A077996 -id:A077996 - cp's OEIS frontend

A151798 a(0)=1, a(1)=2, a(n)=4 for n>=2.

1, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

David Applegate, Jun 29 2009

A010709 preceded by 1, 2.
Partial sums give A131098.
The INVERT transform gives A077996 without A077996(0). The Motzkin transform gives A105696 without A105696(0). Decimal expansion of 28/225=0.12444... . - R. J. Mathar, Jun 29 2009
Continued fraction expansion of 1 + sqrt(1/5). - Arkadiusz Wesolowski, Mar 30 2012
The number of solutions x (mod 2^(n+1)) of x^2 = 1 (mod 2^(n+1)), namely x = 1 (n=0), x = -1, 1 (n=1) and x = -1, 1, 2^n-1, 2^n+1 (n at least 2). - Christopher J. Smyth, May 15 2014
Also, the number of n-step self-avoiding walks on the L-lattice with no non-contiguous adjacencies (see A322419 for details of L-lattice). - Sean A. Irvine, Jul 29 2020

David Applegate, The movie version
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010709, A131098, A077996, A105696.

[ n le 1 select n+1 else 4: n in [0..104] ];

f[n_] := Fold[#2*Floor[#1/#2 + 1/2] &, n, Reverse@ Range[n - 1]]; Array[f, 55]

Vec((1+x+2*x^2)/(1-x) + O(x^100)) \\ Altug Alkan, Jan 19 2016

G.f.: (1+x+2*x^2)/(1-x).
E.g.f. A(x)=x*B(x) satisfies the differential equation B'(x)=1+x+x^2+B(x). - Vladimir Kruchinin, Jan 19 2011
E.g.f.: 4*exp(x) - 2*x - 3. - Elmo R. Oliveira, Aug 06 2024

A224232 a(n) = n! if n <= 3, otherwise a(n) = 2*(a(n-1) + a(n-3)) + a(n-2).

Original entry on oeis.org

1, 1, 2, 6, 16, 42, 112, 298, 792, 2106, 5600, 14890, 39592, 105274, 279920, 744298, 1979064, 5262266, 13992192, 37204778, 98926280, 263041722, 699419280, 1859732842, 4944968408, 13148508218, 34961450528, 92961346090, 247181159144, 657246565434, 1747596982192, 4646802848106, 12355695809272, 32853388431034, 87356078367552, 232276936784682

Offset: 0

N. J. A. Sloane, Apr 11 2013

Also the number of permutations that are sortable after two passes through a pop stack. (See the Pudwell-Smith link.) - Lara Pudwell, Jun 01 2017

Colin Barker, Table of n, a(n) for n = 0..1000
G. Aleksandrowich et al., Permutations with forbidden patterns and polyominoes on a twisted cylinder of width 3, Discrete Math., 313 (2013), 1078-1086.
Anders Claesson and Bjarki Ágúst Guðmundsson, Enumerating permutations sortable by k passes through a pop-stack, arXiv:1710.04978 [math.CO], 2017.
Lara Pudwell and Rebecca Smith, Sorting with Pop Stacks, Special Session on Algebraic and Enumerative Combinatorics with Applications, AMS Central Section Spring Meeting, 2017.
Lara Pudwell and Rebecca Smith, Two-stack-sorting with pop stacks, arXiv:1801.05005 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (2,1,2).

Cf. A077996, A293774, A293775, A293776, A293784.

CoefficientList[Series[(x^3 + x^2 + x - 1)/(2 x^3 + x^2 + 2 x - 1), {x, 0, 35}], x] (* Michael De Vlieger, Jun 01 2017 *)
LinearRecurrence[{2,1,2},{1,1,2,6},40] (* Harvey P. Dale, Aug 28 2023 *)

Vec((x^3+x^2+x-1)/(2*x^3+x^2+2*x-1) + O(x^100)) \\ Colin Barker, Jun 07 2015

G.f.: (x^3 + x^2 + x - 1) / (2*x^3 + x^2 + 2*x - 1). - Colin Barker, Jun 07 2015

a(n) = (b(n) + b(n-1))/2 for b(n) = A077996(n). - Hanzhang Fang, Aug 27 2022

A356563 Sums of powers of roots of x^3 - 2*x^2 - x - 2.

Original entry on oeis.org

3, 2, 6, 20, 50, 132, 354, 940, 2498, 6644, 17666, 46972, 124898, 332100, 883042, 2347980, 6243202, 16600468, 44140098, 117367068, 312075170, 829797604, 2206404514, 5866756972, 15599513666, 41478593332, 110290214274

Offset: 0

Greg Dresden and Hanzhang Fang, Aug 12 2022

The three roots of x^3 - 2*x^2 - x - 2 are c1=2.65896708... = A348909+1, c2=-0.32948354... + 0.80225455...*i, and c3=-0.32948354... - 0.80225455...*i.

a(n) can also be determined by Vieta's formulas and Newton's identities. For example, a(3) by definition is c1^3 + c2^3 + c3^3, and from Newton's identities this equals e1^3 - 3*e1*e2 + 3*e3 for e1, e2, e3 the elementary symmetric polynomials of x^3 - x^2 - x - 3. From Vieta's formulas we have e1 = 2, e2 = -1, and e3 = 2, giving us e1^3 - 3*e1*e2 + 3*e3 = 8 + 6 + 6 = 20, as expected.

			For n=3, a(3) = (2.65896708...)^3 + (-0.32948354... + 0.80225455...*i)^3 + (-0.32948354... - 0.80225455...*i)^3 = 20.

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

Cf. A077996, A348909 (c1-1).

LinearRecurrence[{2, 1, 2}, {3, 2, 6}, 30]

a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3).
G.f.: (3 - 4 x - x^2)/(1 - 2 x - x^2 - 2 x^3).
2*a(n) = 7*b(n) - b(n+1) for b(n) = A077996(n).

cp's OEIS Frontend

A151798 a(0)=1, a(1)=2, a(n)=4 for n>=2.

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A224232 a(n) = n! if n <= 3, otherwise a(n) = 2*(a(n-1) + a(n-3)) + a(n-2).

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Mathematica

PARI

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A356563 Sums of powers of roots of x^3 - 2*x^2 - x - 2.

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Author

Keywords

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Links

Crossrefs

Programs

Mathematica

Formula