A079859 -id:A079859 - cp's OEIS frontend

A087447 a(0) = a(1) = 1; for n > 1, a(n) = (n+2)*2^(n-2).

1, 1, 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

Offset: 0

Paul Barry, Sep 05 2003

Binomial transform of A005408 (with interpolated zeros). Binomial transform is A087448. a(n+2) = 2*A045623(n+1); a(n+1) = A001792(n) + (0^n - (-2)^n)/2. The sequence 1,4,10,... given by 2^n*(n+3)/2 - 0^n/2 is the binomial transform of 1,3,3,5,5,...
Equals real part of binomial transform of [1, 2*i, 3, 4*i, 5, 6*i, ...]; i=sqrt(-1). - Gary W. Adamson, Sep 21 2008
An elephant sequence, see A175655. For the central square 24 A[5] vectors, with decimal values between 27 and 432, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A057711 (without the leading 0). - Johannes W. Meijer, Aug 15 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Essentially same as A079859.

Cf. A001792, A005408, A045623, A057711, A087448, A175655.

Join[{1, 1}, Table[(n + 2) 2^(n - 2), {n, 2, 30}]]  (* Harvey P. Dale, Feb 22 2011 *)

def A087447(n): return n+2<1 else 1 # Chai Wah Wu, Oct 03 2024

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*(2k+1). - Paul Barry, Nov 29 2004
From Colin Barker, Mar 23 2012: (Start)
G.f.: (1-x)*(1-2*x+2*x^2)/(1-2*x)^2.
a(n) = 4*a(n-1) - 4*a(n-2) for n > 3. (End)
E.g.f.: (1 - x + exp(2*x)*(1 + x))/2. - Stefano Spezia, Jan 31 2023

Definition corrected (by a factor of 2) by R. J. Mathar, Feb 21 2009

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

Original entry on oeis.org

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)

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)

A049089 Array read by antidiagonals: T(1,j)=2j+2 i>=1, T(i,1)=2i+2 i>=1, T(i,j)=T(i-1,j-1)+T(i-1,j).

Original entry on oeis.org

4, 6, 6, 8, 10, 8, 10, 16, 14, 10, 12, 24, 24, 18, 12, 14, 34, 40, 32, 22, 14, 16, 46, 64, 56, 40, 26, 16, 18, 60, 98, 96, 72, 48, 30, 18, 20, 76, 144, 160, 128, 88, 56, 34, 20, 22, 94, 204, 258, 224, 160, 104, 64, 38, 22, 24, 114, 280, 402, 384, 288, 192, 120, 72, 42, 24, 26, 136, 374, 606, 642

Offset: 1

Benoit Cloitre, Jun 17 2003

			Table begins:
   4,   6,   8,  10,  12, ...
   6,  10,  14,  18,  22, ...
   8,  16,  24,  32,  40, ...
  10,  24,  40,  56,  72, ...
  12,  34,  64,  96, 128, ...
  ...

Cf. A079859.

Offset corrected by Sean A. Irvine, Jul 18 2021

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

A087447 a(0) = a(1) = 1; for n > 1, a(n) = (n+2)*2^(n-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A049089 Array read by antidiagonals: T(1,j)=2j+2 i>=1, T(i,1)=2i+2 i>=1, T(i,j)=T(i-1,j-1)+T(i-1,j).

Views

Author

Keywords

Examples

Crossrefs

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

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.