A078836 -id:A078836 - cp's OEIS frontend

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

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

A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2n.

Original entry on oeis.org

10, 22, 48, 104, 224, 480, 1024, 2176, 4608, 9728, 20480, 43008, 90112, 188416, 393216, 819200, 1703936, 3538944, 7340032, 15204352, 31457280, 65011712, 134217728, 276824064, 570425344, 1174405120, 2415919104, 4966055936

Offset: 8

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 k's 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 >= 2*k.

			a(8)=10 since the palindromic compositions of 15 that contain a 7 are 7+1+7, 4+7+4, 1+3+7+3+1, 3+1+7+1+3, 2+2+7+2+2, 1+1+1+1+7+1+1+1+1, 1+1+2+7+2+1+1, 1+2+1+7+1+2+1 and 2+1+1+7+1+1+2, for a total of 10 7's.

Vincenzo Librandi, Table of n, a(n) for n = 8..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.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A057711, A001792, A079859, A061256, A079862, A079863, A111297.

[(2+n)*2^(n-8) : n in [8..40]]; // Vincenzo Librandi, Sep 22 2011

Table[(2 + i)*2^(i - 8), {i, 8, 50}]
LinearRecurrence[{4,-4},{10,22},50] (* Harvey P. Dale, Jun 04 2025 *)

Vec(-2*x^8*(9*x-5)/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Dec 16 2014

a(n) = (2+n)*2^(n-8).
a(n) = 2*A111297(n-6). - Colin Barker, Dec 16 2014
a(n) = 4*a(n-1) - 4*a(n-2). - Colin Barker, Dec 16 2014
G.f.: -2*x^8*(9*x-5) / (2*x-1)^2. - Colin Barker, Dec 16 2014
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=8} 1/a(n) = 1024*log(2) - 447047/630.
Sum_{n>=8} (-1)^n/a(n) = 261617/630 - 1024*log(3/2). (End)

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)

A129954 Second differences of A129952.

Original entry on oeis.org

1, 3, 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

Offset: 0

Paul Curtz, Jun 10 2007

First differences of A129953: a(n) = A129953(n+1) - A129953(n).

Essentially the same as A078836: a(n) = A078836(n+4) for n > 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A129952, A129953, A078836.

m:=16; S:=&cat[ [ 1, 2*i ]: i in [0..m] ]; T:=[ &+[ Binomial(j-1, k-1)*S[k]: k in [1..j] ]: j in [1..2*m] ]; U:=[ T[n+1]-T[n]: n in[1..2*m-1] ]; [ U[n+1]-U[n]: n in[1..2*m-2] ]; // Klaus Brockhaus, Jun 17 2007

{m=29; print1(1, ",", 3, ","); for(n=2, m, print1((n+4)*2^(n-2), ","))} \\ Klaus Brockhaus, Jun 17 2007

def A129954(n): return n+4<1 else 2*n+1 # Chai Wah Wu, Oct 03 2024

a(0) = 1, a(1) = 3; for n > 1, a(n) = (n+4)*2^(n-2).
G.f.: (1-x)*(1-2*x^2)/(1-2*x)^2.
Binomial transform of [1, 2, 1, 4, 1, 6, 1, 8, ...]. - Gary W. Adamson, Sep 29 2007
E.g.f.: (x + exp(2*x)*(2 + x))/2. - Stefano Spezia, Oct 04 2024

Edited and extended by Klaus Brockhaus, Jun 17 2007

A196410 a(n) = n*2^(n-5).

Original entry on oeis.org

5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, 2883584, 6029312, 12582912, 26214400, 54525952, 113246208, 234881024, 486539264, 1006632960, 2080374784, 4294967296, 8858370048, 18253611008, 37580963840, 77309411328, 158913789952

Offset: 5

Vincenzo Librandi, Oct 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 5..3000
Index entries for linear recurrences with constant coefficients, signature (4, -4).

Cf. A045623, A078836.

Magma
```
[n*2^(n-5): n in [5..40]]
```

Table[n*2^(n-5), {n, 5, 37}] (* Amiram Eldar, Jan 12 2021 *)

G.f.: x^5*( 5-8*x ) / (2*x-1)^2 . - R. J. Mathar, Oct 13 2011
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=5} 1/a(n) = 32*log(2) - 131/6.
Sum_{n>=5} (-1)^(n+1)/a(n) = 32*log(3/2) - 77/6. (End)

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 1-8 of 8 results.

cp's OEIS Frontend

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

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A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2*n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2*n.

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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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A129954 Second differences of A129952.

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A196410 a(n) = n*2^(n-5).

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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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A079861 a(n) is the number of occurrences of 7's in the palindromic compositions of 2n-1, or also, the number of occurrences of 8's in the palindromic compositions of 2n.

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.