])); A[n])};

Formula

0 = a(n)*(16*a(n+1) + 10*a(n+2)) + a(n+1)*(-11*a(n+1) + a(n+2)) for all n in Z.
a(n) = a(1-n) * 4^(2*n-1) for all n in Z.
a(n) = b(n+1) * b(n) * b(n-1) * b(n-2) for all n in Z where b = A247487.
From Chai Wah Wu, Jun 09 2022: (Start)
a(n) = 11*a(n-1) - 66*a(n-2) + 264*a(n-3) - 704*a(n-4) + 1024*a(n-5) for n > 5.
G.f.: x*(-256*x^4 + 144*x^3 - 45*x^2 + 9*x - 1)/((4*x - 1)*(16*x^2 - 6*x + 1)*(16*x^2 - x + 1)). (End)
Showing 1-3 of 3 results.

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