A036563 -id:A036563 - cp's OEIS frontend

A095264 a(n) = 2^(n+2) - 3*n - 4.

1, 6, 19, 48, 109, 234, 487, 996, 2017, 4062, 8155, 16344, 32725, 65490, 131023, 262092, 524233, 1048518, 2097091, 4194240, 8388541, 16777146, 33554359, 67108788, 134217649, 268435374, 536870827, 1073741736, 2147483557, 4294967202, 8589934495, 17179869084, 34359738265

Offset: 1

Gary W. Adamson, May 31 2004

A sequence derived from a 3rd-order matrix generator.
The number of positive 3-strand braids of degree at most n. - R. J. Mathar, May 04 2006
Define a triangle T by T(n,n) = n*(n+1)/2, T(n,1) = n*(n-1) + 1, and T(r,c) = T(r-1,c-1) + T(r-1,c). Its rows are 1; 3,3; 7,6,6; 13,13,12,10; 21,26,25,22,15; etc. The sum of the terms in the n-th row is a(n). - J. M. Bergot, May 03 2013

			a(5) = 109 = 2^7 - 3*5 - 4.
a(5) = 109 since M^5 * [1 0 0] = [1 5 109].
a(7) = 487 = 4*234 - 5*109 + 2*48.

P. Dehornoy, Combinatorics of normal sequences of braids, arXiv:math/0511114 [math.CO], 2005.
Shishuo Fu, Zhicong Lin, and Yaling Wang, Refined Wilf-equivalences by Comtet statistics, arXiv:2009.04269 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A036563, A125232, A126277.

a[n_] := (MatrixPower[{{1, 0, 0}, {1, 1, 0}, {1, 3, 2}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 30}] (* Robert G. Wilson v, Jun 05 2004 *)
Table[2^(n+2)-3n-4,{n,40}] (* or *) LinearRecurrence[{4,-5,2},{1,6,19},40] (* Harvey P. Dale, Sep 24 2021 *)

Let M = [1 0 0 / 1 1 0 / 1 3 2], then M^n * [1 0 0] = [1 n a(n)]. The characteristic polynomial of M is x^3 - 4*x^2 + 5*x - 2.
a(n+3) = 4*a(n+2) - 5*a(n+1) + 2*a(n).
a(n) = Sum_{i=2..n+1} A036563(i) [A036563 is 2^i-3]. - Gerald McGarvey, Jun 28 2004
Row sums of A125232; 5th diagonal from the right of A126277; binomial transform of [1, 5, 8, 8, 8, ...]. - Gary W. Adamson, Dec 23 2006
a(n) = 2*a(n-1) + (3n-2). - Gary W. Adamson, Sep 30 2007
G.f.: -x*(1+2*x)/((2*x-1)*(x-1)^2). - R. J. Mathar, Nov 18 2007
E.g.f.: exp(x)*(4*exp(x) - 3*x - 4). - Elmo R. Oliveira, Apr 01 2025

Edited, corrected and extended by Robert G. Wilson v, Jun 05 2004

More terms from Elmo R. Oliveira, Apr 01 2025

A130459 A059268 * A097806.

Original entry on oeis.org

1, 3, 2, 3, 6, 4, 3, 6, 12, 8, 3, 6, 12, 24, 16, 3, 6, 12, 24, 48, 32, 3, 6, 12, 24, 48, 96, 64, 3, 6, 12, 24, 48, 96, 192, 128, 3, 6, 12, 24, 48, 96, 192, 384, 256

Offset: 1

Gary W. Adamson, May 26 2007

Row sums = A036563 starting (1, 5, 13, 29, 61, 125, ...).

			First few rows of the triangle:
  1;
  3, 2;
  3, 6,  4;
  3, 6, 12,  8;
  3, 6, 12, 24, 16;
  3, 6, 12, 24, 48, 32;
  ...

Cf. A059268, A097806, A036563.

A059268 * A097806 as infinite lower triangular matrices. A059268 = [1; 1,2; 1,2,4; ...]. A097806 = the pairwise operator.

A135857 Partial sums triangle based on A016777. Riordan convolution triangle ((1 + 2*x)/(1-x)^2, x/(1-x)).

Original entry on oeis.org

1, 4, 1, 7, 5, 1, 10, 12, 6, 1, 13, 22, 18, 7, 1, 16, 35, 40, 25, 8, 1, 19, 51, 75, 65, 33, 9, 1, 22, 70, 126, 140, 98, 42, 10, 1, 25, 92, 196, 266, 238, 140, 52, 11, 1, 28, 117, 288, 462, 504, 378, 192, 63, 12, 1

Offset: 0

Gary W. Adamson, Dec 01 2007

A007318 * a bidiagonal matrix with all 1's in the main diagonal and all 3's in the subdiagonal.
Row sums give A036563(n+2), n >= 0.
From Wolfdieter Lang, Mar 23 2015: (Start)
This is the triangle of iterated partial sums of A016777. Such iterated partial sums of arithmetic progression sequences have been considered by Narayana Pandit (see the Mar 20 2015 comment on A000580 where the MacTutor History of Mathematics archive link and the Gottwald et al. reference, p. 338, are given).
This is therefore the Riordan triangle ((1+2*x)/(1-x)^2, x/(1-x)) with o.g.f. of the columns ((1+2*x)/(1-x)^2)*(x/(1-x))^k, k >= 0.
The column sequences are A016777, A000326, A002411, A001296, A051836, A051923, A050494, A053367, A053310, for k = 0..8.
The alternating row sums are A122553(n) = {1, repeat(3)}.
The Riordan A-sequence is A(y) = 1 + y (implying the Pascal triangle recurrence for k >= 1).
The Riordan Z-sequence is A256096, leading to a recurrence for T(n,0) given in the formula section. See the link "Sheffer a- and z-sequences" under A006232 also for Riordan A- and Z-sequences with references. (End)
When the first column (k = 0) is removed from this triangle, the result is A125232. - Georg Fischer, Jul 26 2023

			The triangle T(n, k) begins:
n\k  0   1   2    3    4    5    6   7   8  9 10 11
0:   1
1:   4   1
2:   7   5   1
3:  10  12   6    1
4:  13  22  18    7    1
5:  16  35  40   25    8    1
6:  19  51  75   65   33    9    1
7:  22  70 126  140   98   42   10   1
8:  25  92 196  266  238  140   52  11   1
9:  28 117 288  462  504  378  192  63  12  1
10: 31 145 405  750  966  882  570 255  75 13  1
11: 34 176 550 1155 1716 1848 1452 825 330 88 14  1
... reformatted and extended by _Wolfdieter Lang_, Mar 23 2015
From _Wolfdieter Lang_, Mar 23 2015: (Start)
T(3, 1) = T(2, 0) + T(2, 1) = 7 + 5 = 12 (Pascal, from the A-sequence given above).
T(4, 0) = 4*T(3, 0) - 9*T(3, 1) + 27*T(3, 2) - 81* T(3, 3) = 4*10 - 9*12 + 27*6 - 81*1 = 13, from the Z-sequence given above and in A256096.
T(4, 0) = 2*T(3, 0) - T(2, 0) = 2*10 - 7 = 13.
(End)

Columns k=0-8 give: A016777, A000326, A002411, A001296, A051836, A051923, A050494, A053367, A053310.
Cf. A036563, A125232.
Cf. A110813. - Philippe Deléham, Feb 08 2009

Binomial transform of an infinite lower triangular matrix with all 1's in the main diagonal and all 3's in the subdiagonal; i.e., by columns - every column = (1, 3, 0, 0, 0, ...).
T(n,k) = (3n-2k+1)*binomial(n+1,k+1)/(n+1). - Philippe Deléham, Feb 08 2009
From Wolfdieter Lang, Mar 23 2015: (Start)
O.g.f. for row polynomials: (1 + 2*z)/((1- z*(1 + x))*(1 - z)) (see the Riordan property from the comment).
O.g.f. for column k (without leading zeros): (1 + 2*x)/(1-x)^(2+k), k >= 0, (Riordan property).
T(n, k) = T(n-1, k-1) + T(n-1, k) for k >= 1. From the Riordan A-sequence given above in a comment.
T(n, 0) = Sum_{j=0..n} Z(j)*T(n-1, j), for n >= 1, from the Riordan Z-sequence A256096 mentioned above in a comment. Of course, T(n, 0) = 2*T(n-1, 0) - T(n-2, 0) for n >= 2 (see A016777).
(End)

Edited. Offset is 0 from the old name and the Philippe Deléham formula. New name, old name as first comment. - Wolfdieter Lang, Mar 23 2015

A204203 Triangle based on (0,1/4,1) averaging array.

Original entry on oeis.org

1, 1, 5, 1, 6, 13, 1, 7, 19, 29, 1, 8, 26, 48, 61, 1, 9, 34, 74, 109, 125, 1, 10, 43, 108, 183, 234, 253, 1, 11, 53, 151, 291, 417, 487, 509, 1, 12, 64, 204, 442, 708, 904, 996, 1021, 1, 13, 76, 268, 646, 1150, 1612, 1900, 2017, 2045, 1, 14, 89, 344, 914

Offset: 1

Clark Kimberling, Jan 12 2012

See A204201 for a discussion and guide to other averaging arrays.

			First six rows:
1
1...5
1...6...13
1...7...19...29
1...8...26...48...61
1...9...34...74...109...125

Cf. A204201.

a = 0; r = 1/4; b = 1;  t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]    (* averaging array *)
u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]  (* A204203 triangle *)
Flatten[u]    (* A204203 sequence *)

From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A036563(n+1).
Sum_{k=1..n} T(n,k) = A014480(n-1).
T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=1, T(2,1)=1, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. (End)

A226616 Smallest positive integer k for which 1 is in a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

Original entry on oeis.org

1, 5, 13, 29, 11, 17, 253, 509, 145, 43, 55, 355, 137, 1129, 1007, 131069, 97, 643, 41, 553, 281, 8388605, 4069, 4793489, 3817, 1843, 59, 113, 1301, 2155, 9397, 289, 131153, 3247, 949, 127, 77

Offset: 2

Geoffrey H. Morley, Jul 02 2013

nonn

A cycle is called primitive if its elements are not a common multiple of the elements of another cycle.
The 3x+k function T_k is defined by T_k(x) = x/2 if x is even, (3x+k)/2 if x is odd, where k is odd.
For primitive cycles, GCD(k,6)=1.
For n>1, T_k has a primitive cycle of length n which includes 1 when k = A036563(n) = 2^n-3. So a(n) <= 2^n-3.

Cf. A226607, A226609, A226617, A226618.

A246168 a(n) = 2^n - 10.

Original entry on oeis.org

-9, -8, -6, -2, 6, 22, 54, 118, 246, 502, 1014, 2038, 4086, 8182, 16374, 32758, 65526, 131062, 262134, 524278, 1048566, 2097142, 4194294, 8388598, 16777206, 33554422, 67108854, 134217718, 268435446, 536870902, 1073741814, 2147483638

Offset: 0

Vincenzo Librandi, Aug 18 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form 2^n-k: A000079 (k=0), A000225 (k=1), A000918 (k=2), A036563 (k=3), A028399 (k=4), A168616 (k=5), A131130 (k=6), A048490 (k=7), A159741 (k=8), A185346 (k=9), this sequence (k=10).

Magma
```
[2^n-10: n in [0..40]];
```

Table[2^n - 10, {n, 0, 35}] (* or *) CoefficientList[Series[(-9 + 19 x)/(1 - 3 x + 2 x^2), {x, 0, 35}], x]
LinearRecurrence[{3,-2},{-9,-8},50] (* Harvey P. Dale, Jan 11 2024 *)

vector(50, n, 2^(n-1)-10) \\ Derek Orr, Aug 18 2014

G.f.: (-9+19*x)/(1-3*x+2*x^2).
a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = A000079(n) - 10.
From Elmo R. Oliveira, Dec 21 2023: (Start)
a(n) = 2*a(n-1) + 10 for n>0.
E.g.f.: exp(x)*(exp(x) - 10). (End)

A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

Original entry on oeis.org

0, 2, -1, 8, 1, -3, 26, 7, -1, -7, 80, 25, 5, -5, -15, 242, 79, 23, 1, -13, -31, 728, 241, 77, 19, -7, -29, -63, 2186, 727, 239, 73, 11, -23, -61, -127, 6560, 2185, 725, 235, 65, -5, -55, -125, -255, 19682, 6559, 2183, 721, 227, 49, -37, -119, -253, -511, 59048, 19681, 6557, 2179, 713, 211, 17, -101, -247, -509, -1023

Offset: 0

K. G. Stier, Jan 22 2015

Table shows differences of a given power of 3 to the powers of 2 (columns), and differences of the powers of 3 to a given power of 2 (rows), respectively.

Note that positive terms (table's upper right area) and negative terms (lower left area) are separated by an imaginary line with slope -log(3)/log(2) = -1.5849625.. (see A020857). This "border zone" of the table is of interest in terms of how close powers of 3 and powers of 2 can get: i.e., those T(n,k) where k/n is a good rational approximation to log(3)/log(2), see A254351 for numerators k and respective A060528 for denominators n.

			Table begins
   0    2   8  26  80..
  -1    1   7  25  79..
  -3   -1   5  23  73..
  -7   -5   1  19  65..
  -15 -13  -7  11  49..
  ..   ..  ..  ..  ..

K. G. Stier, Table of n, a(n) for n = 0..5150

Row 0 (=3^n-1) is A024023.
Row 1 (=3^n-2) is A058481.
Row 2 (=3^n-4) is A168611.
Column 0 (=1-2^n) is (-1)A000225.
Column 1 (=3-2^n) is (-1)A036563.
Column 2 (=9-2^n) is (-1)A185346.
Column 3 (=27-2^n) is (-1)A220087.
0,0-Diagonal (=3^n-2^n) is A001047.
1,0-Diagonal (=3^n-2^(n-1)) for n>0 is A083313 or A064686.
0,1-Diagonal (=3^n-2^(n+1)) is A003063.
0,2-Diagonal (=3^n-2^(n+2)) is A214091.

Table[3^(n-k) - 2^k, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 18 2017 *)

for(i=0, 10, {
     for(j=0, i, print1((3^(i-j)-2^j),", "))
});

A263018 If n is the i-th positive integer with binary weight j, then a(n) is the j-th positive integer with binary weight i.

Original entry on oeis.org

1, 3, 2, 7, 5, 11, 4, 15, 23, 47, 6, 95, 13, 27, 8, 31, 191, 383, 55, 767, 111, 223, 9, 1535, 447, 895, 14, 1791, 29, 59, 16, 63, 3071, 6143, 3583, 12287, 7167, 14335, 119, 24575, 28671, 57343, 239, 114687, 479, 959, 10, 49151, 229375, 458751, 1919, 917503

Offset: 1

Paul Tek, Oct 07 2015

Binary weight is given by A000120.
This is a self-inverse permutation of the natural numbers.
The positive terms in the sequence A036563 give the fixed points.
A000120(n) = A263017(a(n)) for any n>0.
A263017(n) = A000120(a(n)) for any n>0.
a(2^(n+1)-1) = 2^n for any n>0.
a(2^n) = 2^(n+1)-1 for any n>0.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, PERL program for this sequence

Cf. A000120, A036563, A263017.

a(n) = {j = hammingweight(n); v = vector(n, k, hammingweight(k)); i = #select(x->x==j, v); nb = 0; k = 0; while(nb != j, k++; if (hammingweight(k) == i, nb++)); k;} \\ Michel Marcus, Oct 16 2015

A331372 Decimal expansion of Sum_{k>=1} 1/(2^k - 3).

Original entry on oeis.org

3, 4, 3, 6, 7, 3, 4, 3, 3, 1, 8, 1, 7, 6, 9, 0, 1, 8, 5, 4, 4, 4, 8, 2, 8, 3, 3, 3, 8, 1, 2, 4, 1, 2, 0, 6, 1, 8, 8, 8, 0, 7, 1, 7, 6, 4, 8, 6, 7, 8, 3, 8, 4, 8, 6, 5, 1, 1, 0, 5, 9, 2, 1, 7, 4, 5, 5, 0, 0, 9, 5, 4, 1, 2, 4, 1, 8, 0, 9, 7, 4, 9, 5, 2, 6, 7, 8

Offset: 0

Amiram Eldar, May 03 2020

Erdős and Graham (1980) asked whether this constant is irrational, and Borwein (1991) proved that it is indeed irrational.

			0.34367343318176901854448283338124120618880717648678...

Paul Erdős, Some of my favourite unsolved problems, in A. Baker, B. Bollobás and A. Hajnal (eds.), A tribute to Paul Erdős, Cambridge University Press, 1990, p. 470.

Peter B. Borwein, On the irrationality of Sigma (1/(q^n + r)), Journal of Number Theory, Vol. 37, No. 3 (1991), pp. 253-259.
Paul Erdős and Ronald L. Graham, Old and new problems and results in combinatorial number theory, L'enseignement Mathématique, Université de Genève, 1980, p. 62.

Cf. A036563 (2^n-3), A065442.

RealDigits[Sum[1/(2^k - 3), {k, 1, 400}], 10, 100][[1]]

suminf(k=1, 1/(2^k - 3)) \\ Michel Marcus, May 03 2020

A344920 The Worpitzky transform of the squares.

Original entry on oeis.org

0, -1, 5, -13, 29, -61, 125, -253, 509, -1021, 2045, -4093, 8189, -16381, 32765, -65533, 131069, -262141, 524285, -1048573, 2097149, -4194301, 8388605, -16777213, 33554429, -67108861, 134217725, -268435453, 536870909, -1073741821, 2147483645, -4294967293

Offset: 0

Peter Luschny, Jun 24 2021

The Worpitzky transform maps a sequence A to a sequence B, where B(n) = Sum_{k=0..n} A163626(n, k)*A(k). (If A(n) = 1/(n + 1) then B(n) are the Bernoulli numbers (with B(1) = 1/2.))

Also row 2 in A371761. Can be generated by the signed Akiyama-Tanigawa algorithm for powers (see the Python script). - Peter Luschny, Apr 12 2024

Index entries for linear recurrences with constant coefficients, signature (-3,-2).

Up to shift and sign: even bisection A267921, odd bisection A141725.

Cf. A163626, A036563, A371761.

gf := (exp(x) - 1)*(exp(x) - 2)*exp(-2*x): ser := series(gf, x, 36):
seq(n!*coeff(ser, x, n), n = 0..31);

W[n_, k_] := (-1)^k k! StirlingS2[n + 1, k + 1];
WT[a_, len_] := Table[Sum[W[n, k] a[k], {k, 0, n}], {n, 0, len-1}];
WT[#^2 &, 32] (* The Worpitzky transform applied to the squares. *)

# Using the Akiyama-Tanigawa algorithm for powers from A371761.
print([(-1)**n * v for (n, v) in enumerate(ATPowList(2, 32))])
# Peter Luschny, Apr 12 2024

a(n) = n! * [x^n] (exp(x) - 1)*(exp(x) - 2)*exp(-2*x).
a(n) = (-1)^(n + 1)*(3 - 2^(n + 1)) for n >= 1. - Hugo Pfoertner, Jun 24 2021
a(n) = [x^n] x*(2*x - 1)/(2*x^2 + 3*x + 1). - Stefano Spezia, Jun 24 2021

cp's OEIS Frontend

A095264 a(n) = 2^(n+2) - 3*n - 4.

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A130459 A059268 * A097806.

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A135857 Partial sums triangle based on A016777. Riordan convolution triangle ((1 + 2*x)/(1-x)^2, x/(1-x)).

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Extensions

A204203 Triangle based on (0,1/4,1) averaging array.

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Mathematica

Formula

A226616 Smallest positive integer k for which 1 is in a primitive cycle of n positive integers (n>1) under iteration by the Collatz-like 3x+k function.

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A246168 a(n) = 2^n - 10.

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A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

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A263018 If n is the i-th positive integer with binary weight j, then a(n) is the j-th positive integer with binary weight i.

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PARI

A331372 Decimal expansion of Sum_{k>=1} 1/(2^k - 3).

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