A176085 -id:A176085 - cp's OEIS frontend

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.

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

Reinhard Zumkeller, Jun 12 2005

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

			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;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Hacéne Belbachir and László Szalay, On the Arithmetic Triangles, Šiauliai Mathematical Seminar, Vol. 9 (17), 2014. See Fig. 1 p. 18.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Pascal's Triangle.
Wikipedia, Fibonacci number.
Wikipedia, Pascal's triangle.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A074829, A108037.

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

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

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

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

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 *)

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

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

A257838 Main diagonal of iterated partial sums array of Fibonacci numbers (starting with the first partial sums).

Original entry on oeis.org

0, 1, 4, 16, 63, 247, 967, 3785, 14820, 58060, 227612, 892926, 3505386, 13770404, 54129602, 212904952, 837885495, 3299264407, 12997784803, 51230474669, 202014314769, 796928589755, 3145066003589, 12416625685891, 49037912997003, 193734379979677, 765632076098287, 3026670770970925, 11968378998073935

Offset: 0

Luciano Ancora, May 10 2015

The array used here starts in row n=0 with the first partial sums of A000045. The array which starts with the Fibonacci numbers in row k=0 is shown in A136431. The diagonal of that array is given in A176085. - Wolfdieter Lang, Jun 03 2015

			This sequence is the main diagonal of the following array (see the comment and Example field of A136431):
0, 1, 2,  4,  7,  12, ...  A000071
0, 1, 3,  7, 14,  26, ...  A001924
0, 1, 4, 11, 25,  51, ...  A014162
0, 1, 5, 16, 41,  92, ...  A014166
0, 1, 6, 22, 63, 155, ...  A053739
0, 1, 7, 29, 92, 247, ...  A053295

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000045, A000071, A001924, A014162, A014166, A136431, A176085.

Table[DifferenceRoot[Function[{a, n},{(2*n + 4*n^2)*a[n] + (2 + 7*n + 15*n^2)*a[1 + n] + (8 - 6*n - 8*n^2)*a[2 + n] + (-2 + n + n^2)*a[3 + n] == 0, a[1] == 0, a[2] == 1, a[3] == 4, a[4] == 16}]][n], {n, 30}]

a(n):=sum(binomial(2*n-k,n-k)*fib(k),k,0,n); /* Vladimir Kruchinin, Oct 09 2016 */

x='x+O('x^50); concat([0], Vec(-(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x))) \\ G. C. Greubel, Apr 08 2017

a(n) = F^{n+1}(n), n >= 0, with the k-th iterated partial sum F^{k} of the Fibonacci number A000045. - Wolfdieter Lang, Jun 03 2015
Conjecture: n*(n-3)*a(n) +2*(-4*n^2+13*n-6)*a(n-1) +(15*n^2-53*n+48)*a(n-2) +2*(2*n-3)*(n-2)*a(n-3)=0. - R. J. Mathar, Dec 10 2015
G.f.: -(4*x+sqrt(1-4*x)-1)/(8*x^2+sqrt(1-4*x)*(8*x-2)-2*x). - Vladimir Kruchinin, Oct 09 2016
a(n) = Sum_{k=0..n} binomial(2*n-k,n-k)*F(k), where F(k) = A000045(k). - Vladimir Kruchinin, Oct 09 2016
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 09 2016

Name edited by Wolfdieter Lang, Jun 03 2015

cp's OEIS Frontend

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.

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