A056182 -id:A056182 - cp's OEIS frontend

A145563 a(0)=0 and a(n+1) = 3*a(n) + 2^(n+2).

0, 4, 20, 76, 260, 844, 2660, 8236, 25220, 76684, 232100, 700396, 2109380, 6344524, 19066340, 57264556, 171924740, 516036364, 1548633380, 4646948716, 13942943300, 41833024204, 125507461220, 376539160876, 1129651037060, 3389020220044, 10167194877860

Offset: 0

N. J. A. Sloane, Mar 18 2009

nonn

Suggested by a discussion on the Sequence Fans Mailing List; the formula is due to Andrew V. Sutherland.

First differences of A255459. - Klaus Purath, Apr 25 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..170
Index entries for linear recurrences with constant coefficients, signature (5,-6).

Cf. A001047, A056182, A210448, A217764, A245804.

[ 4*(3^n - 2^n): n in [0..50]]; // Vincenzo Librandi, Apr 24 2011

CoefficientList[Series[4x/((1-2x)(1-3x)),{x,0,40}],x] (* or *) RecurrenceTable[{a[0]==0, a[n]==(3a[n-1]+2^(n+1))},a,{n,40}] (* Harvey P. Dale, Apr 24 2011 *)

a(n) = 4*(3^n - 2^n) \\ Felix Fröhlich, Sep 01 2018

From R. J. Mathar, Mar 18 2009: (Start)
a(n) = 4*(3^n - 2^n) = 4*A001047(n).
G.f.: 4*x/((1-2*x)*(1-3*x)). (End)
a(n) = A056182(n)*2. - Omar E. Pol, Mar 18 2009
a(n) = A217764(n,7). - Ross La Haye, Mar 27 2013
From Klaus Purath, Apr 25 2020: (Start)
a(n) = 5*a(n-1) - 6*a(n-2).
a(n) = 2*A210448(n) - A056182(n).
a(n) = (A056182(n) + A245804(n+1))/2. (End)

A260217 Number of base-3 n-digit pandigital numbers.

Original entry on oeis.org

0, 0, 4, 24, 100, 360, 1204, 3864, 12100, 37320, 114004, 346104, 1046500, 3155880, 9500404, 28566744, 85831300, 257756040, 773792404, 2322425784, 6969374500, 20912317800, 62745342004, 188252803224, 564791964100, 1694443001160, 5083463221204, 15250658099064

Offset: 1

Jon E. Schoenfield, Jul 19 2015

From Manfred Boergens, Aug 02 2023: (Start)
a(n+1) is the number of pairs (A,B) where A and B are nonempty subsets of {1,2,...,n} and one of these subsets is a proper subset of the other.
If "proper" is omitted, see A091344.
If empty subsets are included, see A027649 (all subsets) and A056182 (proper subsets). (End)

			a(3)=4 because, in base 3, there are four 3-digit pandigital numbers (11=102_3, 15=120_3, 19=201_3, and 21=210_3).
a(4)=24 because, in base 3, there are 24 4-digit pandigital numbers (1002_3, 1012_3, 1020_3, 1021_3, 1022_3, 1102_3, 1120_3, 1200_3, 1201_3, 1202_3, 1210_3, 1220_3, 2001_3, 2010_3, 2011_3, 2012_3, 2021_3, 2100_3, 2101_3, 2102_3, 2110_3, 2120_3, 2201_3, and 2210_3).

Svenja Huntemann, Values, Temperatures, and Enumeration of Placement Games, Slides, Alberta-Montana Combinatorics and Algorithms Day, Banff, Canada, 23-25 June 2023. See p. 105/109.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000392, A027649, A028243, A056182, A091344, A171102.

[2*3^(n-1) - 2^(n+1) + 2: n in [1..30]]; // Vincenzo Librandi, Jul 20 2015

Table[2 3^(n - 1) - 2^(n + 1) + 2, {n, 30}] (* Vincenzo Librandi, Jul 20 2015 *)

a(n) = 2*A028243(n) = 2*3^(n-1) - 2^(n+1) + 2.
a(n) = 4*A000392(n).
G.f.: 4*x^3/((1-x)*(1-2*x)*(1-3*x)).
E.g.f.: 2/3*((exp(x)-1)^3).

A350771 Triangle read by rows: T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k), 0 <= k <= n-1.

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 7, 12, 12, 7, 15, 32, 36, 32, 15, 31, 80, 100, 100, 80, 31, 63, 192, 270, 280, 270, 192, 63, 127, 448, 714, 770, 770, 714, 448, 127, 255, 1024, 1848, 2128, 2100, 2128, 1848, 1024, 255, 511, 2304, 4680, 5880, 5796, 5796, 5880, 4680, 2304, 511, 1023, 5120, 11610, 16080, 16380, 15624, 16380, 16080, 11610, 5120, 1023

Offset: 1

Ambrosio Valencia-Romero, Jan 14 2022

The elements in T(n,k) result from the product of each element of A350770(n,k) and binomial(n-1,k).

			Triangle begins:
     0;
     1,    1;
     3,    4,     3;
     7,   12,    12,     7;
    15,   32,    36,    32,    15;
    31,   80,   100,   100,    80,    31;
    63,  192,   270,   280,   270,   192,    63;
   127,  448,   714,   770,   770,   714,   448,   127;
   255, 1024,  1848,  2128,  2100,  2128,  1848,  1024,   255;
   511, 2304,  4680,  5880,  5796,  5796,  5880,  4680,  2304,  511;
  1023, 5120, 11610, 16080, 16380, 15624, 16380, 16080, 11610, 5120, 1023;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Ambrosio Valencia-Romero and P. T. Grogan, The strategy dynamics of collective systems: Underlying hindrances beyond two-actor coordination, PLOS ONE 19(4): e0301394 (S1 Appendix).

Column k=0 gives A000225(n-1).

Row sums give A056182(n-1) = 2*A001047(n-1).

T := n -> local k; seq((2^(n - k - 1) + 2^k - 2)*binomial(n - 1, k), k = 0 .. n - 1);
seq(T(n), n = 1 .. 11);

T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k) \\ Andrew Howroyd, Jan 05 2024

T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k).

A071055 Number of 0's in n-th row of triangle in A071038.

Original entry on oeis.org

0, 0, 2, 0, 2, 2, 6, 0, 2, 2, 6, 2, 6, 6, 14, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 2, 6, 6, 14, 6, 14, 14, 30, 6, 14, 14, 30, 14, 30, 30, 62, 0, 2, 2, 6, 2, 6, 6, 14, 2, 6, 6, 14, 6, 14, 14, 30, 2, 6, 6, 14, 6

Offset: 0

Hans Havermann, May 26 2002

nonn

a(n) is also the number of pairs of consecutive entries in the n-th row of Pascal's triangle with opposite parity.
All terms appear to be of the form 2^k - 2 (checked for n <= 10000). - Michael De Vlieger, Mar 02 2015
This appears to be equal to the number of previous values k, from 1..n-1, such that k AND n = k, where 'AND' is binary AND, and where the sequence starts at 1. For example, 1 AND 2 = 0, so a(2) = 0, while 1 AND 3 = 1 and 2 AND 3 = 2, so a(3) = 2. It follows from this that if n = 2^m - 1 then a(n) = n - 1 = 2^m - 2, giving the right border values noted below. - Scott R. Shannon, Apr 19 2023

			From _Omar E. Pol_, Mar 02 2015: (Start)
Also, written as an irregular triangle in which the row lengths are the powers of 2, the sequence begins:
0;
0,2;
0,2,2,6;
0,2,2,6,2,6,6,14;
0,2,2,6,2,6,6,14,2,6,6,14,6,14,14,30;
0,2,2,6,2,6,6,14,2,6,6,14,6,14,14,30,2,6,6,14,6,14,14,30,6,14,14,30,14,30,30,62;
...
It appears that the right border gives the nonnegative terms of A000918.
It appears that the row sums give A056182.
(End)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Rule 182
Index entries for sequences related to cellular automata

Cf. A071042.

Count[#, n_ /; n == 0] & /@
Flatten[CellularAutomaton[182, {{1}, 0}, {{#}}] & /@ Range[0, 100],
1] (* Michael De Vlieger, Mar 02 2015 *)

A011371(n)=my(s); while(n>>=1, s+=n); s
a(n)=my(t=A011371(n)); sum(k=1,n,(A011371(k)+A011371(n-k)==t)!=(A011371(k-1)+A011371(n-k+1)==t)) \\ Charles R Greathouse IV, Mar 02 2015

a(n)=b(n+1), with b(0)=0, b(2n)=b(n), b(2n+1)=2b(n)+2-2[n==0] (conjectured). - Ralf Stephan, Mar 05 2004

a(n) = pext(n, n + 1) (conjectured) where pext is the "parallel bits extract" instruction of the x86 CPU; pext(x, mask) extracts bits from x at the bit locations specified by mask to contiguous low bits. - Falk Hüffner, Jul 26 2019

A326816 a(0) = 0, a(1) = 1, and for n > 1, a(n) = Sum_{k = 0..n} a((n-k) AND k) (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 1, 1, 0, 3, 2, 2, 0, 9, 10, 10, 12, 12, 8, 4, 0, 27, 38, 46, 60, 66, 68, 72, 72, 90, 84, 76, 72, 44, 24, 8, 0, 81, 130, 182, 228, 302, 332, 384, 360, 526, 572, 636, 600, 624, 576, 568, 432, 764, 888, 996, 1008, 972, 936, 888, 864, 712, 560, 408, 320, 144

Offset: 0

Rémy Sigrist, Oct 20 2019

This sequence combines features of A006581 and of A007461.

			a(2) = a(2 AND 0) + a(1 AND 1) + a(0 AND 2) = a(0) + a(1) + a(0) = 1.

Rémy Sigrist, Table of n, a(n) for n = 0..8191

Cf. A006581, A007461, A056182.

a:= proc(n) option remember; `if`(n<2, n,
      add(a(Bits[And](n-k, k)), k=0..n))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Oct 20 2019

a = vector(61); for (n=0, #a-1, print1 (a[1+n] = if (n==0, 0, n==1, 1, sum (k=0, n, a[1+bitand(n-k,k)])) ", "))

a(n) is odd iff n is a power of 2.
a(n) = 0 iff n = 2^k with k = 0 or k = 2.
a(2^k) = 3^(k-1) for any k > 0.
a(2^k+1) = A056182(k-1) for any k > 1.

cp's OEIS Frontend

A145563 a(0)=0 and a(n+1) = 3*a(n) + 2^(n+2).

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A260217 Number of base-3 n-digit pandigital numbers.

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A350771 Triangle read by rows: T(n, k) = (2^(n-k-1) + 2^k - 2)*binomial(n-1, k), 0 <= k <= n-1.

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A071055 Number of 0's in n-th row of triangle in A071038.

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A326816 a(0) = 0, a(1) = 1, and for n > 1, a(n) = Sum_{k = 0..n} a((n-k) AND k) (where AND denotes the bitwise AND operator).

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