A309216 -id:A309216 - cp's OEIS frontend

A309214 a(0)=0; thereafter a(n) = a(n-1)+n if a(n-1) even, otherwise a(n) = a(n-1)-n.

0, 1, -1, -4, 0, 5, -1, -8, 0, 9, -1, -12, 0, 13, -1, -16, 0, 17, -1, -20, 0, 21, -1, -24, 0, 25, -1, -28, 0, 29, -1, -32, 0, 33, -1, -36, 0, 37, -1, -40, 0, 41, -1, -44, 0, 45, -1, -48, 0, 49, -1, -52, 0, 53, -1, -56, 0, 57, -1, -60, 0, 61, -1, -64, 0, 65, -1, -68, 0, 69, -1, -72, 0, 73, -1

Offset: 0

N. J. A. Sloane, Aug 10 2019

A003816 and A309215 have the same terms except for signs.

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

Cf. A003816, A309529, A309215, A309216, A319217.

t:=0;
a:=[t]; M:=100;
for i from 1 to M do
if (t mod 2) = 0 then t:=t+i else t:=t-i; fi;
a:=[op(a),t]; od:
a;

concat(0, Vec(x*(1 - 2*x - x^2) / ((1 - x)*(1 + x^2)^2) + O(x^80))) \\ Colin Barker, Aug 13 2019

a(4t)=0, a(4t+1)=4t+1, a(4t+2)=-1, a(4t+3)=-(4t+4).
From Colin Barker, Aug 13 2019: (Start)
G.f.: x*(1 - 2*x - x^2) / ((1 - x)*(1 + x^2)^2).
a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = (-2 + (1+i)*(-i)^n + (1-i)*i^n + 2*i*((-i)^n-i^n)*n) / 4 where i=sqrt(-1).
(End)
E.g.f.: (1/2)*((1+2*x)*cos(x)-cosh(x)+sin(x)-sinh(x)). - Stefano Spezia, Aug 13 2019 after Colin Barker

A309215 a(0)=0; thereafter a(n) = a(n-1)+n if a(n-1) odd, otherwise a(n) = a(n-1)-n.

Original entry on oeis.org

0, -1, 1, 4, 0, -5, 1, 8, 0, -9, 1, 12, 0, -13, 1, 16, 0, -17, 1, 20, 0, -21, 1, 24, 0, -25, 1, 28, 0, -29, 1, 32, 0, -33, 1, 36, 0, -37, 1, 40, 0, -41, 1, 44, 0, -45, 1, 48, 0, -49, 1, 52, 0, -53, 1, 56, 0, -57, 1, 60, 0, -61, 1, 64, 0, -65, 1, 68, 0, -69, 1, 72, 0, -73, 1, 76, 0, -77

Offset: 0

N. J. A. Sloane, Aug 10 2019

A003816 and A309214 have the same terms except for signs.

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

Cf. A003816, A309529, A309214, A309216, A309217.

t:=0;
a:=[t]; M:=100;
for i from 1 to M do
if (t mod 2) = 1 then t:=t+i else t:=t-i; fi;
a:=[op(a),t]; od:
a;

nxt[{n_,a_}]:={n+1,If[OddQ[a],a+n+1,a-n-1]}; NestList[nxt,{0,0},80][[All,2]] (* Harvey P. Dale, Sep 26 2021 *)

concat(0, Vec(-x*(1 - 2*x - x^2) / ((1 - x)*(1 + x^2)^2) + O(x^40))) \\ Colin Barker, Aug 13 2019

a(4t)=0, a(4t+1)=-(4t+1), a(4t+2)=1, a(4t+3)=4t+4.
From Colin Barker, Aug 13 2019: (Start)
G.f.: -x*(1 - 2*x - x^2) / ((1 - x)*(1 + x^2)^2).
a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = 1/2 - (1/4 - i/4)*((-i)^n+i^(1+n)) - (1/2)*i*((-i)^n-i^n)*(1+n) where i=sqrt(-1).
(End)

A309217 The sequence is {a(n), n>=0}, the concatenation of the binary expansions of the absolute values |a(n)| is {b(n), n>=0}; start with a(0)=0; thereafter a(n) = a(n-1)+n if b(n-1)=0, otherwise a(n) = a(n-1)-n.

Original entry on oeis.org

0, 1, -1, -4, -8, -3, 3, -4, 4, 13, 23, 12, 0, -13, -27, -42, -26, -9, -27, -8, 12, -9, -31, -8, -32, -57, -31, -58, -86, -115, -145, -176, -144, -111, -77, -112, -148, -111, -149, -188, -228, -187, -229, -272, -316, -271, -317, -270, -318, -269, -319, -370, -318, -371, -317, -372, -316

Offset: 0

N. J. A. Sloane, Aug 10 2019

The b-sequence (A309218) is 0; 1; 1; 1, 0, 0; 1, 0, 0, 0; 1, 1; 1, 1; 1, 0, 0; 1, 0, 0; ... Note that we write the binary expansions in human order (as in A309216), with high-order bits on the left.

This is a base-2 analog of A309216.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

Cf. A003816, A309529, A309214, A309215, A309216, A309218.

t:=0;
a:=[0]; b:=[]; M:=100;
for i from 1 to M do
v1:=convert(abs(t),base,2); L:=nops(v1);
v2:=[seq(v1[L-i+1],i=1..L)];
b:=[op(b),op(v2)];
if (b[i] mod 2) = 0 then t:=t+i else t:=t-i; fi;
a:=[op(a),t];
od:
a; # A309217
b; # A309218

cp's OEIS Frontend

A309214 a(0)=0; thereafter a(n) = a(n-1)+n if a(n-1) even, otherwise a(n) = a(n-1)-n.

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A309215 a(0)=0; thereafter a(n) = a(n-1)+n if a(n-1) odd, otherwise a(n) = a(n-1)-n.

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A309217 The sequence is {a(n), n>=0}, the concatenation of the binary expansions of the absolute values |a(n)| is {b(n), n>=0}; start with a(0)=0; thereafter a(n) = a(n-1)+n if b(n-1)=0, otherwise a(n) = a(n-1)-n.

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Maple