A010674 -id:A010674 - cp's OEIS frontend

A115634 Expansion of (1-4*x^2)/(1-x^2).

1, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0, -3, 0

Offset: 0

Paul Barry, Jan 27 2006

Row sums of number triangle A115633.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010674, A115633.

[4*0^n -3*(1+(-1)^n)/2: n in [0..100]]; // G. C. Greubel, Nov 23 2021

Join[{1}, -3*Mod[Range[100] -1, 2]] (* G. C. Greubel, Nov 23 2021 *)
CoefficientList[Series[(1-4x^2)/(1-x^2),{x,0,100}],x] (* or *) LinearRecurrence[{0,1},{1,0,-3},100] (* or *) PadRight[{1},100,{-3,0}] (* Harvey P. Dale, Dec 06 2024 *)

[1]+[-3*((n-1)%2) for n in (1..100)] # G. C. Greubel, Nov 23 2021

a(n) = 4*0^n - 3*(1 + (-1)^n)/2.
a(n) = Sum_{k=0..n} A115633(n, k).
From G. C. Greubel, Nov 23 2021: (Start)
a(n) = 1 if n = 0, otherwise a(n) = -A010674(n-1).
E.g.f.: 4 - 3*cosh(x). (End)

A160019 Triangle: Lodumo_2 applied to each row of Pascal's triangle .

Original entry on oeis.org

1, 1, 3, 1, 0, 3, 1, 3, 5, 7, 1, 0, 2, 4, 3, 1, 3, 0, 2, 5, 7, 1, 0, 3, 2, 5, 4, 7, 1, 3, 5, 7, 9, 11, 13, 15, 1, 0, 2, 4, 6, 8, 10, 12, 3, 1, 3, 0, 2, 4, 6, 8, 10, 5, 7, 1, 0, 3, 2, 4, 6, 8, 10, 5, 12, 7, 1, 3, 5, 7, 0, 2, 4, 6, 9, 11, 13, 15, 1, 0, 2, 4, 3, 6, 8, 10, 5, 12, 14, 16, 7

Offset: 0

Philippe Deléham, Apr 29 2009, May 02 2009

			Triangle begins:
  1;
  1, 3;
  1, 0, 3;
  1, 3, 5, 7;
  1, 0, 2, 4, 3;
  1, 3, 0, 2, 5, 7; ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
OEIS wiki, Lodumo transform

Row sums are A160020.

Cf. A007318, A047999, A261363.

\\ here S(n,k) is A047999.
S(n,k)={bitand(n-k, k)==0}
row(n)={my(v=vector(n+1), b=0); for(k=0, n, if(S(n,k), b++; v[1+k]=2*b-1, v[1+k]=2*(k-b))); v}
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Feb 02 2020

T(n,0)=A000012(n)=1; T(n,1)=A010674(n). - Philippe Deléham, Nov 15 2011

A174971 Periodic sequence: Repeat 3, -3.

Original entry on oeis.org

3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3

Offset: 0

Klaus Brockhaus, Apr 04 2010

Interleaving of A010701 and -A010701; signed version of A010701.
Essentially first differences of A010674.
Inverse binomial transform of 3 followed by A000004.
Second inverse binomial transform of A010701.
Third inverse binomial transform of A007283.
Fourth inverse binomial transform of A000244 without initial term 1.
Fifth inverse binomial transform of A164346.
Sixth inverse binomial transform of A005053 without initial term 1.
Seventh inverse binomial transform of A169604.
Eighth inverse binomial transform of A169634.
Ninth inverse binomial transform of A103333 without initial term 1.
Tenth inverse binomial transform of A013708.
Eleventh inverse binomial transform of A093138 without initial term 1.

Index entries for linear recurrences with constant coefficients, signature (-1).

Cf. A010701 (all 3's sequence), A000004 (all zeros sequence), A007283 (3*2^n), A000244 (powers of 3), A164346 (3*4^n), A005053 (expand (1-2x)/(1-5x)), A169604 (3*6^n), A169634 (3*7^n), A103333 (expand (1-5x)/(1-8x)), A013708 (3^(2n+1)), A093138 (expand (1-7x)/(1-10x)).

&cat[ [3, -3]: n in [0..41] ];
[ 3*(-1)^n: n in [0..83] ];

PadRight[{},120,{3,-3}] (* or *) NestList[-1#&,3,120] (* Harvey P. Dale, Dec 30 2023 *)

a(n)=3*(-1)^n \\ Charles R Greathouse IV, Jun 13 2013

a(n) = 3*(-1)^n.
a(n) = -a(n-1) for n > 0; a(0) = 3.
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = -3.
G.f.: 3/(1+x).

cp's OEIS Frontend

A115634 Expansion of (1-4*x^2)/(1-x^2).

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A160019 Triangle: Lodumo_2 applied to each row of Pascal's triangle .

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A174971 Periodic sequence: Repeat 3, -3.

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Magma

Mathematica

PARI

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