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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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A136391 a(n) = n*F(n) - (n-1)*F(n-1), where the F(j)'s are the Fibonacci numbers (F(0)=0, F(1)=1).

Original entry on oeis.org

1, 1, 4, 6, 13, 23, 43, 77, 138, 244, 429, 749, 1301, 2249, 3872, 6642, 11357, 19363, 32927, 55861, 94566, 159776, 269469, 453721, 762793, 1280593, 2147068, 3595422, 6013933, 10048559, 16773139, 27971549, 46605186, 77587084, 129063117, 214531397, 356346557
Offset: 1

Views

Author

Gary W. Adamson, Dec 28 2007

Keywords

Comments

By definition, the arithmetic mean of a(1) ... a(n) is equal to A000045(n).
Proof of the three-term recurrence formula: a(n+1) - a(n) - a(n-1) = ((n+1)*F(n+1) - n*F(n)) - (n*F(n) - (n-1)*F(n-1)) - ((n-1)*F(n-1) - (n-2)*F(n-2)) = (n+1)*F(n+1) - 2*n*F(n) + (n-2)*F(n-2) = (n+1)*(2*F(n) - F(n-2)) - 2*n*F(n) + (n-2)*F(n-2) = 2*F(n) - 3*F(n-2) = F(n-1) + F(n-3) = L(n-2). - Giuseppe Coppoletta, Sep 01 2014

Examples

			a(6) = 23 = 6*F(6) - 5*F(5) = 6*8 - 5*5 = 48 - 25.

Links

Crossrefs

Cf. A000045, A246715, A354044.

Programs

  • Julia
    # The function 'fibrec' is defined in A354044.
function A136391(n)
    a, b = fibrec(n - 1)
    n*b - (n - 1)*a
end
println([A136391(n) for n in 1:35]) # Peter Luschny, May 18 2022
  • Maple
    with(combinat): seq(n*fibonacci(n)-(n-1)*fibonacci(n-1),n=1..30); # Emeric Deutsch, Jan 01 2008
  • Mathematica
    Table[n Fibonacci[n] - (n-1) Fibonacci[n-1], {n, 1, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)
  • PARI
    Vec(x*(1-x)*(1+x^2)/(1-x-x^2)^2 + O(x^100)) \\ Altug Alkan, Oct 28 2015

Formula

Equals A128064 * A000045.
From R. J. Mathar, Nov 25 2008: (Start)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) = A045925(n) - A045925(n-1).
G.f.: x*(1 - x)*(1 + x^2)/(1 - x - x^2)^2.
a(n) = A014286(n-1) - A014286(n-2), n>3. (End)
Recurrence: a(n+1) = a(n) + a(n-1) + L(n-2) for n>1, where L = A000032 (see proof in Comments section). - Giuseppe Coppoletta, Sep 01 2014
E.g.f.: (exp(x*phi)/phi+exp(-x/phi)*phi)*(x+1)/sqrt(5)-1, where phi=(1+sqrt(5))/2. - Vladimir Reshetnikov, Oct 28 2015
a(n) = F(n-1) + n*F(n-2). - Bruno Berselli, Jul 26 2017

Extensions

More terms from Emeric Deutsch, Jan 01 2008

A128089 Denominators in inverse of triangle A128078 by rows, n * each term in n-th row of A126615.

Original entry on oeis.org

1, 4, 4, 6, 18, 9, 8, 24, 48, 16, 10, 30, 60, 100, 25, 12, 36, 72, 120, 180, 36, 14, 42, 84, 140, 210, 294, 49, 16, 48, 96, 160, 240, 336, 448, 64, 18, 54, 108, 180, 270, 378, 504, 648, 81, 20, 60, 120, 200, 300, 420, 560, 720, 900, 100
Offset: 1

Views

Author

Gary W. Adamson, Feb 14 2007

Keywords

Comments

Row sums = A014820: (1, 8, 33, 96, 225, 456, ...).
Denominators of the inverse of A128078: (1/1; 1/4, 1/4; 1/6, 1/18, 1/9; 1/8, 1/24, 1/48, 1/16; ...).
Row sums of this triangle: 1/1, 1/2, 1/3, ...; e.g., (1/8 + 1/24 + 1/48 + 1/16) = 1/4.

Examples

			First few rows of the triangle:
   1;
   4,  4;
   6, 18,  9;
   8, 24, 48,  16;
  10, 30, 60, 100,  25;
  12, 36, 72, 120, 180,  36;
  14, 42, 84, 140, 210, 294,  49;
  16, 48, 96, 160, 240, 336, 448,  64;
  ...
Row 4 = (8, 24, 48, 16) = 4 * (2, 6, 12, 4); where (2, 6, 12, 4) = row 4 of A126615.

Crossrefs

Cf. A128078, A014820, A006527, A126615, A002260, A128064.

Formula

Denominators in inverse triangular matrix of A128078, where A128078 = A002260 * A128064, = (1; -1, 4; -1, -2, 9; -1, -2, -3, 16; ...).
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