A102301 -id:A102301 - cp's OEIS frontend

A113861 a(n) = (1/9)((6n - 7)*2^(n-1) - (-1)^n).

0, 1, 5, 15, 41, 103, 249, 583, 1337, 3015, 6713, 14791, 32313, 70087, 151097, 324039, 691769, 1470919, 3116601, 6582727, 13864505, 29127111, 61050425, 127693255, 266571321, 555512263, 1155763769, 2401006023, 4980969017, 10319851975, 21355531833, 44142719431

Offset: 1

N. J. A. Sloane, Jan 25 2006

This sequence is connected with the Collatz problem (see the sequences A045883 and A001045). - Michel Lagneau, Jan 13 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
T. Etzion, On the stopping redundancy of Reed-Muller codes, IEEE Trans. Information Theory 52 (11) (2006) 4867-4879; arXiv:cs.IT/0511056.
Index entries for linear recurrences with constant coefficients, signature (3,0,-4).

Cf. A102301.

Cf. A059260, A016813.

Join[{0},Numerator[CoefficientList[Series[(x+1)/((x-1)*(x^2+x-2)),{x,0,40}],x]]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)
LinearRecurrence[{3,0,-4},{0,1,5},40] (* Harvey P. Dale, Jun 16 2022 *)

a(n)=((6*n-7)<<(n-1)-(-1)^n)/9 \\ Charles R Greathouse IV, Jan 13 2012

a(n+1) - 2*a(n) = A001045(n+2), Jacobsthal numbers. - Paul Curtz, Jul 05 2008
3*a(n) - a(n+1) = -1, -2, 4*a(n). - Paul Curtz, Jul 05 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: x^2*(1+2*x)/((1+x)*(1-2*x)^2).
a(n) + a(n+1) = A014480(n-1). (End)
a(n) = 4*a(n-1) - 4*a(n-2) + (-1)^(n+1), n>2. - Gary Detlefs, Dec 19 2010
a(n) = 3*a(n-1) - 4*a(n-3), n>3. - Gary Detlefs, Dec 19 2010
a(n) = n*2^n - A045883(n). - Michel Lagneau, Jan 13 2012
Starting with "1" = triangle A059260 * A016813 as a vector, where A016813 = (4n + 1): [ 1, 5, 9, 13, ...]. - Gary W. Adamson, Mar 06 2012

A115216 "Correlation triangle" for 2^n.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 8, 10, 10, 8, 16, 20, 21, 20, 16, 32, 40, 42, 42, 40, 32, 64, 80, 84, 85, 84, 80, 64, 128, 160, 168, 170, 170, 168, 160, 128, 256, 320, 336, 340, 341, 340, 336, 320, 256, 512, 640, 672, 680, 682, 682, 680, 672, 640, 512, 1024, 1280, 1344, 1360, 1364

Offset: 0

Paul Barry, Jan 16 2006

Row sums are A102301. T(2n,n) gives A002450(n+1). Diagonal sums are A115217.
Construction: Take antidiagonal triangle of MM^T where M is the sequence array for the sequence 2^n.
When formated as a square array, this is the self-fusion matrix (as in Example and Mathematica sections) of the sequence (2^n); for interlacing zeros of associated characteristic polynomials, see A202868.  [Clark Kimberling, Dec 26 2011]

			Triangle begins
  1,
  2, 2,
  4, 5, 4,
  8, 10, 10, 8,
  16, 20, 21, 20, 16,
  32, 40, 42, 42, 40, 32,
  ...
Northwest corner of square matrix:
  1....2....4....8....16
  2....5....10...20...40
  4....10...21...42...85
  8....20...41...85...170
  16...40...84...170..341
  ..

Cf. A003983, A202678.

(* A115216 as a square matrix *)
s[k_] := 2^(k - 1);
U = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[s[k], {k, 1, 12}]];
L = Transpose[U]; M = L.U; TableForm[M]
m[i_, j_] := M[[i]][[j]];
Flatten[Table[m[i, n + 1 - i], {n, 1, 12}, {i, 1, n}]]
f[n_] := Sum[m[i, n], {i, 1, n}] + Sum[m[n, j], {j, 1, n - 1}]
Table[f[n], {n, 1, 12}]
Table[Sqrt[f[n]], {n, 1, 12}]  (* -1+2^n *)
Table[m[n, n], {n, 1, 12}]  (* A002450 *)
(* Clark Kimberling, Dec 26 2011 *)

T(n, k) = Sum_{j=0..n} [j<=k]*2^(k-j)[j<=n-k]*2^(n-k-j).

G.f.: 1/((1-2*x)*(1-2*x*y)*(1-x^2*y)). - Christian G. Bower, Jan 17 2006

A102841 a(n) = ((9n^2 + 33n + 26)*2^n + (-1)^n)/27.

Original entry on oeis.org

1, 5, 19, 61, 179, 493, 1299, 3309, 8211, 19949, 47635, 112109, 260627, 599533, 1366547, 3089901, 6937107, 15476205, 34331155, 75769325, 166451731, 364127725, 793500179, 1723082221, 3729512979, 8048092653, 17319057939

Offset: 0

Creighton Dement, Feb 27 2005

nonn

A floretion-generated sequence relating the number of edges and faces in n-dimensional hypercube.
Equals A001787, (1, 4, 12, 32, 80, ...) convolved with A001045, the Jacobsthal sequence. - Gary W. Adamson, May 23 2009
The sum of the sizes of all inversions in compositions of n. - Arnold Knopfmacher, Jan 22 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-4,8).

Cf. A045883, A001788, A001793, A102301.

Cf. A001787, A001045. - Gary W. Adamson, May 23 2009

[((9*n^2 + 33*n + 26)*2^n + (-1)^n)/27 : n in [0..40]]; // Wesley Ivan Hurt, Jul 03 2020

Table[(1/27)*((9 n^2 + 33 n + 26) 2^n + (-1)^n), {n, 0, 50}] (* or *) LinearRecurrence[{5,-6,-4,8}, {1,5,19,61}, 50] (* G. C. Greubel, Sep 27 2017 *)

G.f.: 1/((1+x)*(1-2*x)^3).
a(n+1) - 2*a(n) = A045883(n+2).
a(n) + a(n+1) = A001788(n+2).
a(n) = 5*a(n-1) - 6*a(n-2) - 4*a(n-3) + 8*a(n-4). - Wesley Ivan Hurt, Jul 03 2020

Corrected by T. D. Noe, Nov 08 2006

A128971 A130125 * A000012.

Original entry on oeis.org

1, 2, 2, 5, 4, 4, 10, 10, 8, 8, 21, 20, 20, 16, 16, 42, 42, 40, 40, 32, 32, 85, 84, 84, 80, 80, 64, 64, 170, 170, 168, 168, 160, 160, 128

Offset: 0

Gary W. Adamson, May 11 2007

Left column = A000975: (1, 2, 5, 10, 21, 42, ...). Row sums = A102301: (1, 4, 13, 36, 93, ...). A130127 = A000012 * A130125.

			First few rows of the triangle:
   1;
   2,  2;
   5,  4,  4;
  10, 10,  8,  8;
  21, 20, 20, 16, 16;
  42, 42, 40, 40, 32, 32;
  85, 84, 84, 80, 80, 64, 64;
  ...

Cf. A000975, A102301, A130127, A000012, A130125.

A130125 * A000012 as infinite lower triangular matrices.

cp's OEIS Frontend

A113861 a(n) = (1/9)*((6*n - 7)*2^(n-1) - (-1)^n).

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A115216 "Correlation triangle" for 2^n.

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A102841 a(n) = ((9*n^2 + 33*n + 26)*2^n + (-1)^n)/27.

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A128971 A130125 * A000012.

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A113861 a(n) = (1/9)((6n - 7)*2^(n-1) - (-1)^n).

A102841 a(n) = ((9n^2 + 33n + 26)*2^n + (-1)^n)/27.