A117568 -id:A117568 - cp's OEIS frontend

A117569 Expansion of (1+x+x^2)/(1+x^2).

1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0

Offset: 0

Paul Barry, Mar 29 2006

Row sums of A117568.

			1 + x - x^3 + x^5 - x^7 + x^9 - x^11 + x^13 - x^15 + x^17 - x^19 + ...

Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A073097, A163804, A163805.

CoefficientList[Series[(1+x+x^2)/(1+x^2),{x,0,120}],x] (* or *) LinearRecurrence[{0,-1},{1,1,0},120] (* or *) PadRight[{1},120,{0,1,0,-1}] (* Harvey P. Dale, Dec 12 2016 *)

a(n) = (n==0) + [0, 1, 0, -1][n%4 + 1] /* Michael Somos, Aug 04 2009 */

a(n) = (n==0) + kronecker( -4, n) /* Michael Somos, Aug 04 2009 */

G.f.: (1-x^3)/((1-x)(1+x^2)); a(n)=0^n+(1-(-1)^n)(cos(pi*n/2)+sin(pi*n/2))/2;
a(n) = A101455(n), n>0. - R. J. Mathar, Aug 10 2008
Expansion of (1 - x^2) * (1 - x^3) / ((1 - x) * (1 - x^4)) in powers of x.
G.f.: 1 / (1 - x / (1 + x / (1 - x / (1 + x)))). - Michael Somos, Apr 02 2012
Euler transform of length 4 sequence [ 1, -1, -1, 1]. - Michael Somos, Aug 04 2009
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v - u * (2 - u) * (2*v - 1).
a(n) is completely multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e if p == 3 (mod 4).
a(2*n) = 0 unless n=0, a(4*n + 3) = -1, a(4*n + 1) = a(0) = 1.
a(-n) = -a(n) unless n=0. a(n+2) = -a(n) unless n=0 or n=-2.
a(n) = -A163805(n) unless n=0. a(n) = (-1)^n * A163805(n). Convolution inverse of A163804.
E.g.f.: 1 + sin(x). - Arkadiusz Wesolowski, Aug 13 2012
a(n) = floor(1/(n+1)) + (1-(-1)^n)/2*(-1)^((n-1)/2). - Tani Akinari, Nov 09 2012

A117567 Riordan array ((1+x^2)/(1-x^3),x).

Original entry on oeis.org

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

Offset: 0

Paul Barry, Mar 29 2006

Sequence array for the sequence F(L((n+2)/3)).

			Triangle begins:
n\k|  0  1  2  3  4  5  6  7  8  9
---+--------------------------------
0  |  1,
1  |  0, 1,
2  |  1, 0, 1,
3  |  1, 1, 0, 1,
4  |  0, 1, 1, 0, 1,
5  |  1, 0, 1, 1, 0, 1,
6  |  1, 1, 0, 1, 1, 0, 1,
7  |  0, 1, 1, 0, 1, 1, 0, 1,
8  |  1, 0, 1, 1, 0, 1, 1, 0, 1,
9  |  1, 1, 0, 1, 1, 0, 1, 1, 0, 1
etc. Row and column numbering added by _Antti Karttunen_, Jan 19 2025

Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Antti Karttunen, Table of n, a(n) for n = 0..101474; the first 450 rows of the triangle
D. Panario, M. Sahin, Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.
Index entries for characteristic functions.

Row sums are A093878. Diagonal sums are A051275. Inverse is A117568.

up_to = 119;
A117567tr0(n,k) = abs(kronecker((n-k+2), 3)); \\ We could also use fibonacci instead of abs
A117567list(up_to) = { my(v = vector(1+up_to), i=0); for(n=0,oo, for(k=0,n, i++; if(i > 1+up_to, return(v)); v[i] = A117567tr0(n,k))); (v); };
v117567 = A117567list(up_to);
A117567(n) = v117567[1+n]; \\ Antti Karttunen, Jan 19 2025

Number triangle T(n,k) = F(L((n-k+2)/3))[k<=n] where L(j/p) is the Legendre symbol of j and p.

In the above, I assume that F stands for Fibonacci sequence (A000045), which in domain {-1, 0, 1} reduces to taking the absolute value of the argument. - Antti Karttunen, Jan 19 2025

Data section extended up to a(119) [15 rows of triangle] by Antti Karttunen, Jan 19 2025

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A117569 Expansion of (1+x+x^2)/(1+x^2).

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A117567 Riordan array ((1+x^2)/(1-x^3),x).

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