A138003 -id:A138003 - cp's OEIS frontend

A138112 a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4), a(0)=a(1)=a(2)=0, a(3)=1, a(4)=3.

0, 0, 0, 1, 3, 5, 5, 0, -13, -34, -55, -55, 0, 144, 377, 610, 610, 0, -1597, -4181, -6765, -6765, 0, 17711, 46368, 75025, 75025, 0, -196418, -514229, -832040, -832040, 0, 2178309, 5702887, 9227465, 9227465, 0, -24157817, -63245986, -102334155, -102334155

Offset: 0

Paul Curtz, May 04 2008

sign

Obeys also the recurrence a(n)=5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+2a(n-5), so the sequence is identical to its fifth differences (cf. A135356). a(n) = A138110(0,n): if A138110 is interpreted as an array with five rows, this is the top row.
The first differences are represented by A100334(n-1).
The 2nd differences are represented by A103311(n).
The 3rd differences are essentially represented by -A138003(n-2).
The 4th differences are represented by -A105371(n).
A102312 contains the absolute values of the terms which occur in pairs, for example a(5)=a(6)=5=A102312(1), a(10)=a(11)= -55 = -A102312(2).
Inverse BINOMIAL transform yields two zeros followed by A105384. - R. J. Mathar, Jul 04 2008

Index entries for linear recurrences with constant coefficients, signature (3, -4, 2, -1).

Cf. A138003, A103311, A105371.

CoefficientList[Series[x^3/(1-3x+4x^2-2x^3+x^4),{x,0,45}],x] (* or *) LinearRecurrence[{3,-4,2,-1},{0,0,0,1},45] (* Harvey P. Dale, Jun 22 2011 *)

O.g.f.: x^3/(1-3x+4x^2-2x^3+x^4). - R. J. Mathar, Jul 04 2008

Edited and extended by R. J. Mathar, Jul 04 2008

A138019 Period 5: repeat [1, 1, 0, -1, -1].

Original entry on oeis.org

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

Offset: 0

Paul Curtz, May 01 2008

			G.f. = 1 + x - x^3 - x^4 + x^5 + x^6 - x^8 - x^9 + x^10 + x^11 - x^13 + ...

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

Cf. A080891, A105368.

A138019 := proc(n)
    op( 1+modp(n,5),[1,1,0,-1,-1]) ;
end proc:
seq(A138019(n),n=0..30) ; # R. J. Mathar, Feb 12 2021

a[ n_] := Sign[2 - Mod[n, 5]]; (* Michael Somos, Jun 17 2015 *)
PadRight[{},120,{1,1,0,-1,-1}] (* Harvey P. Dale, Oct 06 2023 *)

a(n)=sign(2-n%5) /* Jaume Oliver Lafont, Aug 28 2009 */

Inverse binomial transform of A138003.
O.g.f.: (1+x)(x^2+x+1)/(1+x+x^2+x^3+x^4). - R. J. Mathar, Jun 28 2008
Euler transform of length 5 sequence [ 1, -1, -1, 0, 1]. - Michael Somos, Jun 17 2015
G.f.: (1 - x^2 ) * (1 - x^3) / ((1 - x) * (1 - x^5)). - Michael Somos, Jun 17 2015
a(n) = -a(-1-n) = a(n+5) for all n in Z. - Michael Somos, Jun 17 2015

A138110 Table T(d,n) read column by column: the n-th term in the sequence of the d-th differences of A138112, d=0..4.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 0, 1, 0, -1, 0, 1, 1, -1, -1, 1, 2, 0, -2, -1, 3, 2, -2, -3, 0, 5, 0, -5, -3, 3, 5, -5, -8, 0, 8, 0, -13, -8, 8, 13, -13, -21, 0, 21, 13, -34, -21, 21, 34, 0, -55, 0, 55, 34, -34, -55, 55, 89, 0, -89, 0, 144, 89, -89, -144, 144, 233, 0, -233, -144, 377, 233, -233, -377, 0, 610, 0, -610, -377, 377

Offset: 0

Paul Curtz, May 04 2008

Ignoring signs, the sequence contains A000045(2)=1 ten times and each of the following Fibonacci numbers A000045(i>2) four times.

			All 5 rows of the table T(d,n) are:
.0,.0,.0,.1,.3,.5,.5,..0,-13,-34,-55,-55,...0,.144,...
.0,.0,.1,.2,.2,.0,-5,-13,-21,-21,..0,.55,.144,.233,...
.0,.1,.1,.0,-2,-5,-8,.-8,..0,.21,.55,.89,..89,...0,...
.1,.0,-1,-2,-3,-3,.0,..8,.21,.34,.34,..0,.-89,-233,...
-1,-1,-1,-1,.0,.3,.8,.13,.13,..0,-34,-89,-144,-144,...

Cf. A102312 (A049666), A099100, A134489, A134490, A134491.

T(0,n)=A138112(n). T(d,n)= T(d-1,n+1)-T(d-1,n), d=1..4.
T(1,n)=A100334(n-1). T(2,n)=A103311(n). T(3,n) = -A138003(n-2). T(4,n)= -A105371(n).
sum_(d=0..4) T(d,n)=0 (columns sum to zero).

Edited by R. J. Mathar, Jul 04 2008

cp's OEIS Frontend

A138112 a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4), a(0)=a(1)=a(2)=0, a(3)=1, a(4)=3.

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A138019 Period 5: repeat [1, 1, 0, -1, -1].

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A138110 Table T(d,n) read column by column: the n-th term in the sequence of the d-th differences of A138112, d=0..4.

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