A122771 -id:A122771 - cp's OEIS frontend

A123947 Expansion of x^2(1+x-x^2)/(1-2x-4*x^2+x^3+x^4).

0, 1, 3, 9, 29, 90, 284, 890, 2797, 8780, 27574, 86581, 271881, 853732, 2680833, 8418132, 26433983, 83005929, 260648825, 818469251, 2570093890, 8070410030, 25342077544, 79577232067, 249882270390, 784660981474, 2463931734897

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 26 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,4,-1,-1).

Cf. A122773, A122771.

a:=[0,1,3,9];; for n in [5..30] do a[n]:=2*a[n-1]+4*a[n-2]-a[n-3] -a[n-4]; od; a; # G. C. Greubel, Aug 05 2019

R:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x^2*(1+x-x^2)/(1-2*x-4*x^2+x^3+x^4) )); // G. C. Greubel, Aug 05 2019

seq(coeff(series(x^2*(1+x-x^2)/(1-2*x-4*x^2+x^3+x^4), x, n+1), x, n), n = 1..30); # G. C. Greubel, Aug 05 2019

M = {{0,-1,-1,0,1}, {-1,0,0,0,-1}, {-1,0,1,0,-1}, {0,0,-1,0,0}, {1,-1, -1,0,1}}; v[1] = {0,0,0,0,1}; v[n_]:= v[n] = M.v[n-1]; Table[v[n][[1]], {n, 30}]
CoefficientList[Series[x^2*(1+x-x^2)/(1-2*x-4*x^2+x^3+x^4), {x, 0, 30}], x] (* G. C. Greubel, Aug 05 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x^2*(1+x-x^2)/(1-2*x-4*x^2+x^3+x^4))) \\ G. C. Greubel, Aug 05 2019

a=(x^2*(1+x-x^2)/(1-2*x-4*x^2+x^3+x^4)).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Aug 05 2019

A123951 A polynomial of matrices is used to make a triangular sequence. The upper triangular antidiagonal Steinbach matrices are summed over their characteristic polynomial triangular sequences to give a new sequence of matrices: the characteristic polynomials of these new summed matrices are, then, used to make up this triangular sequence.

Original entry on oeis.org

1, 1, -1, -1, -1, 1, -1, -3, 4, -1, 37, -88, 69, -19, 1, 10879, -14344, 6831, -1375, 99, -1, -4322473, -40529664, -17486038, 3188841, -40896, -2346, 1, -11384127259974047, -783824545942228, 1058675233347, 505084925760, -64007100, -32568519, 23164, -1, -121986767767877481129923

Offset: 1

Roger L. Bagula and Gary W. Adamson, Oct 26 2006

Basically everything is done twice. The determinants get very large very fast for these matrices: Table[Det[w[[d]]], {d, 1, Length[w]}] {1, -1, -1, 37, 10879, -4322473, -11384127259974047, -121986767767877481129923, -323621163456130064854374309178100414058036559, 189651898964129252384795657180434913387386019400002936829101989683}

			{1},
{1, -1},
{-1, -1, 1},
{-1, -3, 4, -1},
{37, -88, 69, -19,1},
{10879, -14344, 6831, -1375, 99, -1},
{-4322473, -40529664, -17486038, 3188841, -40896, -2346, 1}

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31

Cf. A122765, A122771.

An[d_] := Table[If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; a = Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d],x], x], {d, 1, 20}]]; w = Join[{{{1}}}, Table[Sum[MatrixPower[a[[n]][[m + 1]]*An[n], m - 1], {m, 0, Length[a[[n]]] - 1}], {n, 2, 10}]]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[w[[d]], x], x], {d, 1, Length[w]}]]; Flatten[%]

p(n,x) = CharacteristicPolynomial(a(i,j)) p(n,x)->t(n,m) b(i,j) = Sum[t(i,j).a(j,k).{j,1,m}] p'(n,x) = CharacteristicPolynomial(b(i,j)) p'(n,x)->t'(n,m).

cp's OEIS Frontend

A123947 Expansion of x^2(1+x-x^2)/(1-2x-4*x^2+x^3+x^4).

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cp's OEIS Frontend

A123947 Expansion of x^2*(1+x-x^2)/(1-2*x-4*x^2+x^3+x^4).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A123947 Expansion of x^2(1+x-x^2)/(1-2x-4*x^2+x^3+x^4).