A117377 -id:A117377 - cp's OEIS frontend

A117378 Expansion of (1-4*x)/(1-x+x^2).

1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3, -4, -1, 3, 4, 1, -3

Offset: 0

Paul Barry, Mar 10 2006

Row sums of number triangle A117377.

Period 6: repeat [1, -3, -4, -1, 3, 4]. - Philippe Deléham, Nov 03 2008

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,-1).

Cf. A117377.

[(1+(-n mod 3))^(n mod 3)*(-1)^Floor((n+2)/3) : n in [0..100]]; // Wesley Ivan Hurt, Aug 31 2014

A117378:=n->(1+(-n mod 3))^(n mod 3)*(-1)^floor((n+2)/3): seq(A117378(n), n=0..100); # Wesley Ivan Hurt, Aug 31 2014

CoefficientList[Series[(1 - 4 x)/(1 - x + x^2), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{1,-1},{1,-3},100] (* Harvey P. Dale, Sep 27 2018 *)

G.f.: (1-4*x)/(1-x+x^2).
a(n) = Sum_{k=0..n} (-1)^(n-k) * ( C(k,n-k) + 4*C(k,n-k-1) ).
a(n) = a(n-1) - a(n-2) for n>1. [Philippe Deléham, Nov 03 2008]
a(n) = (1+(-n mod 3))^(n mod 3)*(-1)^floor((n+2)/3). - Wesley Ivan Hurt, Aug 31 2014
a(n) = (3*cos(n*Pi/3) - 7*sqrt(3)*sin(n*Pi/3))/3. - Wesley Ivan Hurt, Jun 23 2016
E.g.f.: (3*cos(sqrt(3)*x/2) - 7*sqrt(3)*sin(sqrt(3)*x/2))*exp(x/2)/3. - Ilya Gutkovskiy, Jun 27 2016

A117380 Riordan array (1/(1-4xc(x)),xc(x)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, 4, 1, 20, 5, 1, 104, 26, 6, 1, 548, 137, 33, 7, 1, 2904, 726, 178, 41, 8, 1, 15432, 3858, 954, 228, 50, 9, 1, 82128, 20532, 5100, 1242, 288, 60, 10, 1, 437444, 109361, 27233, 6701, 1601, 359, 71, 11, 1, 2331128, 582782, 145338, 35977, 8744, 2043, 442, 83, 12

Offset: 0

Paul Barry, Mar 10 2006

Triangle factors as (1,xc(x))*(1/(1-4x),x). Inverse of A117377. First row is A076035. Second row is A076025(n)-0^n. Row sums are A076025(n+1). Diagonal sums are A117381.

			Triangle begins
1,
4, 1,
20, 5, 1,
104, 26, 6, 1,
548, 137, 33, 7, 1,
2904, 726, 178, 41, 8, 1
Production array begins
4, 1
4, 1, 1
4, 1, 1, 1
4, 1, 1, 1, 1
4, 1, 1, 1, 1, 1
4, 1, 1, 1, 1, 1, 1
4, 1, 1, 1, 1, 1, 1, 1
4, 1, 1, 1, 1, 1, 1, 1, 1
... - _Philippe Deléham_, Mar 05 2013

Number triangle T(0,0)=1, T(n,k)=[k<=n]*sum{j=0..n, (j/(n-j))*C(2n-j,n-j)[k<=j]*4^(j-k)}

A117379 Expansion of (1-4x)/(1-x^2+x^3).

Original entry on oeis.org

1, -4, 1, -5, 5, -6, 10, -11, 16, -21, 27, -37, 48, -64, 85, -112, 149, -197, 261, -346, 458, -607, 804, -1065, 1411, -1869, 2476, -3280, 4345, -5756, 7625, -10101, 13381, -17726, 23482, -31107, 41208

Offset: 0

Paul Barry, Mar 10 2006

Diagonal sums of number triangle A117377.

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

CoefficientList[Series[(1-4x)/(1-x^2+x^3),{x,0,40}],x] (* or *) LinearRecurrence[ {0,1,-1},{1,-4,1},40] (* Harvey P. Dale, Nov 08 2022 *)

a(n)=sum{k=0..floor(n/2), (-1)^n*(C(k,n-2k)+4*C(k, n-2k-1))}

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A117378 Expansion of (1-4*x)/(1-x+x^2).

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A117380 Riordan array (1/(1-4xc(x)),xc(x)), c(x) the g.f. of A000108.

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A117379 Expansion of (1-4x)/(1-x^2+x^3).

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cp's OEIS Frontend

A117378 Expansion of (1-4*x)/(1-x+x^2).

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Crossrefs

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Maple

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A117380 Riordan array (1/(1-4*x*c(x)),xc(x)), c(x) the g.f. of A000108.

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A117379 Expansion of (1-4x)/(1-x^2+x^3).

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Mathematica

Formula

A117380 Riordan array (1/(1-4xc(x)),xc(x)), c(x) the g.f. of A000108.