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 1-6 of 6 results.

A037205 a(n) = (n+1)^n - 1.

Original entry on oeis.org

0, 1, 8, 63, 624, 7775, 117648, 2097151, 43046720, 999999999, 25937424600, 743008370687, 23298085122480, 793714773254143, 29192926025390624, 1152921504606846975, 48661191875666868480, 2185911559738696531967, 104127350297911241532840, 5242879999999999999999999, 278218429446951548637196400, 15519448971100888972574851071
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For n >= 1, a(n) = order of Fibonacci group F(n+1,n).
The terms, written in base n+1, are n digits of value n. For example, a(4) = 624 = 4444 in base 5. - Marc Morgenegg, Nov 30 2016
For n >= 1, in a square grid of side n, this is the number of ways to populate the grid with 1 X 1 blocks (with at least one block) so that no block falls under the effect of gravity. - Paolo Xausa, Apr 12 2021
For n > 1, (n-1)^2 | a(n). - David A. Corneth, Dec 15 2022

References

  • D. L. Johnson, Presentation of Groups, Cambridge, 1976, p. 182.
  • Richard M. Thomas, The Fibonacci groups revisited, in Groups - St. Andrews 1989, Vol. 2, 445-454, London Math. Soc. Lecture Note Ser., 160, Cambridge Univ. Press, Cambridge, 1991.

Links

Crossrefs

A diagonal of A202624.

Programs

  • Magma
    [(n + 1)^n - 1: n in [0..25]]; // G. C. Greubel, Nov 10 2017
  • Mathematica
    Table[(n + 1)^n - 1, {n, 0, 21}] (* or *)
Table[If[n < 1, Length@ #, FromDigits[#, n + 1]] &@ ConstantArray[n, n], {n, 0, 21}] (* Michael De Vlieger, Nov 30 2016 *)
  • PARI
    for(n=0,25, print1((n + 1)^n - 1, ", ")) \\ G. C. Greubel, Nov 10 2017

Formula

a(n) = A000169(n+1) - 1 = A060072(n+1)*(n-1) = A060073(n+1)*(n-1)^2.
E.g.f.: 1/(exp(LambertW(-x)) - x) - exp(x). - Ilya Gutkovskiy, Nov 30 2016
E.g.f.: -exp(x) - 1/(x + x/LambertW(-x)). - Vaclav Kotesovec, Dec 05 2016
a(n) = Sum_{k=1..n} binomial(n,k)*n^k [from Paolo Xausa's comment]. - Joerg Arndt, Apr 12 2021

Extensions

Revised by N. J. A. Sloane, Dec 30 2011

A065530 If n is odd then a(n) = n, else a(n) = n*(n+2).

Original entry on oeis.org

0, 1, 8, 3, 24, 5, 48, 7, 80, 9, 120, 11, 168, 13, 224, 15, 288, 17, 360, 19, 440, 21, 528, 23, 624, 25, 728, 27, 840, 29, 960, 31, 1088, 33, 1224, 35, 1368, 37, 1520, 39, 1680, 41, 1848, 43, 2024, 45, 2208, 47, 2400, 49, 2600, 51, 2808, 53, 3024, 55, 3248, 57
Offset: 0

Views

Author

George E. Antoniou, Dec 02 2001

Keywords

Comments

Order of Fibonacci group F(n+1,2) (0 means group is infinite). - N. J. A. Sloane, Dec 30 2011

References

  • D. L. Johnson, Presentation of Groups, Cambridge, 1976, p. 182.
  • Thomas, Richard M., The Fibonacci groups revisited, in Groups - St. Andrews 1989, Vol. 2, 445-454, London Math. Soc. Lecture Note Ser., 160, Cambridge Univ. Press, Cambridge, 1991.

Links

Crossrefs

A column of A202624.

Programs

  • Mathematica
    Array[If[OddQ[#], #, #*(#+2)] &, 100, 0] (* Paolo Xausa, Feb 22 2024 *)
With[{nn=60},Riffle[Table[n(n+2),{n,0,nn,2}],Range[1,nn+1,2]]] (* or *) LinearRecurrence[{0,3,0,-3,0,1},{0,1,8,3,24,5},100] (* Harvey P. Dale, Sep 27 2024 *)
  • PARI
    a(n) = { if (n%2, n, n*(n + 2)) } \\ Harry J. Smith, Oct 20 2009
  • PARI
    concat(0, Vec(x*(x^4-8*x-1)/((x-1)^3*(x+1)^3) + O(x^100))) \\ Colin Barker, May 02 2015

Formula

O.g.f.: (x+8x^2-x^5)/(1-x^2)^3. - Len Smiley, Dec 04 2001
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>5. - Colin Barker, May 02 2015

A202625 Order of Fibonacci group F(n,3) (or 0 if the group is infinite).

Original entry on oeis.org

0, 8, 2, 63, 0, 5, 342, 0, 8, 999, 0, 11, 2196, 0, 14, 4095, 0, 17, 6858, 0, 20, 10647, 0, 23, 15624, 0, 26, 21951, 0, 29, 29790, 0, 32, 39303, 0, 35, 50652, 0, 38, 63999, 0, 41, 79506, 0, 44, 97335, 0, 47, 117648, 0, 50, 140607, 0, 53, 166374, 0, 56, 195111, 0, 59, 226980, 0, 62, 262143, 0, 65, 300762, 0, 68, 342999, 0, 71, 389016, 0, 74, 438975, 0
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2011

Keywords

Comments

A column of the array described in A202624. See that entry for references.

Crossrefs

Cf. A202624.

Programs

  • Maple
    f:=n-> if n mod 3 = 0 then n-1 elif n mod 3 = 1 then n^3-1 else 0 fi; # Gives all values correctly except that a(2) should be 8.

A202626 Order of Fibonacci group F(n,4) (or 0 if the group is infinite).

Original entry on oeis.org

0, 5, 0, 3, 624, 125, 0, 7, 6560, 4905, 0, 11, 28560, 104845, 0, 15, 83520, 2236945, 0, 19, 194480, 43997205, 0, 23, 390624, 839065625, 0, 27, 707280, 15568306205, 0, 31, 1185920, 283472166945, 0, 35, 1874160, 5085221879845, 0, 39, 2825760, 90160039460905, 0, 43, 4100624, 1583296366510125, 0, 47, 5764800, 27584549361811505, 0, 51, 7890480
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2011

Keywords

Comments

A column of the array described in A202624. See that entry for references.

Crossrefs

Cf. A202624. A202628 is a subsequence.

Programs

  • Maple
    g:=k->(4*k+1)*(2^(4*k+1)+(-1)^k*2^(2*k+1)+1);
f:=n-> if n mod 4 = 0 then n-1 elif n mod 4 = 1 then n^4-1 elif n mod 4 = 2 then g((n-2)/4); else 0; fi;
[seq(f(n),n=1..60)];

A202627 Order of Fibonacci group F(n,5) (or 0 if the group is infinite).

Original entry on oeis.org

0, 11, 22, 0, 4, 7775
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2011

Keywords

Comments

A column of the array described in A202624. See that entry for references.

Crossrefs

Cf. A202624.

A202628 a(n) = (4*n+1)*(2^(4*n+1)+(-1)^n*2^(2*n+1)+1).

Original entry on oeis.org

5, 125, 4905, 104845, 2236945, 43997205, 839065625, 15568306205, 283472166945, 5085221879845, 90160039460905, 1583296366510125, 27584549361811505, 477381553387733045, 8214565750925426745, 140656423431038828605, 2398076730140587458625, 40730410912379868020805, 689465506509001244803145, 11635911013748474608877645
Offset: 0

Views

Author

N. J. A. Sloane, Dec 31 2011

Keywords

Comments

A subsequence of one of the columns of the array described in A202624.

Links

Crossrefs

Cf. A202624.

Programs

  • Mathematica
    Table[(4n+1)(2^(4n+1)+(-1)^n 2^(2n+1)+1),{n,0,20}] (* or *) LinearRecurrence[ {26,-65,-1480,-1040,6656,-4096},{5,125,4905,104845,2236945,43997205},20] (* Harvey P. Dale, May 23 2014 *)

Formula

a(0)=5, a(1)=125, a(2)=4905, a(3)=104845, a(4)=2236945, a(5)=43997205, a(n)=26*a(n-1)-65*a(n-2)-1480*a(n-3)-1040*a(n-4)+ 6656*a(n-5)- 4096*a(n-6). - Harvey P. Dale, May 23 2014
G.f.: 5*(1-x+396*x^2-1432*x^3+4000*x^4+1536*x^5)/(x-1)^2/(4*x+1)^2/(16*x-1)^2 . - R. J. Mathar, Sep 02 2017
Showing 1-6 of 6 results.

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

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