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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A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

Original entry on oeis.org

0, 2, 3, 8, 15, 30, 56, 104, 189, 340, 605, 1068, 1872, 3262, 5655, 9760, 16779, 28746, 49096, 83620, 142065, 240812, 407353, 687768, 1159200, 1950650, 3277611, 5499704, 9216519, 15426870, 25793240, 43080608, 71884197, 119835652
Offset: 0

Views

Author

Paul Barry and Ralf Stephan, Oct 15 2004

Keywords

Comments

A Fibonacci-Lucas convolution.
The number of edges in the Lucas cube L_(n+1) [Klavzar]. - R. J. Mathar, Nov 05 2008
Sums of rows of the triangle in A108037. - Reinhard Zumkeller, Oct 07 2012
a(n-1) is the total binary weight of all bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - Steven Finch, May 26 2020

References

  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 35.

Crossrefs

Equals A010049(n) + A001629(n+1).

Programs

  • Haskell
    a099920 n = a099920_list !! n
    a099920_list = zipWith (*) [1..] a000045_list
    -- Reinhard Zumkeller, Oct 07 2012
    
  • Magma
    [(n+1)*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011
    
  • Mathematica
    Table[(n + 1) Fibonacci[n], {n, 0, 40}] (* Harvey P. Dale, Jan 18 2012 *)
    LinearRecurrence[{2, 1, -2, -1}, {0, 2, 3, 8}, 40] (* Harvey P. Dale, Jan 18 2012 *)
    CoefficientList[Series[(2 - x) x/(-1 + x + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 28 2023 *)
  • PARI
    a(n)=(n+1)*fibonacci(n) \\ Charles R Greathouse IV, Jun 11 2015

Formula

G.f.: x*(2-x)/(1-x-x^2)^2;
a(n) = Sum_{k=0..n} F(n-k)*(L(k-1) + 0^k).
a(n) = Sum_{k=0..n+1} F(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4); a(0)=0, a(1)=2, a(2)=3, a(3)=8. - Harvey P. Dale, Jan 18 2012
a(n) = a(n-1) + a(n-2) + A000032(n-1) (Lucas numbers). - Bob Selcoe, Aug 19 2015
a(n) = 2*A001629(n+1) - A001629(n). - R. J. Mathar, Feb 04 2022

Extensions

Entry revised by N. J. A. Sloane, Jan 23 2006. The offset changed, so some of the formulas may now be slightly off.

A108617 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) for 0 < k < n, T(n,0) = T(n,n) = n-th Fibonacci number.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 3, 2, 3, 5, 6, 5, 3, 5, 8, 11, 11, 8, 5, 8, 13, 19, 22, 19, 13, 8, 13, 21, 32, 41, 41, 32, 21, 13, 21, 34, 53, 73, 82, 73, 53, 34, 21, 34, 55, 87, 126, 155, 155, 126, 87, 55, 34, 55, 89, 142, 213, 281, 310, 281, 213, 142, 89, 55, 89, 144, 231, 355, 494, 591, 591, 494, 355, 231, 144, 89
Offset: 0

Views

Author

Reinhard Zumkeller, Jun 12 2005

Keywords

Comments

Sum of n-th row = 2*A027934(n). - Reinhard Zumkeller, Oct 07 2012

Examples

			Triangle begins:
   0;
   1,   1;
   1,   2,   1;
   2,   3,   3,   2;
   3,   5,   6,   5,   3;
   5,   8,  11,  11,   8,   5;
   8,  13,  19,  22,  19,  13,   8;
  13,  21,  32,  41,  41,  32,  21,  13;
  21,  34,  53,  73,  82,  73,  53,  34,  21;
  34,  55,  87, 126, 155, 155, 126,  87,  55,  34;
  55,  89, 142, 213, 281, 310, 281, 213, 142,  89,  55;
		

Crossrefs

T(2n,n) gives 2*A176085(n).

Programs

  • Haskell
    a108617 n k = a108617_tabl !! n !! k
    a108617_row n = a108617_tabl !! n
    a108617_tabl = [0] : iterate f [1,1] where
       f row@(u:v:_) = zipWith (+) ([v - u] ++ row) (row ++ [v - u])
    -- Reinhard Zumkeller, Oct 07 2012
    
  • Magma
    function T(n,k) // T = A108617
      if k eq 0 or k eq n then return Fibonacci(n);
      else return T(n-1,k-1) + T(n-1,k);
      end if;
    end function;
    [T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 20 2023
    
  • Maple
    A108617 := proc(n,k) option remember;
        if k = 0 or k=n then
            combinat[fibonacci](n) ;
        elif k <0 or k > n then
            0 ;
        else
            procname(n-1,k-1)+procname(n-1,k) ;
        end if;
    end proc: # R. J. Mathar, Oct 05 2012
  • Mathematica
    a[1]:={0}; a[n_]:= a[n]= Join[{Fibonacci[#]}, Map[Total, Partition[a[#],2,1]], {Fibonacci[#]}]&[n-1]; Flatten[Map[a, Range[15]]] (* Peter J. C. Moses, Apr 11 2013 *)
  • SageMath
    def T(n,k): # T = A108617
        if (k==0 or k==n): return fibonacci(n)
        else: return T(n-1,k-1) + T(n-1,k)
    flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 20 2023

Formula

T(n,0) = T(n,n) = A000045(n);
T(n,1) = T(n,n-1) = A000045(n+1) for n>0;
T(n,2) = T(n,n-2) = A000045(n+2) - 2 = A001911(n-1) for n>1;
Sum_{k=0..n} T(n,k) = 2*A027934(n-1) for n>0.
Sum_{k=0..n} (-1)^k*T(n, k) = 2*((n+1 mod 2)*Fibonacci(n-2) + [n=0]). - G. C. Greubel, Oct 20 2023

A108035 Triangle read by rows: n-th row consists of n copies of the n-th nonzero Fibonacci number.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 144, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233, 233
Offset: 1

Views

Author

N. J. A. Sloane, Jun 01 2005

Keywords

Examples

			1; 2,2; 3,3,3; 5,5,5,5; 8,8,8,8,8; ...
		

Crossrefs

Cf. A023607 (row sums).

Programs

  • Haskell
    a108035 n k = a108035_tabl !! (n-1) !! (n-1)
    a108035_row n = a108035_tabl !! (n-1)
    a108035_tabl = zipWith replicate [1..] $ drop 2 a000045_list
    -- Reinhard Zumkeller, Oct 07 2012
    
  • Mathematica
    Flatten[Table[Table[Fibonacci[n],{n-1}],{n,13}]] (* Harvey P. Dale, Jul 18 2015 *)
  • Python
    from math import isqrt
    from sympy import fibonacci
    def A108035(n): return int(fibonacci(1+(m:=isqrt(k:=n<<1))+(k>m*(m+1)))) # Chai Wah Wu, Nov 07 2024

Formula

G.f.: (1+x+y)/((1-x-x^2)*(1-y-y^2)). [U coordinates]

Extensions

Definition clarified by N. J. A. Sloane, Nov 09 2024
Showing 1-3 of 3 results.