A355021 -id:A355021 - cp's OEIS frontend

A100218 Riordan array ((1-2*x)/(1-x), (1-x)).

1, -1, 1, -1, -2, 1, -1, 0, -3, 1, -1, 0, 2, -4, 1, -1, 0, 0, 5, -5, 1, -1, 0, 0, -2, 9, -6, 1, -1, 0, 0, 0, -7, 14, -7, 1, -1, 0, 0, 0, 2, -16, 20, -8, 1, -1, 0, 0, 0, 0, 9, -30, 27, -9, 1, -1, 0, 0, 0, 0, -2, 25, -50, 35, -10, 1, -1, 0, 0, 0, 0, 0, -11, 55, -77, 44, -11, 1, -1, 0, 0, 0, 0, 0, 2, -36, 105, -112, 54, -12, 1

Offset: 0

Paul Barry, Nov 08 2004

			Triangle begins as:
   1;
  -1,  1;
  -1, -2,  1;
  -1,  0, -3,  1;
  -1,  0,  2, -4,  1;
  -1,  0,  0,  5, -5,   1;
  -1,  0,  0, -2,  9,  -6,   1;
  -1,  0,  0,  0, -7,  14,  -7,  1;
  -1,  0,  0,  0,  2, -16,  20, -8,  1;
  -1,  0,  0,  0,  0,   9, -30, 27, -9,  1;

G. C. Greubel, Rows n = 0..49 of the triangle, flattened, flattened
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

Cf. A000007, A000027, A077961, A080956, A098601.
Cf. A154955, A157142, A280560, A355021.
Row sums are A100219.
Matrix inverse of A100100.
Apart from signs, same as A098599.
Very similar to triangle A111125.

A100218:= func< n,k | n eq 0 select 1 else (-1)^(n+k)*(Binomial(k,n-k) + Binomial(k-1,n-k-1)) >;
[A100218(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Mar 28 2024

T[0,0]:= 1; T[1,1]:= 1; T[1,0]:= -1; T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, T[n- 1,k] +T[n-1,k-1] -2*T[n-2,k-1] +T[n-3,k-1]]; Table[T[n,k], {n,0,14}, {k,0,n} ]//Flatten (* G. C. Greubel, Mar 13 2017 *)

def A100218(n,k): return 1 if n==0 else (-1)^(n+k)*(binomial(k,n-k) + binomial(k-1,n-k-1))
flatten([[A100218(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Mar 28 2024

Sum_{k=0..n} T(n, k) = A100219(n) (row sums).
Number triangle T(n, k) = (-1)^(n-k)*(binomial(k, n-k) + binomial(k-1, n-k-1)), with T(0, 0) = 1. - Paul Barry, Nov 09 2004
T(n,k) = T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 09 2014
From G. C. Greubel, Mar 28 2024: (Start)
T(n, n-1) = A000027(n), n >= 1.
T(n, n-2) = -A080956(n-1), n >= 2.
T(2*n, n) = A280560(n).
T(2*n-1, n) = A157142(n-1), n >= 1.
T(2*n+1, n) = -A000007(n) = A154955(n+2).
T(3*n, n) = T(4*n, n) = A000007(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A355021(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A098601(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = -1 + 2*A077961(n) + A077961(n-2). (End)
From Peter Bala, Apr 28 2024: (Start)
This Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x*(1 - x) and hence belongs to the hitting time subgroup of the Riordan group (see Peart and Woan for properties of this subgroup).
T(n,k) = [x^(n-k)] (1/c(x))^n, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. In general the (n, k)-th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)

A355020 a(n) = (-1)^n * A000045(n) + 1.

Original entry on oeis.org

1, 0, 2, -1, 4, -4, 9, -12, 22, -33, 56, -88, 145, -232, 378, -609, 988, -1596, 2585, -4180, 6766, -10945, 17712, -28656, 46369, -75024, 121394, -196417, 317812, -514228, 832041, -1346268, 2178310, -3524577, 5702888, -9227464, 14930353, -24157816, 39088170

Offset: 0

Clark Kimberling, Jun 21 2022

There are the partial sums of F(1) - F(2) + F(3) - F(4) + F(5) - ... .
Closely related (Lucas, A000032) partial sums of L(1) - L(2) + L(3) - L(4) + L(5) - ... are given by A355021.
Apart from signs, same as A008346 and A119282.

			a(0) = 1;
a(1) = 1 - 1 = 0;
a(2) = 1 - 1 + 2 = 2;
a(3) = 1 - 1 + 2 - 3 = -1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,-1).

Cf. A000045, A008346, A119282, A355018, A355019, A355021.

[1 - Fibonacci(-n): n in [0..50]]; // G. C. Greubel, Mar 17 2024

f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
Table[(-1)^n f[n] + 1, {n, 0, 40}]   (* this sequence *)
Table[(-1)^n g[n] - 1, {n, 0, 40}]   (* A355021 *)
1 - Fibonacci[-Range[0, 50]] (* G. C. Greubel, Mar 17 2024 *)

a(n) = (-1)^n*fibonacci(n) + 1; \\ Michel Marcus, Jun 24 2022

[1 - fibonacci(-n) for n in range(51)] # G. C. Greubel, Mar 17 2024

a(n) = 2*a(n-2) - a(n-3) for n > 2.

G.f.: 1/(1 - 2*x^2 + x^3).

A355018 Partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ..., where F = A000045 and L = A000032.

Original entry on oeis.org

1, 0, 1, -2, 0, -4, -1, -8, -3, -14, -6, -24, -11, -40, -19, -66, -32, -108, -53, -176, -87, -286, -142, -464, -231, -752, -375, -1218, -608, -1972, -985, -3192, -1595, -5166, -2582, -8360, -4179, -13528, -6763, -21890, -10944, -35420, -17709, -57312, -28655

Offset: 0

Clark Kimberling, Jun 16 2022

The closely related partial sums of L(1) - F(1) + L(2) - F(2) + L(3) - F(3) + .... are given by A355019.

			a(0) = 1
a(1) = 1 - 1 = 0
a(2) = 1 - 1 + 1 = 1
a(3) = 1 - 1 + 1 - 3 = -2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1).

Cf. A000032, A000045, A355019, A355020, A355021.

F:=Fibonacci; [2 - (((n+1) mod 2)*F(Floor((n+2)/2)) + 2*(n mod 2)*F(Floor((n+3)/2))) : n in [0..60]]; // G. C. Greubel, Mar 17 2024

f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1[n_] := If[OddQ[n], 2 - 2 f[(n + 3)/2], 2 - f[(n + 2)/2]]
f2 = Table[f1[n], {n, 0, 20}]  (* this sequence *)
g1[n_] := If[OddQ[n], -2 + 2 f[(n + 3)/2], -2 + f[(n + 8)/2]]
g2 = Table[g1[n], {n, 0, 20}]  (* A355019 *)
LinearRecurrence[{1,1,-1,1,-1}, {1,0,1,-2,0}, 61] (* G. C. Greubel, Mar 17 2024 *)

f=fibonacci; [2 - (((n+1)%2)*f(((n+2)//2)) +2*(n%2)*f((n+3)//2)) for n in range(61)] # G. C. Greubel, Mar 17 2024

a(n) = 2 - 2*F((n+3)/2) if n is odd, a(n) = 2 - F((n+2)/2) if n is even, where F = A000045 (Fibonacci numbers).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n >= 5.
G.f.: (1 - x - 2*x^3)/((1 - x)*(1 - x^2 - x^4)).
From G. C. Greubel, Mar 17 2024: (Start)
a(n) = (1/2)*Sum_{j=0..n} ( (1+(-1)^j)*Fibonacci(floor((j+3)/2)) - (1 - (-1)^j)*Lucas(floor((j+1)/2)) ).
a(n) = 2 - (1/2)*( (1+(-1)^n)*Fibonacci(floor((n+2)/2)) + 2*(1-(-1)^n)* Fibonacci(floor((n+3)/2)) ). (End)

A355019 Partial sums of L(1) - F(1) + L(2) - F(2) + L(3) - F(3) + ..., where L = A000032 and F = A000045.

Original entry on oeis.org

1, 0, 3, 2, 6, 4, 11, 8, 19, 14, 32, 24, 53, 40, 87, 66, 142, 108, 231, 176, 375, 286, 608, 464, 985, 752, 1595, 1218, 2582, 1972, 4179, 3192, 6763, 5166, 10944, 8360, 17709, 13528, 28655, 21890, 46366, 35420, 75023, 57312, 121391, 92734, 196416, 150048

Offset: 0

Clark Kimberling, Jun 16 2022

The closely related partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ... are given by A355018.

			a(0) = 1
a(1) = 1 - 1 = 0
a(2) = 1 - 1 + 3 = 3
a(3) = 1 - 1 + 3  - 1 = 2.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1).

Cf. A000032, A000045, A355018, A355020, A355021.

F:=Fibonacci; [(((n+1) mod 2)*F(Floor(n/2)+4) + 2*(n mod 2)*F(Floor((n+3)/2))) - 2: n in [0..60]]; // G. C. Greubel, Mar 17 2024

f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1[n_] := If[OddQ[n], 2 - 2 f[(n + 3)/2], 2 - f[(n + 2)/2]]
f2 = Table[f1[n], {n, 0, 20}]  (* A355018 *)
g1[n_] := If[OddQ[n], -2 + 2 f[(n + 3)/2], -2 + f[(n + 8)/2]]
g2 = Table[g1[n], {n, 0, 20}]  (* this sequence *)
LinearRecurrence[{1,1,-1,1,-1}, {1,0,3,2,6}, 61] (* G. C. Greubel, Mar 17 2024 *)

f=fibonacci; [(((n+1)%2)*f((n//2)+4) +2*(n%2)*f((n+3)//2)) -2 for n in range(61)] # G. C. Greubel, Mar 17 2024

a(n) = -2 + 2 F((n+3)/2) if n is odd, a(n) = - 2 + F((n+8)/2) if n is even, where F = A000045 (Fibonacci numbers).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n >= 5.
G.f.: (1 - x + 2*x^2)/((1 - x)*(1 - x^2 - x^4)).
From G. C. Greubel, Mar 17 2024: (Start)
a(n) = (1/2)*Sum_{j=0..n} ( (1+(-1)^j)*Lucas(floor(j/2) +1) - (1-(-1)^j) *Fibonacci(floor((j+1)/2)) ).
a(n) = (1/2)*( (1+(-1)^n)*Fibonacci(floor(n/2) +4) + 2*(1-(-1)^n)* Fibonacci(floor((n+3)/2)) ) - 2. (End)

cp's OEIS Frontend

A100218 Riordan array ((1-2*x)/(1-x), (1-x)).

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A355020 a(n) = (-1)^n * A000045(n) + 1.

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A355018 Partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ..., where F = A000045 and L = A000032.

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A355019 Partial sums of L(1) - F(1) + L(2) - F(2) + L(3) - F(3) + ..., where L = A000032 and F = A000045.

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