A077899 -id:A077899 - cp's OEIS frontend

A175722 a(n) = -a(n-1) + a(n-2) - F(-n) + 1, a(0) = 1, a(1) = -1, where F() = Fibonacci numbers A000045.

1, -1, 4, -6, 14, -24, 47, -83, 152, -268, 476, -832, 1453, -2517, 4348, -7474, 12810, -21880, 37275, -63335, 107376, -181656, 306744, -517056, 870169, -1462249, 2453812, -4112478, 6884102, -11510808, 19226951, -32084027, 53489288, -89097892, 148290068

Offset: 0

Roger L. Bagula, Dec 04 2010

			G.f. = 1 - x + 4*x^2 - 6*x^3 + 14*x^4 - 24*x^5 + 47*x^6 - 83*x^7 + 152*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,3,1,-3,1).

Cf. m=1: A077899, m large: A077925.

Cf. A000032, A000045, A006478.

List([0..40], n-> 1 + (-1)^n*(n*Lucas(1,-1,n+1)[2] + 7*Fibonacci(n))/5 ); # G. C. Greubel, Dec 04 2019

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1+x-x^2)^2))); // G. C. Greubel, Aug 14 2018

with(combinat); seq( 1 + (-1)^n*(n*fibonacci(n+2) + (n+7)*fibonacci(n))/5, n=0..40); # G. C. Greubel, Dec 04 2019

f[x_, m_] = ExpandAll[(x -x^(m+1))*(1-x-x^2) -(1 -2*x +x^(m+1))];
g[x_, n_] = ExpandAll[x^(m + 3)*f[1/x, m]];
a = Table[Table[SeriesCoefficient[Series[1/g[x, m], {x, 0, 20}], n], {n, 0, 20}], {m, 1, 20}]
CoefficientList[Series[1/((1-x)(1+x-x^2)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 13 2014 *)
RecurrenceTable[{a[0]==1,a[1]==-1,a[n]==-a[n-1]+a[n-2]-Fibonacci[-n]+1},a,{n,40}] (* Harvey P. Dale, May 12 2018 *)
Table[1 + (-1)^n*(n*LucasL[n+1] + 7*Fibonacci[n])/5, {n,0,40}] (* G. C. Greubel, Dec 04 2019 *)

{a(n) = if( n<0, polcoeff( x^5 / ((1 - x) * (1 - x - x^2)^2) + x * O(x^-n), -n), polcoeff( 1 / ((1 - x) * (1 + x - x^2)^2) + x * O(x^n), n))}; /* Michael Somos, Mar 11 2014 */

vector(41, n, my(f=fibonacci); 1 -(-1)^n*((n-1)*f(n+1) +(n+6)*f(n-1))/5 ) \\ G. C. Greubel, Dec 04 2019

[1 + (-1)^n*(n*lucas_number2(n+1, 1,-1) + 7*fibonacci(n))/5 for n in (0..40)] # G. C. Greubel, Dec 04 2019

G.f.: 1/(- x^m + 1 - x^(1 + m) + x + 3*x^(2 + m) - 2*x^2 - x^(3 + m)) for m=2.
G.f.: 1 / ((1 - x) * (1 + x - x^2)^2). - Michael Somos, Mar 11 2014
a(n) = A006478(-2-n) for all n in Z. - Michael Somos, Mar 11 2014
a(n) = 1 + (-1)^n*(n*Lucas(n+1) + 7*Fibonacci(n))/5. - G. C. Greubel, Dec 04 2019
E.g.f.: exp(-x/2)*(25*exp(3*x/2) - 15*x*cosh(sqrt(5)*x/2) + sqrt(5)*(5*x - 14)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Jul 24 2022

A175721 Array T(n,m) read by antidiagonals: the coefficient of [x^m] of 1/(-x^n + 1 - x^(1+n) + x + 3x^(2+n) - 2x^2 - x^(3+n)) in row n, column 1 <= m.

Original entry on oeis.org

0, 3, -1, -3, 4, -1, 10, -6, 3, -1, -18, 14, -4, 3, -1, 42, -24, 10, -5, 3, -1, -87, 47, -19, 12, -5, 3, -1, 190, -83, 42, -22, 11, -5, 3, -1, -405, 152, -84, 45, -20, 11, -5, 3, -1

Offset: 1

Roger L. Bagula, Dec 04 2010

Antidiagonal sums are: 0, 2, 0, 6, -6, 25, -50, 135, -304, ...

For large n, the rows approach A077925.

			The array starts in row n=1 with columns m >= 1 as
   0, 3, -3, 10, -18, 42, -87, 190, -405, 873, ... A077899
  -1, 4, -6, 14, -24, 47, -83, 152, -268, 476, ... A175722
  -1, 3, -4, 10, -19, 42, -84, 174, -353, 726, ...
  -1, 3, -5, 12, -22, 45, -87, 174, -340, 670, ...
  -1, 3, -5, 11, -20, 42, -83, 169, -339, 686, ...
  -1, 3, -5, 11, -21, 44, -86, 173, -343, 685, ...
  -1, 3, -5, 11, -21, 43, -84, 170, -339, 681, ...
  -1, 3, -5, 11, -21, 43, -85, 172, -342, 685, ...
  -1, 3, -5, 11, -21, 43, -85, 171, -340, 682, ...
  -1, 3, -5, 11, -21, 43, -85, 171, -341, 684, ...

Cf. A000045, A077899, A077925, A175722.

A175721 := proc(m,k) 1/(-x^m + 1 - x^(1 + m) + x + 3*x^(2 + m) - 2* x^2 - x^(3 + m)) ; coeftayl(%,x=0,k) ; end proc: # R. J. Mathar, Dec 22 2010

f[x_, m_] = ExpandAll[(x - x^(m + 1))*(1 - x - x^2) - (1 - 2*x + x^(m + 1))];
g[x_, n_] = ExpandAll[x^(m + 3)*f[1/x, m]];
a = Table[Table[SeriesCoefficient[
      Series[1/g[x, m], {x, 0, 10}], n], {n, 0, 10}], {m, 1, 10}];
Table[Table[a[[m, n - m + 1]], {m, 1, n - 1}], {n, 1, 10}]
Flatten[%]

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

cp's OEIS Frontend

A175722 a(n) = -a(n-1) + a(n-2) - F(-n) + 1, a(0) = 1, a(1) = -1, where F() = Fibonacci numbers A000045.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A175721 Array T(n,m) read by antidiagonals: the coefficient of [x^m] of 1/(-x^n + 1 - x^(1+n) + x + 3x^(2+n) - 2x^2 - x^(3+n)) in row n, column 1 <= m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

cp's OEIS Frontend

A175722 a(n) = -a(n-1) + a(n-2) - F(-n) + 1, a(0) = 1, a(1) = -1, where F() = Fibonacci numbers A000045.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A175721 Array T(n,m) read by antidiagonals: the coefficient of [x^m] of 1/(-x^n + 1 - x^(1+n) + x + 3*x^(2+n) - 2*x^2 - x^(3+n)) in row n, column 1 <= m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A175721 Array T(n,m) read by antidiagonals: the coefficient of [x^m] of 1/(-x^n + 1 - x^(1+n) + x + 3x^(2+n) - 2x^2 - x^(3+n)) in row n, column 1 <= m.