A087168 -id:A087168 - cp's OEIS frontend

A247560 a(n) = 3a(n-1) - 4a(n-2) with a(0) = a(1) = 1.

1, 1, -1, -7, -17, -23, -1, 89, 271, 457, 287, -967, -4049, -8279, -8641, 7193, 56143, 139657, 194399, 24569, -703889, -2209943, -3814273, -2603047, 7447951, 32756041, 68476319, 74404793, -50690897, -449691863, -1146312001, -1640168551, -335257649, 5554901257

Offset: 0

Michael Somos, Sep 19 2014

			G.f. = 1 + x - x^2 - 7*x^3 - 17*x^4 - 23*x^5 - x^6 + 89*x^7 + 271*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2, pp. 251-267. See Theorem 2, p. 259.
M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
Index entries for linear recurrences with constant coefficients, signature (3,-4).

Cf. A087168, A247487, A247564, A247565.

a247560 n = a247560_list !! n
a247560_list = 1 : 1 : zipWith (-) (map (* 3) $ tail a247560_list)
                                   (map (* 4) a247560_list)
-- Reinhard Zumkeller, Sep 20 2014

I:=[1, 1]; [n le 2 select I[n] else 3*Self(n-1) - 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 04 2018

A247560:=n->simplify((1/14*I)*sqrt(7)*((3/2+(1/2*I)*sqrt(7))^n-(3/2-(1/2*I)*sqrt(7))^n)+1/2*((3/2+(1/2*I)*sqrt(7))^n+(3/2-(1/2*I)*sqrt(7))^n)): seq(A247560(n), n=0..40); # Wesley Ivan Hurt, Oct 02 2014

a[ n_] := Re[ (1 - 1/Sqrt[-7]) (3 + Sqrt[-7])^n / 2^n];
LinearRecurrence[{3,-4},{1,1},40] (* Harvey P. Dale, Jun 13 2017 *)

PARI
```
{a(n) = real( (1 + quadgen(-7))^n )};
```

[((1-1/sqrt(-7))*(3+sqrt(-7))^n/2^n).real() for n in range(34)] # Peter Luschny, Oct 02 2014 (after Somos)

G.f.: (1 - 2*x) / (1 - 3*x + 4*x^2).
a(n) = 3*a(n-1) - 4*a(n-2) for all n in Z.
a(n) = a(-1-n) * 2^(2*n + 1)  for all n in Z.
a(n) = (-1)^n * A087168(n) for all n in Z.
A247565(n) = 2^n + a(n) for all n in Z.
a(n) = A247487(2*n + 1) = A247564(2*n + 1) for all n in Z.

A247487 Expansion of (2 + x + x^2 + x^3 - x^4 - 2x^5 - 4x^6 - 8x^7) / (1 - x^4 + 16x^8) in powers of x.

Original entry on oeis.org

2, 1, 1, 1, 1, -1, -3, -7, -31, -17, -19, -23, -47, -1, 29, 89, 449, 271, 333, 457, 1201, 287, -131, -967, -5983, -4049, -5459, -8279, -25199, -8641, -3363, 7193, 70529, 56143, 83981, 139657, 473713, 194399, 137789, 24569, -654751, -703889, -1205907, -2209943

Offset: 0

Michael Somos, Sep 18 2014

			G.f. = 2 + x + x^2 + x^3 + x^4 - x^5 - 3*x^6 - 7*x^7 - 31*x^8 - 17*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,-16).

CF. A087168, A247518.

a:=[2,1,1,1,1,-1,-3,-7];; for n in [9..50] do a[n]:=a[n-4]-16*a[n-8]; od; a; # Muniru A Asiru, Aug 05 2018

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2+x+x^2+x^3-x^4-2*x^5-4*x^6-8*x^7)/(1-x^4+16*x^8))); // G. C. Greubel, Aug 05 2018

seq(coeff(series((2+x+x^2+x^3-x^4-2*x^5-4*x^6-8*x^7)/(1-x^4+16*x^8), x,n+1),x,n),n=0..50); # Muniru A Asiru, Aug 05 2018

CoefficientList[Series[(2+x+x^2+x^3-x^4-2*x^5-4*x^6-8*x^7)/(1 -x^4 + 16*x^8), {x, 0, 60}], x] (* G. C. Greubel, Aug 05 2018 *)

{a(n) = if( n<0, n=-n; 2^-n, 1) * polcoeff( (2 + x + x^2 + x^3 - x^4 - 2*x^5 - 4*x^6 - 8*x^7) / (1 - x^4 + 16*x^8) + x * O(x^n), n)};

a(n) = a(-n) * 2^n = a(n-4) - 16*a(n-8) for all n in Z.
a(2*n + 1) = (-1)^n * A087168(n).
A247518(n) = a(n+1) * a(n) * a(n-1) * a(n-2) for all n in Z.

A202672 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A087062 based on (1,1,1,1,...); by antidiagonals.

Original entry on oeis.org

1, -1, 1, -3, 1, 1, -5, 6, -1, 1, -7, 15, -10, 1, 1, -9, 28, -35, 15, -1, 1, -11, 45, -84, 70, -21, 1, 1, -13, 66, -165, 210, -126, 28, -1, 1, -15, 91, -286, 495, -462, 210, -36, 1, 1, -17, 120, -455, 1001, -1287, 924, -330, 45, -1, 1, -19, 153

Offset: 1

Clark Kimberling, Dec 22 2011

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix of A087062. The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

Closely related to A076756; however, for example, successive rows of A076756 are (1,-3,1), (-1,5,-6,1), compared to rows (1,-3,1), (1,-5,6,-1) of A202672.

			The 1st principal submatrix (ps) of A087062 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,1},{1,2}}, with p(2)=1-3x+x^2 and zero-set {0.381..., 2.618...}.
...
The 3rd ps is {{1,1,1},{1,2,2},{1,2,3}}, with p(3)=1-5x+6x^2-x^3 and zero-set {0.283..., 0.426..., 8.290...}.
...
Top of the array:
1...-1
1...-3....1
1...-5....6....-1
1...-7...15...-10....1
1...-9...28...-35...15...-1

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.
A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Cf. A087062, A202673 (based on n), A202671 (based on n^2), A202605 (based on Fibonacci numbers), A076756.

U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[1, {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]
TableForm[Table[c[n], {n, 1, 10}]]
Table[(F[k] /. x -> -2), {k, 1, 30}] (* A007583 *)
Table[(F[k] /. x -> 2), {k, 1, 30}]  (* A087168 *)

A174565 Expansion of g.f. (1+3x)/((1-x)(1+3x+4x^2)).

Original entry on oeis.org

1, 1, -3, 9, -11, 1, 45, -135, 229, -143, -483, 2025, -4139, 4321, 3597, -28071, 69829, -97199, 12285, 351945, -1104971, 1907137, -1301523, -3723975, 16378021, -34238159, 37202397, 25345449, -224845931, 573156001, -820084275, 167628825, 2777450629, -9002867183

Offset: 0

Ralf Stephan, Nov 29 2010

Bisection of A107443.

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

a087168:=func< n | &+[ Binomial(n+k, 2*k)*(-2)^(n-k): k in [0..n] ] >; [ (1-a087168(n+1))/2: n in [0..35] ];

a(n) = (1/2)*(1 - A087168(n+1)).

A120581 Hankel transform of sum{k=0..n, C(2k,k)*2^k}.

Original entry on oeis.org

1, 4, -64, -28672, -17825792, -24696061952, -4398046511104, 6413125869375586304, 1279761316857673852911616, 565738597953418785628899770368, 372547299599873882347063111621541888, -5264848781000759906705331926184317767647232

Offset: 0

Paul Barry, Jun 15 2006

Cf. A120580.

a(n)=4^(n+C(n,2))(-1)^n*sum{k=0..n, C(n+k,2k)(-2)^(n-k)}; a(n)=8^n*4^C(n,2)*sum{k=0..n, C(n+k,2k)(-2)^(-k)}; a(n)=4^n*4^C(n,2)(-1)^n*A087168(n);

cp's OEIS Frontend

A247560 a(n) = 3*a(n-1) - 4*a(n-2) with a(0) = a(1) = 1.

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A247487 Expansion of (2 + x + x^2 + x^3 - x^4 - 2*x^5 - 4*x^6 - 8*x^7) / (1 - x^4 + 16*x^8) in powers of x.

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A202672 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A087062 based on (1,1,1,1,...); by antidiagonals.

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Mathematica

A174565 Expansion of g.f. (1+3*x)/((1-x)*(1+3*x+4*x^2)).

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Comments

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Programs

Magma

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A120581 Hankel transform of sum{k=0..n, C(2k,k)*2^k}.

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Author

Keywords

Crossrefs

Formula

A247560 a(n) = 3a(n-1) - 4a(n-2) with a(0) = a(1) = 1.

A247487 Expansion of (2 + x + x^2 + x^3 - x^4 - 2x^5 - 4x^6 - 8x^7) / (1 - x^4 + 16x^8) in powers of x.

A174565 Expansion of g.f. (1+3x)/((1-x)(1+3x+4x^2)).