A079861 -id:A079861 - cp's OEIS frontend

A078836 a(n) = n*2^(n-6).

6, 14, 32, 72, 160, 352, 768, 1664, 3584, 7680, 16384, 34816, 73728, 155648, 327680, 688128, 1441792, 3014656, 6291456, 13107200, 27262976, 56623104, 117440512, 243269632, 503316480, 1040187392, 2147483648, 4429185024, 9126805504, 18790481920, 38654705664

Offset: 6

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 17 2003

a(n) is the number of occurrences of 5s in the palindromic compositions of 2n-1 = the number of occurrences of 6s in the palindromic compositions of 2n.
This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A079859, A079861 - A079863. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.
Also the number of independent vertex sets and vertex covers in the (n-4)-sun graph. - Eric W. Weisstein, Sep 27 2017

			a(6) = 6 since the palindromic compositions of 11 that contain a 5 are 3+5+3, 1+2+5+2+1, 2+1+5+1+2, 1+1+1+5+1+1+1 and 5+1+5, for a total of 6 5s. The palindromic compositions of 12 that contain a 6 are 3+6+3, 1+2+6+2+1, 2+1+6+1+2, 1+1+1+6+1+1+1 and 6+6.

Vincenzo Librandi, Table of n, a(n) for n = 6..3000
Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Sun Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A079859, A079861, A079862, A079863.

[n*2^(n-6): n in [6..40]]; // Vincenzo Librandi, Oct 04 2011

Table[n 2^(n - 6), {m, 6, 50}]
LinearRecurrence[{4, -4}, {6, 14}, 20] (* Eric W. Weisstein, Sep 27 2017 *)
CoefficientList[Series[-2 (-3 + 5 x)/(-1 + 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Sep 27 2017 *)

a(n)=n<<(n-6) \\ Charles R Greathouse IV, Oct 03 2011

Vec(-2*x^6*(5*x-3)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Sep 29 2015

def a(n): return n << (n-6)
print([a(n) for n in range(6, 37)]) # Michael S. Branicky, Jun 14 2021

From Colin Barker, Sep 29 2015: (Start)
a(n) = 2*A045891(n-4).
a(n) = 4*a(n-1) - 4*a(n-2) for n > 7.
G.f.: -2*x^6*(5*x-3) / (2*x-1)^2.
(End)
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=6} 1/a(n) = 64*log(2) - 661/15.
Sum_{n>=6} (-1)^n/a(n) = 391/15 - 64*log(3/2). (End)

A079859 a(n) = n*2^(n-4).

Original entry on oeis.org

4, 10, 24, 56, 128, 288, 640, 1408, 3072, 6656, 14336, 30720, 65536, 139264, 294912, 622592, 1310720, 2752512, 5767168, 12058624, 25165824, 52428800, 109051904, 226492416, 469762048, 973078528, 2013265920, 4160749568, 8589934592, 17716740096, 36507222016

Offset: 4

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

a(n) = the number of occurrences of 3s in the palindromic compositions of m = 2*n-1 = the number of occurrences of 4s in the palindromic compositions of k = 2*n.
This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862, A079863. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.
Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (01;1). 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
a(n) appears to be the coefficient of Pi^n in the closed-form expression for the expected value of X^n, where X is the area of a spherical triangle formed by three random points on the unit sphere. (The n*2^(n-4) formula applies when n=2,3 as well, and produces fractional coefficients.) - Drake Thomas, Jan 24 2021

			a(4)=4 since the palindromic compositions of 7 that contain a 3 are 2+3+2, 1+1+3+1+1 and 3+1+3, for a total of 4 3s. The palindromic compositions of 8 that contain a 4 are 2+4+2, 1+1+4+1+1 and 4+4.

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
Phyllis Chinn, Ralph Grimaldi and Silvia Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin., Vol. 69 (2003), pp. 65-78.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), A21, 20pp.
Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
Math StackExchange, The distribution of areas of a random triangle on the sphere.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A078836, A079861, A079862.

Main diagonal of A049089.

[n*2^(n-4) : n in [4..40]]; // Vincenzo Librandi, Sep 22 2011

Mathematica
```
Table[i*2^(i - 4), {i, 4, 50}]
```

Vec(-2*x^4*(3*x-2)/(2*x-1)^2 + O(x^50)) \\ Colin Barker, Sep 29 2015

a(n) = n*2^(n-4);
vector(40, n, a(n+3)) \\ Altug Alkan, Sep 29 2015

O.g.f.: 2*x^4*(2-3*x)/(1-2*x)^2. a(n) = 2*A045623(n-3). - R. J. Mathar, Jun 13 2008
a(n) = 4*a(n-1) - 4*a(n-2) for n>5. - Colin Barker, Sep 29 2015
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=4} 1/a(n) = 16*log(2) - 32/3.
Sum_{n>=4} (-1)^n/a(n) = 20/3 - 16*log(3/2). (End)
E.g.f.: x*(exp(2*x) - 1 - 2*x - 2*x^2)/8. - Stefano Spezia, Apr 06 2021

A079862 a(i) = the number of occurrences of 9's in the palindromic compositions of n=2i-1 = the number of occurrences of 10's in the palindromic compositions of n=2i.

Original entry on oeis.org

18, 38, 80, 168, 352, 736, 1536, 3200, 6656, 13824, 28672, 59392, 122880, 253952, 524288, 1081344, 2228224, 4587520, 9437184, 19398656, 39845888, 81788928, 167772160, 343932928, 704643072, 1442840576, 2952790016, 6039797760, 12348030976, 25232932864

Offset: 10

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.

			a(10) = 18 since the palindromic compositions of 19 that contain a 9 are 9+1+9 and the 16 compositions of the form c+9+(reverse of c), where c represents a composition of 5.

Colin Barker, Table of n, a(n) for n = 10..1000
P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin. 69 (2003), 65-78.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A079859, A078836, A079861, A079863.

Mathematica
```
Table[(8 + i)*2^(i - 10), {i, 10, 50}]
```

Vec(-2*x^10*(17*x-9)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Sep 29 2015

a(n) = (n+8)*2^(n-10).
From Colin Barker, Sep 29 2015: (Start)
a(n) = 2*A159697(n-10).
a(n) = 4*a(n-1) - 4*a(n-2) for n>11.
G.f.: -2*x^10*(17*x-9) / (2*x-1)^2.
(End)

A079863 a(n) is the number of occurrences of 11s in the palindromic compositions of m=2n-1 = the number of occurrences of 12s in the palindromic compositions of m=2n.

Original entry on oeis.org

34, 70, 144, 296, 608, 1248, 2560, 5248, 10752, 22016, 45056, 92160, 188416, 385024, 786432, 1605632, 3276800, 6684672, 13631488, 27787264, 56623104, 115343360, 234881024, 478150656, 973078528, 1979711488, 4026531840, 8187281408, 16642998272, 33822867456

Offset: 12

Silvia Heubach (sheubac(AT)calstatela.edu), Jan 11 2003

This sequence is part of a family of sequences, namely R(n,k), the number of ks in palindromic compositions of n. See also A057711, A001792, A078836, A079861, A079862. General formula: R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k) if n and k have different parity and R(n,k)=2^(floor(n/2) - k) * (2 + floor(n/2) - k + 2^(floor((k+1)/2 - 1)) otherwise, for n >= 2k.

			a(12) = 34 since the palindromic compositions of 23 that contain a 11 are 11+1+11 and the 32 compositions of the form c+11+(reverse of c), where c represents a composition of 6.

Colin Barker, Table of n, a(n) for n = 12..1000
P. Chinn, R. Grimaldi and S. Heubach, The frequency of summands of a particular size in Palindromic Compositions, Ars Combin. 69 (2003), 65-78.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A079859, A078836, A079861, A079862.

Table[(22 + i)*2^(i - 12), {i, 12, 50}]
LinearRecurrence[{4,-4},{34,70},30] (* Harvey P. Dale, Jan 30 2017 *)

Vec(-2*x^12*(33*x-17)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Sep 29 2015

a(n)=(n+22)<<(n-12) \\ Charles R Greathouse IV, Sep 29 2015

a(n) = (n+22)*2^(n-12).
From Colin Barker, Sep 29 2015: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) for n>13.
G.f.: -2*x^12*(33*x-17) / (2*x-1)^2.
(End)

A079028 a(0) = 1, a(n) = (n + 4)*4^(n-1) for n >= 1.

Original entry on oeis.org

1, 5, 24, 112, 512, 2304, 10240, 45056, 196608, 851968, 3670016, 15728640, 67108864, 285212672, 1207959552, 5100273664, 21474836480, 90194313216, 377957122048, 1580547964928, 6597069766656, 27487790694400, 114349209288704, 474989023199232, 1970324836974592, 8162774324609024

Offset: 0

Benoit Cloitre, Feb 01 2003

a(n) = det(M(n)) where M(n) is the n X n matrix defined by m(i,i) = 5, m(i,j) = i/j.
Main diagonal of array defined by m(1,j) = j; m(i,1) = i and m(i,j) = m(i-1,j) + 3*m(i-1,j-1).
4th binomial transform of (1,1,0,0,0,0,...). - Paul Barry, Mar 07 2003
Number of independent vertex subsets of the graph obtained by attaching two pendant edges to each vertex of the complete graph K_n (see A235113). Example: a(1)=5; indeed, K_1 is the one vertex graph and after attaching two pendant vertices we obtain the path graph ABC; the independent vertex subsets are: empty, {A}, {B}, {C}, and {A,C}. - Emeric Deutsch, Jan 13 2014
Row sums of A235113.

F. Disanto, A. Frosini, R. Pinzani and S. Rinaldi, A closed formula for the number of convex permutominoes, arXiv:math/0702550 [math.CO], 2007.
Index entries for linear recurrences with constant coefficients, signature (8,-16).

Cf. A001792, A006234, A081105, A006234, A235113.

Cf. A002697, A034007, A079861, A045891, A087449.

LinearRecurrence[{8, -16}, {1, 5}, 22] (* Jean-François Alcover, Nov 06 2018 *)

[lucas_number2(n, 4, 0)*n/2^10 for n in range(4, 26)] # Zerinvary Lajos, Mar 13 2009

a(n) = 8*a(n-1)-16*a(n-2), a(0) = 1, a(1) = 5. - Paul Barry, Mar 07 2003
G.f.: (1 - 3*x)/(1 - 4*x)^2. - Philippe Deléham, Dec 11 2008
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=0} 1/a(n) = 1024*log(4/3) - 880/3.
Sum_{n>=0} (-1)^n/a(n) = 688/3 - 1024*log(5/4). (End)
E.g.f.: exp(4*x)*(1 + x). - Stefano Spezia, Mar 05 2023

More terms from Stefano Spezia, Mar 05 2023

A227978 a(0)=1, a(1)=2; for n>1, a(n) = n*(2^n+4)/4.

Original entry on oeis.org

1, 2, 4, 9, 20, 45, 102, 231, 520, 1161, 2570, 5643, 12300, 26637, 57358, 122895, 262160, 557073, 1179666, 2490387, 5242900, 11010069, 23068694, 48234519, 100663320, 209715225, 436207642, 905969691, 1879048220, 3892314141, 8053063710, 16642998303

Offset: 0

Paul Curtz, Oct 07 2013

The inverse binomial transform of A176328/A176591 (see Comments field in A228827) begins: 1, -2, 25/6, -9, 599/30, -45, 4285/42, -231, 15599/30, -1161, 169625/66, ... Consider these values without sign and the fractions rounded to the nearest integer, the sequence lists the resulting numbers.
Differences table of a(n):
  1, 2,  4,  9, 20,  45, 102, 231,  520, 1161, ...
  1, 2,  5, 11, 25,  57, 129, 289,  641, 1409, ... After 2: 2^m*(m+4)+1.
  1, 3,  6, 14, 32,  72, 160, 352,  768, 1664, ... A078836 (after 3).
  2, 3,  8, 18, 40,  88, 192, 416,  896, 1920, ... A129955.
  1, 5, 10, 22, 48, 104, 224, 480, 1024, 2176, ... A079861 (after 5).
  4, 5, 12, 26, 56, 120, 256, 544, 1152, 2432, ... After 5: 2^m*(m+12).
  1, 7, 14, 30, 64, 136, 288, 608, 1280, 2688, ... After 7: 2^m*(m+14).
  6, 7, 16, 34, 72, 152, 320, 672, 1408, 2944, ..., etc.
(n-1)*a(n)-n*a(n-1) = A001788(n-1) for n>1. [Bruno Berselli, Oct 11 2013]

Bruno Berselli, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A001788, A079862, A079863.

[1,2] cat [n*(2^n+4)/4: n in [2..40]]; // Bruno Berselli, Oct 11 2013

Join[{1, 2}, Table[n (2^n + 4)/4, {n, 2, 40}]] (* Bruno Berselli, Oct 11 2013 *)

a(n) = if (n == 0, 1, if (n == 1, 2, n*(2^n+4)/4)); \\ Michel Marcus, Oct 11 2013

a(2n+2) = A229135(n+1); a(2n-1) = -A228767(n) for n>0.
a(n) = 6*a(n-1) -13*a(n-2) +12*a(n-3) -4*a(n-4) for n>5.
G.f.: (1-4*x+5*x^2-x^3-2*x^4+2*x^5)/((1-x)^2*(1-2*x)^2). - Colin Barker, Oct 09 2013

More terms from Colin Barker, Oct 09 2013

A239631 Triangular array read by rows: T(n,k) is the number of parts equal to k over all palindromic compositions of n, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 6, 3, 0, 1, 8, 2, 1, 0, 1, 16, 8, 2, 1, 0, 1, 20, 6, 4, 0, 1, 0, 1, 40, 20, 6, 4, 0, 1, 0, 1, 48, 16, 10, 2, 2, 0, 1, 0, 1, 96, 48, 16, 10, 2, 2, 0, 1, 0, 1, 112, 40, 24, 6, 6, 0, 2, 0, 1, 0, 1, 224, 112, 40, 24, 6, 6, 0, 2, 0, 1, 0, 1

Offset: 1

Geoffrey Critzer, Mar 22 2014

			1,
2,   1,
3,   0,  1,
6,   3,  0,  1,
8,   2,  1,  0, 1,
16,  8,  2,  1, 0, 1,
20,  6,  4,  0, 1, 0, 1,
40,  20, 6,  4, 0, 1, 0, 1,
48,  16, 10, 2, 2, 0, 1, 0, 1,
96,  48, 16, 10,2, 2, 0, 1, 0, 1,
112, 40, 24, 6, 6, 0, 2, 0, 1, 0, 1
In the palindromic compositions of 5: 5, 1+3+1, 2+1+2, 1+1+1+1+1  there are T(5,1)=8 ones, T(5,2)=2 twos, and T(5,3)=1 three and T(5,5)=1 five.

Phyllis Zweig Chinn, Ralph Grimaldi and Silvia Heubach, The Frequency of Summands of a Particular Size in Palindromic Compositions, Ars Combinatoria 69 (2003), 65-78.

Cf. A016116, A079859, A078836, A079861, A079862, A079863, A239632 (row sums).

nn=15;Table[Take[Drop[Transpose[Map[PadRight[#,nn+1]&,Level[Table[r=Solve[p==1/(1-x)-x^n+y x^n+(x^2/(1-x^2)-x^(2n)+y^2x^(2n))p,p];CoefficientList[Series[D[p/.r,y]/.y->1,{x,0,nn}],x],{n,1,nn}],{2}]]],1][[n]],n],{n,1,nn}]//Grid

Explicit formulas for T(n,k) given in reference [Chinn, Grimaldi, Heubach] as Theorem 6:

T(n,k) = 0 if n

T(n,k) = 2^(floor(n/2)-k)*(2 + floor(n/2) - k) if n>=2k and n!=k (mod 2);

T(n,k) = 1 if n=k;

T(n,k) = 2^((n-k)/2-1) if k

T(n,k) = 2^(floor(n/2)-k)*(2 + floor(n/2) - k + 2^floor((k+1)/2-1)) if n>=2k and n==k (mod 2).

O.g.f. for column k: x^k/(1-F(x^2)) + 2*x^(2*k)*(1 + F(x))/(1 - F(x^2))^2 where F(x)= x/(1-x).

Showing 1-7 of 7 results.

cp's OEIS Frontend

A078836 a(n) = n*2^(n-6).

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A079859 a(n) = n*2^(n-4).

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A079862 a(i) = the number of occurrences of 9's in the palindromic compositions of n=2*i-1 = the number of occurrences of 10's in the palindromic compositions of n=2*i.

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A079863 a(n) is the number of occurrences of 11s in the palindromic compositions of m=2*n-1 = the number of occurrences of 12s in the palindromic compositions of m=2*n.

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A079028 a(0) = 1, a(n) = (n + 4)*4^(n-1) for n >= 1.

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A227978 a(0)=1, a(1)=2; for n>1, a(n) = n*(2^n+4)/4.

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A079862 a(i) = the number of occurrences of 9's in the palindromic compositions of n=2i-1 = the number of occurrences of 10's in the palindromic compositions of n=2i.

A079863 a(n) is the number of occurrences of 11s in the palindromic compositions of m=2n-1 = the number of occurrences of 12s in the palindromic compositions of m=2n.