A087288 -id:A087288 - cp's OEIS frontend

A052531 If n is even then 2^n+1 otherwise 2^n.

2, 2, 5, 8, 17, 32, 65, 128, 257, 512, 1025, 2048, 4097, 8192, 16385, 32768, 65537, 131072, 262145, 524288, 1048577, 2097152, 4194305, 8388608, 16777217, 33554432, 67108865, 134217728, 268435457, 536870912, 1073741825, 2147483648

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
IBM Research, Control-flow graphs, Ponder This Challenge, October 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 461
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A001045, A042950, A052929, A062510, A087288, A280345.

a:=[2,2,5];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, May 09 2019

[2^n + (1+(-1)^n)/2: n in [0..30]]; // G. C. Greubel, May 09 2019

spec:= [S,{S=Union(Sequence(Union(Z,Z)),Sequence(Prod(Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(2^n + (1+(-1)^n)/2, n=0..30); # G. C. Greubel, Oct 17 2019

2^# + (1 - Mod[#, 2]) & /@ Range[0, 40] (* Peter Pein, Jan 11 2008 *)
Table[If[EvenQ[n], 2^n + 1, 2^n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010, modified by G. C. Greubel, May 09 2019 *)
Table[2^n + Boole[EvenQ[n]], {n, 0, 31}] (* Alonso del Arte, May 09 2019 *)

my(x='x+O('x^40)); Vec((2-2*x-x^2)/((1-x^2)*(1-2*x))) \\ G. C. Greubel, May 09 2019

PARI
```
a(n) = 1<David A. Corneth, Oct 18 2019
```

[2^n + (1+(-1)^n)/2 for n in (0..30)] # G. C. Greubel, May 09 2019

G.f.: (2 - 2*x - x^2)/( (1-x^2)*(1-2*x) ).
a(n) = a(n-1) + 2*a(n-2) - 1, with a(0) = 2, a(1) = 2, a(2) = 5.
a(n) = 2^n + Sum_{alpha = RootOf(-1+x^2)} alpha^(-n)/2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), with a(0) = 2, a(1) = 2, a(2) = 5. - G. C. Greubel, May 09 2019
a(n) = 2^n + (1 + (-1)^n)/2. - G. C. Greubel, Oct 17 2019
E.g.f.: exp(2*x) + cosh(x). - Stefano Spezia, Oct 18 2019

More terms from James Sellers, Jun 05 2000

Better definition from Peter Pein (petsie(AT)dordos.net), Jan 11 2008

A137241 Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

Original entry on oeis.org

0, 3, 2, 1, 2, 0, 2, 1, -2, 3, 0, -4, 4, -1, -6, 5, -2, -8, 6, -3, -10, 7, -4, -12, 8, -5, -14, 9, -6, -16, 10, -7, -18, 11, -8, -20, 12, -9, -22, 13, -10, -24, 14, -11, -26, 15, -12, -28, 16, -13, -30, 17, -14, -32, 18, -15, -34, 19, -16, -36, 20, -17, -38, 21, -18, -40

Offset: 0

Paul Curtz, Mar 09 2008

The entries are the coefficients in a family of Jacobsthal recurrences: a(n)=k*a(n-1)+(3-k)*a(n-2)+(2-2k)*a(n-3).
Examples for k=0 are in A001045 and A113954. Examples for k=1 are A001045, A078008.
Examples for k=2 are A000975, A087288, A084639, A000012 and A001045.
Examples for k=3 are A045883, A059570. Examples for k=4 are A094705 and A015518.

			The triples (k,3-k,2-2k) are (0,3,2), (1,2,0), (2,1,-2), (3,0,-4),...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 2, 0, 0, -1).

CoefficientList[Series[x*(3 + 2*x + x^2 - 4*x^3 - 4*x^4)/((x - 1)^2*(1 + x + x^2)^2), {x, 0, 50}], x] (* G. C. Greubel, Sep 28 2017 *)
Table[{n,3-n,2-2n},{n,0,30}]//Flatten (* or *) LinearRecurrence[ {0,0,2,0,0,-1},{0,3,2,1,2,0},100] (* Harvey P. Dale, Jun 23 2019 *)

x='x+O('x^50); Vec(x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x +x^2 )^2)) \\ G. C. Greubel, Sep 28 2017

From R. J. Mathar, Feb 25 2009: (Start)
a(n) = 2*a(n-3) - a(n-6).
G.f.: x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x+x^2)^2). (End)

Edited by R. J. Mathar, Jun 28 2008

A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

Original entry on oeis.org

4, 10, 34, 130, 514, 2050, 8194, 32770, 131074, 524290, 2097154, 8388610, 33554434, 134217730, 536870914, 2147483650, 8589934594, 34359738370, 137438953474, 549755813890, 2199023255554, 8796093022210, 35184372088834, 140737488355330, 562949953421314

Offset: 0

Peter M. Chema, Feb 28 2017

Number of vertices required to make a Sierpinski tetrahedron or tetrix of side length 2^n. The sum of the vertices (balls) plus line segments (rods) of one tetrix equals the vertices of its larger, adjacent iteration. See formula.
Equivalently, the number of vertices in the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 17 2017
Also the independence number of the (n+2)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 29 2021
Final digit alternates 4 and 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph
Eric Weisstein's World of Mathematics, Tetrix
Eric Weisstein's World of Mathematics, Vertex Count
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Subsequence of A016957.
Cf. A052539, A279511, A279512.
First bisection of A052548, A087288; second bisection of A049332, A133140, A135440.
Cf. A002023 (edge count).

Table[2 4^n + 2, {n, 0, 30}] (* Bruno Berselli, Feb 28 2017 *)
2 (4^Range[0, 20] + 1) (* Eric W. Weisstein, Aug 17 2017 *)
LinearRecurrence[{5, -4}, {4, 10}, 20] (* Eric W. Weisstein, Aug 17 2017 *)
CoefficientList[Series[-((2 (-2 + 5 x))/(1 - 5 x + 4 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 17 2017 *)

a(n)=2*4^n+2 \\ Charles R Greathouse IV, Feb 28 2017

Vec(2*(2 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Mar 02 2017

def a(n): return 2*4**n + 2
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 29 2021

G.f.: 2*(2 - 5*x)/((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
a(n+1) = a(n) + A002023(n).
a(n) = 2*A052539(n) = A188161(n) - 1 = A087289(n) + 1 = A056469(2*n+2) = A261723(4*n+1).
E.g.f.: 2*(exp(4*x) + exp(x)). - G. C. Greubel, Aug 17 2017

Entry revised by Editors of OEIS, Mar 01 2017

A140359 a(n) = 2a(n-1) + a(n-2) - 2a(n-3).

Original entry on oeis.org

1, 1, 6, 11, 26, 51, 106, 211, 426, 851, 1706, 3411, 6826, 13651, 27306, 54611, 109226, 218451, 436906, 873811, 1747626, 3495251, 6990506, 13981011, 27962026, 55924051, 111848106, 223696211, 447392426, 894784851, 1789569706, 3579139411

Offset: 0

Paul Curtz, Jun 24 2008

nonn

This is the sequence A(1,1;1,2;3) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A001045, A087288, A137241, A140360.

[(5*2^(n+1) -9 + 5*(-1)^n)/6: n in [0..50]]; // G. C. Greubel, Oct 10 2017

Table[(5*2^(n+1) -9 + 5*(-1)^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 10 2017 *)
LinearRecurrence[{2,1,-2},{1,1,6},40] (* Harvey P. Dale, Mar 24 2021 *)

for(n=0,50, print1((5*2^(n+1) -9 + 5*(-1)^n)/6, ", ")) \\ G. C. Greubel, Oct 10 2017

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n+1) - a(n) = 5*A001045(n), Jacobsthal numbers.
a(n+1) - 2*a(n) = (-1)^(n+1)* A010685(n).
From R. J. Mathar, Jul 10 2008: (Start)
O.g.f.: (1-x+3*x^2)/((x-1)*(2*x-1)*(1+x)).
a(n) = (5*2^(n+1) - 9 + 5*(-1)^n)/6. (End)
a(n) = a(n-1) + 2*a(n-2) +3, n>1 - Gary Detlefs, Jun 20 2010

Extended by R. J. Mathar, Jul 10 2008

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A052531 If n is even then 2^n+1 otherwise 2^n.

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A137241 Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

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A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

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A140359 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).

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A140359 a(n) = 2a(n-1) + a(n-2) - 2a(n-3).