A016741 -id:A016741 - cp's OEIS frontend

A115361 Inverse of matrix (1,x)-(x,x^2) (expressed in Riordan array notation).

1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Jan 21 2006

Row sums are the 'ruler function' A001511. Columns are stretched Fredholm-Rueppel sequences (A036987). Inverse is A115359.
Eigensequence of triangle A115361 = A018819 starting with offset 1: (1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, ...). - Gary W. Adamson, Nov 21 2009
From Gary W. Adamson, Nov 27 2009: (Start)
A115361 * [1, 2, 3, ...] = A129527 = (1, 3, 3, 7, 5, 9, 7, 15, ...).
(A115361)^(-1) * [1, 2, 3, ...] = A115359 * [1, 2, 3, ...] = A026741 starting /Q (1, 1, 3, 2, 5, 3, 7, 4, 9, ...). (End)
This is the lower-left triangular part of the inverse of the infinite matrix A_{ij} = [i=j] - [i=2j], its upper-right part (above / right to the diagonal) being zero. The n-th row has 1 in column n/2^i, i = 0, 1, ... as long as this is an integer. - M. F. Hasler, May 13 2018
The rows are the reversed binary expansions of A127804. - Tilman Piesk, Jun 10 2025

			Triangle begins:
  1;
  1,1;
  0,0,1;
  1,1,0,1;
  0,0,0,0,1;
  0,0,1,0,0,1;
  0,0,0,0,0,0,1;
  1,1,0,1,0,0,0,1;
  0,0,0,0,0,0,0,0,1;
  0,0,0,0,1,0,0,0,0,1;
  0,0,0,0,0,0,0,0,0,0,1;

Tilman Piesk, Illustration of first 32 rows.

Cf. A016741, A018819, A129527. - Gary W. Adamson, Nov 21 2009

Cf. A036987, A209229, A127804.

A115361 := proc(n,k)
    for j from 0 do
        if k+(2*j-1)*(k+1) > n then
            return 0 ;
        elif k+(2^j-1)*(k+1) = n then
            return 1 ;
        end if;
    end do;
end proc: # R. J. Mathar, Jul 14 2012

(*recurrence*)
Clear[t]
t[1, 1] = 1;
t[n_, k_] :=
t[n, k] =
  If[k == 1, Sum[t[n, k + i], {i, 1, 2 - 1}],
   If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 14}]] (* Mats Granvik, Jun 26 2014 *)

tabl(nn) = {T = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = T^(-1); for (n=1, nn, for (k=1, n, print1(Ti[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

A115361_row(n,v=vector(n))={until(bittest(n,0)||!n\=2,v[n]=1);v} \\ Yields the n-th row (of length n). - M. F. Hasler, May 13 2018

T(n,k)={if(n%k, 0, my(e=valuation(n/k,2)); n/k==1<Andrew Howroyd, Aug 03 2018

# translation of Maple code by R. J. Mathar
def a115361(n, k):
    j = 0
    while True:
        if k + (2*j - 1) * (k + 1) > n:
            return 0
        elif k + (2**j - 1) * (k + 1) == n:
            return 1
        else:
            j += 1  #  Tilman Piesk, Jun 10 2025

Number triangle whose k-th column has g.f. x^k*sum{j>=0} x^((2^j-1)*(k+1)).

T(n,k) = A209229((n+1)/(k+1)) for k+1 divides n+1, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 05 2018

A052492 Initial pile sizes that guarantee a win for player 2 in a variant of Fibonacci Nim where the players may not take one stone.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 10, 15, 16, 24, 25, 38, 39, 59, 60, 90, 91, 137, 138, 207, 208, 312, 313, 470, 471, 707, 708, 1062, 1063, 1595, 1596, 2394, 2395, 3593, 3594, 5391, 5392, 8088, 8089, 12134, 12135, 18203, 18204, 27306, 27307

Offset: 1

Felix Goldberg (sgefelix(AT)t2.technion.ac.il), Mar 16 2000

nonn

			a(5)=6 because player 1 can't take 1 stone; if player 1 takes 2 stones or more, player 2 grabs the rest.

Tal Kubo, Generalized Fibonacci Nim.

Cf. A016741.

A129555 A054523 * A129372.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 2, 1, 0, 1, 5, 0, 0, 0, 1, 3, 3, 1, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 9, 0, 3, 0, 0, 0, 0, 0, 1, 5, 5, 0, 0, 1, 0, 0, 0, 0, 1

Offset: 0

Gary W. Adamson, Apr 20 2007

Row sums = A002131: (1, 2, 4, 4, 6, 8, 8, 8, 13, 12, ...). Left column = A016741: (1, 1, 3, 2, 5, 3, 7, ...).

			First few rows of the triangle:
  1;
  1, 1;
  3, 0, 1;
  2, 1, 0, 1;
  5, 0, 0, 0, 1;
  3, 3, 1, 0, 0, 1;
  7, 0, 0, 0, 0, 0, 1;
  4, 2, 0, 1, 0, 0, 0, 1;
  9, 0, 3, 0, 0, 0, 0, 0, 1;
  ...

Cf. A054523, A129372, A002131, A026741.

A054523 * A129372 as infinite lower triangular matrices.

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A115361 Inverse of matrix (1,x)-(x,x^2) (expressed in Riordan array notation).

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A052492 Initial pile sizes that guarantee a win for player 2 in a variant of Fibonacci Nim where the players may not take one stone.

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A129555 A054523 * A129372.

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