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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A210729 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 8, 16, 31, 55, 95, 160, 266, 438, 717, 1169, 1901, 3086, 5004, 8108, 13131, 21259, 34411, 55692, 90126, 145842, 235993, 381861, 617881, 999770, 1617680, 2617480, 4235191, 6852703, 11087927, 17940664, 29028626, 46969326, 75997989, 122967353
Offset: 0

Views

Author

Alex Ratushnyak, May 10 2012

Keywords

Links

Crossrefs

Cf. A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).
Cf. A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).
Cf. A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.
Cf. A104161: a(n)=a(n-1)+a(n-2)+n, a(0)=1,a(1)=2 (except the first term).
Cf. A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.
Cf. A210728: a(n)=a(n-1)+a(n-2)+n+2, a(0)=1,a(1)=2.
Cf. A000032, A000045.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> 2*F(n+3)+3*F(n+1)-n-6); # G. C. Greubel, Jul 09 2019
  • Magma
    [3*Fibonacci(n+1)+2*Fibonacci(n+3)-n-6: n in [0..40]]; // Vincenzo Librandi, Jul 18 2013
  • Mathematica
    Table[3*Fibonacci[n+1]+2*Fibonacci[n+3]-n-6,{n,0,40}] (* Vaclav Kotesovec, May 13 2012 *)
  • PARI
    vector(40, n, n--; f=fibonacci; 2*f(n+3)+3*f(n+1)-n-6) \\ G. C. Greubel, Jul 09 2019
  • Python
    prpr, prev = 1,2
for n in range(2, 99):
    current = prev+prpr+n+3
    print(prpr, end=',')
    prpr = prev
    prev = current
  • Sage
    f=fibonacci; [2*f(n+3)+3*f(n+1)-n-6 for n in (0..40)] # G. C. Greubel, Jul 09 2019

Formula

G.f.: (1-x+4*x^2-3*x^3)/((1-x-x^2)*(1-x)^2).
a(n) = 3*Fibonacci(n+1)+2*Fibonacci(n+3)-n-6. - Vaclav Kotesovec, May 13 2012
a(n) = 2*Lucas(n+2) + Fibonacci(n+1) - (n+6). - G. C. Greubel, Jul 09 2019

A117666 Expansion of (1-3*x+x^2)*(1-x-x^2)/((1+x+x^2)*(1-x+x^2)*(1-x)^2).

Original entry on oeis.org

1, -2, -3, 2, 3, 2, 3, 0, -1, 4, 5, 4, 5, 2, 1, 6, 7, 6, 7, 4, 3, 8, 9, 8, 9, 6, 5, 10, 11, 10, 11, 8, 7, 12, 13, 12, 13, 10, 9, 14, 15, 14, 15, 12, 11, 16, 17, 16, 17, 14, 13, 18, 19, 18, 19, 16, 15, 20, 21, 20, 21, 18, 17, 22, 23, 22, 23, 20, 19, 24, 25, 24, 25, 22, 21, 26, 27, 26, 27, 24, 23, 28, 29, 28, 29, 26, 25
Offset: 0

Views

Author

Creighton Dement, Apr 11 2006

Keywords

Links

Crossrefs

Cf. A104161, A001595.

Programs

  • GAP
    a:=[1,-2,-3,2,3,2];; for n in [7..100] do a[n]:=2*a[n-1] -2*a[n-2] +2*a[n-3] -2*a[n-4] +2*a[n-5] -a[n-6]; od; a; # G. C. Greubel, Jul 13 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1-3*x+x^2)*(1-x-x^2)/((1+x^2+x^4)*(1-x)^2) )); // G. C. Greubel, Jul 13 2019
  • Mathematica
    CoefficientList[Series[(1-3*x+x^2)*(1-x-x^2)/((1+x^2+x^4)*(1-x)^2), {x, 0, 100}], x] (* G. C. Greubel, Jul 13 2019 *)
  • PARI
    Vec((1-3*x+x^2)*(1-x-x^2)/((1-x)^2*(1+x^2+x^4)) + O(x^100)) \\ Colin Barker, May 18 2019
  • Sage
    ((1-3*x+x^2)*(1-x-x^2)/((1+x^2+x^4)*(1-x)^2)).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Jul 13 2019

Formula

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) for n>5. - Colin Barker, May 18 2019

A131247 2*A052509 - A000012.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 3, 1, 1, 7, 7, 3, 1, 1, 9, 13, 7, 3, 1, 1, 11, 21, 15, 7, 3, 1, 1, 13, 31, 29, 15, 7, 3, 1, 1, 15, 43, 51, 31, 15, 7, 3, 1, 1, 17, 57, 83, 61, 31, 15, 7, 3, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 23 2007

Keywords

Comments

Row sums = A104161 starting (1, 2, 5, 10, 19, 34, 59, ...). Reversal, A131248 is generated from 2*A004070 - A000012.

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  3,  1;
  1,  7,  7,  3,  1;
  1,  9, 13,  7,  3,  1;
  1, 11, 21, 15,  7,  3,  1;
  ...

Crossrefs

Cf. A052509, A104161, A004070, A131248.

Formula

2*A052509 - A000012, where A000012 = (1; 1,1; 1,1,1; ...).

A131248 2*A004070 - A000012.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 3, 7, 7, 1, 1, 3, 7, 13, 9, 1, 1, 3, 7, 15, 21, 11, 1, 1, 3, 7, 15, 29, 31, 13, 1, 1, 3, 7, 15, 31, 51, 43, 15, 1, 1, 3, 7, 15, 31, 61, 83, 57, 17, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 23 2007

Keywords

Comments

Row sums = A104161: (1, 2, 5, 10, 19, 34, 59, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  3,  1;
  1,  3,  5,  1;
  1,  3,  7,  7,  1;
  1,  3,  7, 13,  9,  1;
  1,  3,  7, 15, 21, 11,  1;
  1,  3,  7, 15, 29, 31, 13,  1;
  ...

Crossrefs

Cf. A004070, A000012, A104161, A131247.

Formula

2*A004070 - A000012 as infinite lower triangular matrices. Reversal triangle of A131247, read by rows.

A210675 a(n)=a(n-1)+a(n-2)+n+4, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 7, 15, 30, 54, 94, 159, 265, 437, 716, 1168, 1900, 3085, 5003, 8107, 13130, 21258, 34410, 55691, 90125, 145841, 235992, 381860, 617880, 999769, 1617679, 2617479, 4235190, 6852702, 11087926, 17940663, 29028625, 46969325, 75997988, 122967352, 198965380
Offset: 0

Views

Author

Alex Ratushnyak, May 09 2012

Keywords

Links

Crossrefs

Cf. A210673: a(n)=a(n-1)+a(n-2)+n-4, a(0)=0,a(1)=1.
Cf. A066982: a(n)=a(n-1)+a(n-2)+n-2, a(0)=0,a(1)=1 (except the first term).
Cf. A104161: a(n)=a(n-1)+a(n-2)+n-1, a(0)=0,a(1)=1.
Cf. A001924: a(n)=a(n-1)+a(n-2)+n, a(0)=0,a(1)=1.
Cf. A192760: a(n)=a(n-1)+a(n-2)+n+1, a(0)=0,a(1)=1.
Cf. A192761: a(n)=a(n-1)+a(n-2)+n+2, a(0)=0,a(1)=1.
Cf. A192762: a(n)=a(n-1)+a(n-2)+n+3, a(0)=0,a(1)=1.

Programs

Formula

a(n) = 3*a(n-1)-2*a(n-2)-a(n-3)+a(n-4). G.f.: x*(4*x^2-4*x-1) / ((x-1)^2*(x^2+x-1)). - Colin Barker, May 31 2013
Previous Showing 11-15 of 15 results.

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