A175164 -id:A175164 - cp's OEIS frontend

A028399 a(n) = 2^n - 4.

0, 4, 12, 28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, 65532, 131068, 262140, 524284, 1048572, 2097148, 4194300, 8388604, 16777212, 33554428, 67108860, 134217724, 268435452, 536870908, 1073741820, 2147483644, 4294967292, 8589934588, 17179869180

Offset: 2

N. J. A. Sloane

Number of permutations of [n] with 2 sequences.
Number of 2 X n binary matrices that avoid simultaneously the right angled numbered polyomino patterns (ranpp) (00;1) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1Sergey Kitaev, Nov 11 2004
The number of edges in the dual Edwards-Venn diagram graph with n-1 digits when n>2.
a(n) (n>=6) is the number of vertices in the molecular graph NS2[n-5], defined pictorially in the Ashrafi et al. reference (Fig. 2, where NS2[2] is shown). - Emeric Deutsch, May 16 2018
From Petros Hadjicostas, Aug 08 2019: (Start)
With regard to the comment above about a(n) being the "number of permutations of [n] with 2 sequences", we refer to Ex. 13 (pp. 260-261) of Comtet (1974), who uses the definition of a "séquence" given by André in several of his papers in the 19th century.
In the terminology of array A059427, these so-called "séquences" in permutations (defined by Comtet and André) are called "alternating runs" (or just "runs"). We discuss these so-called "séquences" below.
If b = (b_1, b_2, ..., b_n) is a permutation of [n], written in one-line notation (not in cycle notation), a "séquence" in a permutation of length l >= 2 (according to Comtet) is a maximal interval of integers {i, i+1, ..., i+l-1} for some i (where 1 <= i <= n-l+1) such that b_i < b_{i+1} < ... < b_{i+l-1} or b_i > b_{i+1} > ... > b_{i+l-1}. (The word "maximal" means that, in the first case, we have b_{i-1} > b_i and b_{i+l} < b_{i+l-1}, while in the second case, we have b_{i-1} < b_i and b_{i+l} > b_{i+l-1} provided b_{i-1} and b_{i+l} can be defined.)
When defining a "séquence", André (1884) actually refers to the list of terms (b_i, b_{i+1}, ..., b_{i+l-1}) rather than the corresponding index set {i, i+1, ..., i+l-1} (which is essentially the same thing).
For more details about these so-called "séquences" (or "alternate runs"), see the comments and examples for sequence A000708.
(End)
For n >= 1, a(n+2) is the number of shortest paths from (0,0) of a square grid to {(x,y): |x|+|y| = n} along the grid line. - Jianing Song, Aug 23 2021

			From _Petros Hadjicostas_, Aug 08 2019: (Start)
We have a(3) = 4 because each of the following permutations of [3] has the following so-called "séquences" ("alternate runs"):
   123 -> 123 (one),
   132 -> 13, 32 (two),
   213 -> 21, 13 (two),
   231 -> 23, 31 (two),
   312 -> 31, 12 (two),
   321 -> 321 (one).
Recall that a so-called "séquence" ("alternate run") must start with a "maximum" and end with "minimum", or vice versa, and it should not contain any other maxima and minima in between. Two consecutive such "séquences" ("alternate runs") have exactly one minimum or exactly one maximum in common.
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 261.
A. W. F. Edwards, Cogwheels of the Mind, Johns Hopkins University Press, 2004, p. 82.

Muniru A Asiru, Table of n, a(n) for n = 2..700
Désiré André, Sur les permutations alternées, J. Math. Pur. Appl., 7 (1881), 167-184.
Désiré André, Étude sur les maxima, minima et séquences des permutations, Ann. Sci. Ecole Norm. Sup., 3, no. 1 (1884), 121-135.
Désiré André, Mémoire sur les permutations quasi-alternées, Journal de mathématiques pures et appliquées 5e série, tome 1 (1895), 315-350.
Désiré André, Mémoire sur les séquences des permutations circulaires, Bulletin de la S. M. F., tome 23 (1895), pp. 122-184.
Ali Reza Ashrafi and Parisa Nikzad, Kekulé index and bounds of energy for nanostar dendrimers, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (3rd line of Table 2 is a(n+1)).
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Column k = 2 of A059427.
Row n = 2 of A371064.
Cf. A000225, A000918, A052548, A140504, A175164.
Cf. A000203, A003945.

a:=List([2..40], n->2^n-4); # Muniru A Asiru, May 17 2018

seq(2^n-4, n=2..40); # Muniru A Asiru, May 17 2018

2^Range[2,40]-4 (* Harvey P. Dale, Jul 05 2019 *)

PARI
```
a(n)=if(n<2, 0, 2^n-4)
```

def A028399(n): return (1<Chai Wah Wu, Jun 27 2023

O.g.f.: 4*x^3/((1-x)*(1-2*x)). - R. J. Mathar, Aug 07 2008
From Reinhard Zumkeller, Feb 28 2010: (Start)
a(n) = A175164(2*n)/A140504(n+2);
a(2*n) = A052548(n)*A000918(n) for n > 0;
a(n) = A173787(n,2). (End)
a(n) = a(n-1) + 2^(n-1) (with a(2)=0). - Vincenzo Librandi, Nov 22 2010
a(n) = 4*A000225(n-2). - R. J. Mathar, Dec 15 2015
E.g.f.: 3 + 2*x - 4*exp(x) + exp(2*x). - Stefano Spezia, Apr 06 2021
a(n) = sigma(A003945(n-2)) for n>=3. - Flávio V. Fernandes, Apr 20 2021

Additional comments from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 02 2001

A140504 a(n) = 2^n + 4.

Original entry on oeis.org

5, 6, 8, 12, 20, 36, 68, 132, 260, 516, 1028, 2052, 4100, 8196, 16388, 32772, 65540, 131076, 262148, 524292, 1048580, 2097156, 4194308, 8388612, 16777220, 33554436, 67108868, 134217732, 268435460, 536870916, 1073741828

Offset: 0

Paul Curtz, Jun 30 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000051 (m=0), A052548 (m=2), this sequence (m=4), A153973 (m=6), A231643 (m=5), A175161 (m=8), A175162 (m=16), A175163 (m=32).

[2^n +4: n in [0..30]]; // G. C. Greubel, Jul 08 2021

Table[2^n + 4, {n, 0, 30}] (* Stefan Steinerberger, Aug 04 2008 *)
LinearRecurrence[{3,-2}, {5,6}, 40] (* Harvey P. Dale, May 26 2018 *)

a(n)=2^n+4 \\ Charles R Greathouse IV, Dec 21 2011

[2^n + 4 for n in range(0,31)] # Zerinvary Lajos, May 31 2009

G.f.: (5 - 9*x)/((1 - x)*(1 - 2*x)). - Jaume Oliver Lafont, Aug 30 2009
a(n) = 2*a(n-1) - 4 with a(0) = 5. - Vincenzo Librandi, Nov 24 2009
From Reinhard Zumkeller, Feb 28 2010: (Start)
a(n) = A173786(n,2) for n > 1.
a(n+2)*A028399(n) = A175164(2*n). (End)
From G. C. Greubel, Jul 08 2021: (Start)
a(n) = m*(2^(n-2) + 1), with m = 4.
E.g.f.: exp(2*x) + 4*exp(x). (End)

More terms from Stefan Steinerberger, Aug 04 2008

A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 7, 6, 4, 0, 15, 14, 12, 8, 0, 31, 30, 28, 24, 16, 0, 63, 62, 60, 56, 48, 32, 0, 127, 126, 124, 120, 112, 96, 64, 0, 255, 254, 252, 248, 240, 224, 192, 128, 0, 511, 510, 508, 504, 496, 480, 448, 384, 256, 0, 1023, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 0

Offset: 0

Reinhard Zumkeller, Feb 28 2010

			Triangle begins as:
   0;
   1,  0;
   3,  2,  0;
   7,  6,  4,  0;
  15, 14, 12,  8,  0;
  31, 30, 28, 24, 16, 0;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

[2^n -2^k: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 13 2021

Table[2^n -2^k, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 13 2021 *)

flatten([[2^n -2^k for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 13 2021

A000120(T(n,k)) = A025581(n,k).
Row sums give A000337.
Central terms give A020522.
T(2*n+1, n) = A006516(n+1).
T(2*n+3, n+2) = A059153(n).
T(n, k) = A140513(n,k) - A173786(n,k), 0 <= k <= n.
T(n, k) = A173786(n,k) - A059268(n+1,k+1), 0 < k <= n.
T(2*n, 2*k) = T(n,k) * A173786(n,k), 0 <= k <= n.
T(n, 0) = A000225(n).
T(n, 1) = A000918(n) for n>0.
T(n, 2) = A028399(n) for n>1.
T(n, 3) = A159741(n-3) for n>3.
T(n, 4) = A175164(n-4) for n>4.
T(n, 5) = A175165(n-5) for n>5.
T(n, 6) = A175166(n-6) for n>6.
T(n, n-4) = A110286(n-4) for n>3.
T(n, n-3) = A005009(n-3) for n>2.
T(n, n-2) = A007283(n-2) for n>1.
T(n, n-1) = A000079(n-1) for n>0.
T(n, n) = A000004(n).

A175166 a(n) = 64*(2^n - 1).

Original entry on oeis.org

0, 64, 192, 448, 960, 1984, 4032, 8128, 16320, 32704, 65472, 131008, 262080, 524224, 1048512, 2097088, 4194240, 8388544, 16777152, 33554368, 67108800, 134217664, 268435392, 536870848, 1073741760

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), A175165 (m=32), this sequence (m=64).

Cf. A173787, A175161.

I:=[0,64]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

LinearRecurrence[{3,-2},{0,64},30] (* Harvey P. Dale, Apr 08 2015 *)

def A175166(n): return (1<Chai Wah Wu, Jun 27 2023

[64*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+6) - 64.
a(n) = A173787(n+6, 6).
a(2*n) = A175161(n)*A159741(n) for n > 0.
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=64. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 64*x/((1-x)*(1-2*x)).
E.g.f.: 64*(exp(2*x) - exp(x)). (End)

A175165 a(n) = 32*(2^n - 1).

Original entry on oeis.org

0, 32, 96, 224, 480, 992, 2016, 4064, 8160, 16352, 32736, 65504, 131040, 262112, 524256, 1048544, 2097120, 4194272, 8388576, 16777184, 33554400, 67108832, 134217696, 268435424, 536870880

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), this sequence (m=32), A175166 (m=64).

Cf. A173787.

I:=[0,32]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

32(2^Range[0,30] -1) (* or *) LinearRecurrence[{3,-2},{0,32},30] (* Harvey P. Dale, Mar 23 2015 *)

def A175165(n): return (1<Chai Wah Wu, Jun 27 2023

[32*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+5) - 32.
a(n) = A173787(n+5, 5).
a(n) = 3*a(n-1) - 2*a(n-2); a(0)=0, a(1)=32. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 32*x/((1-x)*(1-2*x)).
E.g.f.: 32*(exp(2*x) - exp(x)). (End)

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A028399 a(n) = 2^n - 4.

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A140504 a(n) = 2^n + 4.

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A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

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A175166 a(n) = 64*(2^n - 1).

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A175165 a(n) = 32*(2^n - 1).

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