A023001 -id:A023001 - cp's OEIS frontend

A333813 a(n) = 2^(1 + floor(n*log_2(3))) - (3^n + 1).

0, 0, 6, 4, 46, 12, 294, 1908, 1630, 13084, 6486, 84996, 517134, 502828, 3605638, 2428308, 24062142, 5077564, 149450422, 985222180, 808182894, 6719515980, 2978678758, 43295774644, 267326277406, 252223018332, 1856180682774, 1170495537220

Offset: 0

Ctibor O. Zizka, Apr 06 2020

nonn

For integers X, Y, let a(n) = (X^(t+1) - 1) / (X - 1) - Y^n, where t = floor(n*log_X(Y)) . This sequence is for X = 2, Y = 3.

			a(0) = 2^(1 + floor(0*log_2(3))) - (3^0 + 1) = 0; a(4) = 2^(1 + floor(4*log_2(3))) - (3^4 + 1) = 46.

Cf. A000225, A024036.

Examples for integers X = Y from {2, 3, 4, 5, 6, 7, 8, 9, 10} are A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275. Examples for X = 2, Y = 4 are A024036; for X = 2, Y = 8, A024088; and for X = 3, Y = 9, A191681.

Table[2^(1+Floor[n Log2[3]])-(3^n+1),{n,0,30}] (* Harvey P. Dale, Sep 04 2023 *)

a(n) = 2^(1 + floor(n*log_2(3))) - (3^n + 1).

A360965 Array T(n,m) = (2^(n*m)-1)/(2^m-1) read by antidiagonals, n,m>=1.

Original entry on oeis.org

1, 1, 3, 1, 5, 7, 1, 9, 21, 15, 1, 17, 73, 85, 31, 1, 33, 273, 585, 341, 63, 1, 65, 1057, 4369, 4681, 1365, 127, 1, 129, 4161, 33825, 69905, 37449, 5461, 255, 1, 257, 16513, 266305, 1082401, 1118481, 299593, 21845, 511, 1, 513, 65793, 2113665, 17043521, 34636833, 17895697, 2396745, 87381, 1023

Offset: 1

R. J. Mathar, Feb 27 2023

			The array starts in rows n>=1 and columns m>=1 as
   1    1    1     1       1
   3    5    9    17      33
   7   21   73   273    1057
  15   85  585  4369   33825
  31  341 4681 69905 1082401

Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, JIS vol 19 (2016) #16.3.1 table 1.

Cf. A000225 (first col), A002450 (2nd col), A023001 (3rd col)

T(n,m) = (2^(n*m)-1)/(2^m-1) for n>1.

A378931 Triangle read by rows, based on products of Jacobsthal numbers (A001045).

Original entry on oeis.org

1, -1, 3, -2, -9, 15, -4, -18, -25, 55, -8, -36, -50, -121, 231, -16, -72, -100, -242, -441, 903, -32, -144, -200, -484, -882, -1849, 3655, -64, -288, -400, -968, -1764, -3698, -7225, 14535, -128, -576, -800, -1936, -3528, -7396, -14450, -29241, 58311, -256, -1152, -1600, -3872, -7056, -14792, -28900, -58482, -116281, 232903

Offset: 1

Werner Schulte, Dec 11 2024

Let M = T^(-1) be matrix inverse of T seen as a lower triangular matrix. M is a harmonic triangle with M(n, k) = 1 / A084175(n) if k = n, and 1 / A378676(k) if 1 <= k < n. Triangle M(n, k) for 1 <= k <= n starts:
  1/1
  1/3  1/3
  1/3  1/5  1/15
  1/3  1/5  1/33   1/55
  1/3  1/5  1/33  1/105  1/231
  1/3  1/5  1/33  1/105  1/473  1/903
  etc.
  Sum_{k=1..n} M(n, k) * 2^(k-1) = 1.
  Sum_{k=1..n} M(n, k) * (-2)^(k-1) = (-1)^(n-1) / A001045(n+1).
  Sum_{k=1..n} 2^(k-1) / M(n, k) = (8^n - 1) / 7 = A023001(n).

			Triangle T(n, k) for 1 <= k <= n starts:
n\k :     1     2     3      4      5      6       7       8      9
===================================================================
  1 :     1
  2 :    -1     3
  3 :    -2    -9    15
  4 :    -4   -18   -25     55
  5 :    -8   -36   -50   -121    231
  6 :   -16   -72  -100   -242   -441    903
  7 :   -32  -144  -200   -484   -882  -1849    3655
  8 :   -64  -288  -400   -968  -1764  -3698   -7225   14535
  9 :  -128  -576  -800  -1936  -3528  -7396  -14450  -29241  58311
  etc.

Cf. A001045, A023001, A378676.

A084175 (main diagonal), A139818 (1st subdiagonal), A000079 (column 1 and row sums).

T[n_,k_]:=If[k==n, (2*4^n-(-2)^n-1)/9, -2^(n-1-k)*(2^(k+1)+(-1)^k)^2/9]; Table[T[n,k],{n,10},{k,n}]//Flatten (* Stefano Spezia, Dec 11 2024 *)

T(n,k)=if(k==n,(2*4^n-(-2)^n-1)/9,-2^(n-1-k)*(2^(k+1)+(-1)^k)^2/9)

T(n, n) = (2 * 4^n - (-2)^n - 1) / 9 = A084175(n), and T(n, k) = -2^(n-1-k) * (2^(k+1) + (-1)^k)^2 / 9 for 1 <= k < n.

G.f.: x*t * (1 - 3*t - 6*x*t^2 + 8*x^2*t^3) / ((1 - 2*t) * (1 - x*t) * (1 + 2*x*t) * (1 - 4*x*t)).

cp's OEIS Frontend

A333813 a(n) = 2^(1 + floor(n*log_2(3))) - (3^n + 1).

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A360965 Array T(n,m) = (2^(n*m)-1)/(2^m-1) read by antidiagonals, n,m>=1.

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A378931 Triangle read by rows, based on products of Jacobsthal numbers (A001045).

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