A060546 -id:A060546 - cp's OEIS frontend

A320770 a(n) = (-1)^floor(n/4) * 2^floor(n/2).

1, 1, 2, 2, -4, -4, -8, -8, 16, 16, 32, 32, -64, -64, -128, -128, 256, 256, 512, 512, -1024, -1024, -2048, -2048, 4096, 4096, 8192, 8192, -16384, -16384, -32768, -32768, 65536, 65536, 131072, 131072, -262144, -262144, -524288, -524288, 1048576, 1048576

Offset: 0

Michael Somos, Oct 20 2018

			G.f. = 1 + x + 2*x^2 + 2*x^3 - 4*x^4 - 4*x^5 - 8*x^6 - 8*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-4).

Cf. A016116.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

[(-1)^Floor(n/4)* 2^Floor(n/2): n in [0..50]]; // G. C. Greubel, Oct 27 2018

a[ n_] := (-1)^Quotient[n, 4] * 2^Quotient[n, 2];

{a(n) = (-1)^floor(n/4) * 2^floor(n/2)};

def A320770(n): return -(1<<(n>>1)) if n&4 else 1<<(n>>1) # Chai Wah Wu, Jan 18 2023

G.f.: (1 + x) * (1 + 2*x^2) / (1 + 4*x^4).
G.f.: A(x) = 1/(1 - x/(1 - x/(1 + 2*x/(1 - 4*x/(1 + 3*x/(1 + 5*x/(3 - 2*x))))))).
a(n) = (-1)^floor(n/2) * 2 * a(n-2) = -4 * a(n-4) for all n in Z.
a(n) = c(n) * (-2)^n * a(-n) for all n in Z where c(4*k+2) = -1 else 1.
a(n) = a(n+1) = (1+I)^n * (-I)^(n/2) * (-1)^floor(n/4) if n = 2*k.
a(n) =  (-1)^floor(n/4) * A016116(n).
E.g.f.: cosh(x)*(cos(x) + sin(x)) + sin(x)*sinh(x). - Stefano Spezia, Feb 04 2023

A343177 a(0)=4; if n > 0 is even then a(n) = 2^(n/2+1)+3, otherwise a(n) = 3*(2^((n-1)/2)+1).

Original entry on oeis.org

4, 6, 7, 9, 11, 15, 19, 27, 35, 51, 67, 99, 131, 195, 259, 387, 515, 771, 1027, 1539, 2051, 3075, 4099, 6147, 8195, 12291, 16387, 24579, 32771, 49155, 65539, 98307, 131075, 196611, 262147, 393219, 524291, 786435, 1048579, 1572867, 2097155, 3145731, 4194307, 6291459

Offset: 0

N. J. A. Sloane, Apr 26 2021

Number of edges along the boundary of the graph G(n) described in A342759.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A342759.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

f:=n->if n = 0 then 4 elif (n mod 2) = 0 then 2^(n/2+1)+3 else 3*(2^((n-1)/2)+1); fi;
[seq(f(n),n=0..40)];

LinearRecurrence[{1, 2, -2}, {4, 6, 7, 9}, 50] (* or *)
A343177[n_] := Which[n == 0, 4, OddQ[n], 3*(2^((n-1)/2)+1), True, 2^(n/2+1)+3];
Array[A343177, 50, 0] (* Paolo Xausa, Feb 02 2024 *)

G.f.: (4 + 2*x - 7*x^2 - 2*x^3)/((1 - x)*(1 - 2*x^2)). - Stefano Spezia, Feb 04 2023

E.g.f.: 3*cosh(x) + 2*cosh(sqrt(2)*x) + 3*sinh(x) + 3*sinh(sqrt(2)*x)/sqrt(2) - 1. - Stefano Spezia, Jul 25 2024

A354785 Numbers of the form 32^k or 92^k.

Original entry on oeis.org

3, 6, 9, 12, 18, 24, 36, 48, 72, 96, 144, 192, 288, 384, 576, 768, 1152, 1536, 2304, 3072, 4608, 6144, 9216, 12288, 18432, 24576, 36864, 49152, 73728, 98304, 147456, 196608, 294912, 393216, 589824, 786432, 1179648, 1572864, 2359296, 3145728, 4718592, 6291456, 9437184, 12582912, 18874368, 25165824, 37748736, 50331648

Offset: 1

N. J. A. Sloane, Jul 12 2022

Index entries for linear recurrences with constant coefficients, signature (0,2).

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283.

seq[max_] := Union[Table[3*2^n, {n, 0, Floor[Log2[max/3]]}], Table[9*2^n, {n, 0, Floor[Log2[max/9]]}]]; seq[10^8] (* Amiram Eldar, Jan 16 2024 *)

Sum_{n>=1} 1/a(n) = 8/9. - Amiram Eldar, Jan 16 2024

G.f.: (3*x^2+6*x+3)/(1-2*x^2). - Georg Fischer, Apr 10 2025

A036991 Numbers k with the property that in the binary expansion of k, reading from right to left, the number of 0's never exceeds the number of 1's.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 13, 15, 19, 21, 23, 27, 29, 31, 39, 43, 45, 47, 51, 53, 55, 59, 61, 63, 71, 75, 77, 79, 83, 85, 87, 91, 93, 95, 103, 107, 109, 111, 115, 117, 119, 123, 125, 127, 143, 151, 155, 157, 159, 167, 171, 173, 175, 179, 181, 183, 187, 189, 191, 199, 203

Offset: 1

N. J. A. Sloane

List of binary words that correspond to a valid pairing of parentheses. - Joerg Arndt, Nov 27 2004
This sequence includes as subsequences A000225, A002450, A007583, A036994, A052940, A112627, A113836, A113841, A290114; and also A015521 (without 0), A083713 (without 0), A086224 (without 6), A182512 (without 0). - Gennady Eremin, Nov 27 2021 and Aug 26 2023
Partial differences are powers of 2 (cf. A367626, A367627). - Gennady Eremin, Dec 23 2021
This is the sequence A030101(A014486(n)), n >= 0, sorted into ascending order. See A014486 for more references, illustrations, etc., concerning Dyck paths and other associated structures enumerated by the Catalan numbers. - Antti Karttunen, Sep 25 2023
The terms in this sequence with a given length in base 2 are counted by A001405. For example, the number of terms of bit length k=5 (these are 19, 21, 23, 27, 29, and 31) is equal to A001405(k-1) = A001405(4) = 6. - Gennady Eremin, Nov 07 2023

			From _Joerg Arndt_, Dec 05 2021: (Start)
List of binary words with parentheses for those in the sequence (indicated by P). The binary words are scanned starting from the least significant bit, while the parentheses words are written left to right:
     Binary   Parentheses (if the value is in the sequence)
00:  ..... P  [empty string]
01:  ....1 P   ()
02:  ...1.
03:  ...11 P   (())
04:  ..1..
05:  ..1.1 P   ()()
06:  ..11.
07:  ..111 P   ((()))
08:  .1...
09:  .1..1
10:  .1.1.
11:  .1.11 P   (()())
12:  .11..
13:  .11.1 P   ()(())
14:  .111.
15:  .1111 P   (((())))
16:  1....
17:  1...1
18:  1..1.
19:  1..11 P   (())()
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..13496 (all terms < 2^16; first 1000 terms from Reinhard Zumkeller)
Joerg Arndt, Matters Computational (The Fxtbook), section 1.28, pp. 78-80.
Gennady Eremin, Dyck Numbers, IV. Nested patterns in OEIS A036991, arXiv:2306.10318, 2023. See pp. 1-2, 4, 7-11, 13-14.
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp 2-5, 14.
Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See pp. 3, 11.
Index entries for sequences related to binary expansion of n

Cf. A350577 (primes subsequence).
See also A014486, A030101, A036988, A036990, A036992. A036994 is a subset (requires the count of zeros to be strictly less than the count of 1's).
See also A030308, A000225, A002450, A007583, A350346, A367625, A367626 & A367627 (first differences).

a036991 n = a036991_list !! (n-1)
a036991_list = filter ((p 1) . a030308_row) [0..] where
   p     []    = True
   p ones (0:bs) = ones > 1 && p (ones - 1) bs
   p ones (1:bs) = p (ones + 1) bs
-- Reinhard Zumkeller, Jul 31 2013

q:= proc(n) local l, t, i; l:= Bits[Split](n); t:=0;
      for i to nops(l) do t:= t-1+2*l[i];
        if t<0 then return false fi
      od: true
    end:
select(q, [$0..300])[];  # Alois P. Heinz, Oct 09 2019

moreOnesRLQ[n_Integer] := Module[{digits, len, flag = True, iter = 1, ones = 0, zeros = 0}, digits = Reverse[IntegerDigits[n, 2]]; len = Length[digits]; While[flag && iter < len, If[digits[[iter]] == 1, ones++, zeros++]; flag = ones >= zeros; iter++]; flag]; Select[Range[0, 203], moreOnesRLQ] (* Alonso del Arte, Sep 21 2011 *)
Join[{0},Select[Range[210],Min[Accumulate[Reverse[IntegerDigits[#,2]]/.{0->-1}]]>-1&]] (* Harvey P. Dale, Apr 18 2014 *)

select( {is_A036991(n,c=1)=!n||!until(!n>>=1,(c-=(-1)^bittest(n,0))||return)}, [0..99]) \\ M. F. Hasler, Nov 26 2021

def ok(n):
    if n == 0: return True # by definition
    count = {"0": 0, "1": 0}
    for bit in bin(n)[:1:-1]:
        count[bit] += 1
        if count["0"] > count["1"]: return False
    return True
print([k for k in range(204) if ok(k)]) # Michael S. Branicky, Nov 25 2021

from itertools import count, islice
def A036991_gen(): # generator of terms
    yield 0
    for n in count(1):
        s = bin(n)[2:]
        c, l = 0, len(s)
        for i in range(l):
            c += int(s[l-i-1])
            if 2*c <= i:
                break
        else:
            yield n
A036991_list = list(islice(A036991_gen(),20)) # Chai Wah Wu, Dec 30 2021

If a(n) = A000225(k) for some k, then a(n+1) = a(n) + A060546(k). - Gennady Eremin, Nov 07 2023

More terms from Erich Friedman
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar
Offset corrected and example adjusted accordingly by Reinhard Zumkeller, Jul 31 2013

A083906 Table read by rows: T(n, k) is the number of length n binary words with exactly k inversions.

Original entry on oeis.org

1, 2, 3, 1, 4, 2, 2, 5, 3, 4, 3, 1, 6, 4, 6, 6, 6, 2, 2, 7, 5, 8, 9, 11, 9, 7, 4, 3, 1, 8, 6, 10, 12, 16, 16, 18, 12, 12, 8, 6, 2, 2, 9, 7, 12, 15, 21, 23, 29, 27, 26, 23, 21, 15, 13, 7, 4, 3, 1, 10, 8, 14, 18, 26, 30, 40, 42, 48, 44, 46, 40, 40, 30, 26, 18, 14, 8, 6, 2, 2

Offset: 0

Alford Arnold, Jun 19 2003

There are A033638(n) values in the n-th row, compliant with the order of the polynomial.
In the example for n=6 detailed below, the orders of [6, k]_q are 1, 6, 9, 10, 9, 6, 1 for k = 0..6,
the maximum order 10 defining the row length.
Note that 1 6 9 10 9 6 1 and related distributions are antidiagonals of A077028.
A083480 is a variation illustrating a relationship with numeric partitions, A000041.
The rows are formed by the nonzero entries of the columns of A049597.
The coefficient of q^j in the Gaussian polynomial [n, m]q is the number of binary words on alphabet {0,1} of length n having m 1's and j inversions. Hence T(n, k) is the number of length n binary words with exactly k inversions. - _Geoffrey Critzer, May 14 2017
If n is even the n-th row converges to n+1, n-1, n-4, ..., 19, 13, 7, 4, 3, 1 which is A029552 reversed, and if n is odd the sequence is twice A098613. - Michael Somos, Jun 25 2017

			When viewed as an array with A033638(r) entries per row, the table begins:
. 1 ............... : 1
. 2 ............... : 2
. 3 1 ............. : 3 + q = (1) + (1+q) + (1)
. 4 2 2 ........... : 4 + 2q + 2q^2 = 1 + (1+q+q^2) + (1+q+q^2) + 1
. 5 3 4 3 1 ....... : 5 + 3q + 4q^2 + 3q^3 + q^4
. 6 4 6 6 6 2 2
. 7 5 8 9 11 9 7 4 3 1
. 8 6 10 12 16 16 18 12 12 8 6 2 2
. 9 7 12 15 21 23 29 27 26 23 21 15 13 7 4 3 1
...
The second but last row is from the sum over 7 q-polynomials coefficients:
. 1 ....... : 1 = [6,0]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,1]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,2]_q
. 1 1 2 3 3 3 3 2 1 1 ....... : 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9 = [6,3]_q
. 1 1 2 2 3 2 2 1 1 ....... : 1+q+2q^2+2q^3+3q^4+2q^5+2q^6+q^7+q^8 = [6,4]_q
. 1 1 1 1 1 1 ....... : 1+q+q^2+q^3+q^4+q^5 = [6,5]_q
. 1 ....... : 1 = [6,6]_q

George E. Andrews, 'Theory of Partitions', 1976, page 242.

Seiichi Manyama, Rows n = 0..48, flattened
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Alexander Gruber, "The Egg:" Bizarre behavior of the roots of a family of polynomials Mathematics StackExchange Oct 04 2012
MathOverflow, Partition numbers and Gaussian binomial coefficient
Eric Weisstein, q-Binomial Coefficient, Mathworld.
Wikipedia, q-binomial
Index entries for sequences related to binary linear codes

Cf. A000025, A000034, A000041, A016116, A029552, A033638, A060546, A063746, A077028, A083479, A083480, A098613, A260460.

R:=PowerSeriesRing(Rationals(), 100);
qBinom:= func< n,k,x | n eq 0 or k eq 0 select 1 else (&*[(1-x^(n-j))/(1-x^(j+1)): j in [0..k-1]]) >;
A083906:= func< n,k | Coefficient(R!((&+[qBinom(n,k,x): k in [0..n]]) ), k) >;
[A083906(n,k): k in [0..Floor(n^2/4)], n in [0..12]]; // G. C. Greubel, Feb 13 2024

QBinomial := proc(n,m,q) local i ; factor( mul((1-q^(n-i))/(1-q^(i+1)),i=0..m-1) ) ; expand(%) ; end:
A083906 := proc(n,k) add( QBinomial(n,m,q),m=0..n ) ; coeftayl(%,q=0,k) ; end:
for n from 0 to 10 do for k from 0 to A033638(n)-1 do printf("%d,",A083906(n,k)) ; od: od: # R. J. Mathar, May 28 2009
T := proc(n, k) if n < 0 or k < 0 or k > floor(n^2/4) then return 0 fi;
if n < 2 then return n + 1 fi; 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) end:
seq(print(seq(T(n, k), k = 0..floor((n/2)^2))), n = 0..8);  # Peter Luschny, Feb 16 2024

Table[CoefficientList[Total[Table[FunctionExpand[QBinomial[n, k, q]], {k, 0, n}]],q], {n, 0, 10}] // Grid (* Geoffrey Critzer, May 14 2017 *)

{T(n, k) = polcoeff(sum(m=0, n, prod(k=0, m-1, (x^n - x^k) / (x^m - x^k))), k)}; /* Michael Somos, Jun 25 2017 */

def T(n,k): # T = A083906
    if k<0 or k> (n^2//4): return 0
    elif n<2 : return n+1
    else: return 2*T(n-1, k) - T(n-2, k) + T(n-2, k-n+1)
flatten([[T(n,k) for k in range(int(n^2//4)+1)] for n in range(13)]) # G. C. Greubel, Feb 13 2024

T(n, k) is the coefficient [q^k] of the Sum_{m=0..n} [n, m]_q over q-Binomial coefficients.
Row sums: Sum_{k=0..floor(n^2/4)} T(n,k) = 2^n.
For n >= k, T(n+1,k) = T(n, k) + A000041(k). - Geoffrey Critzer, Feb 12 2021
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A060546(n). - G. C. Greubel, Feb 13 2024
From Mikhail Kurkov, Feb 14 2024: (Start)
T(n, k) = 2*T(n-1, k) - T(n-2, k) + T(n-2, k - n + 1) for n >= 2 and 0 <= k <= floor(n^2/4).
Sum_{i=0..n} T(n-i, i) = A000041(n+1). Note that upper limit of the summation can be reduced to A083479(n) = (n+2) - ceiling(sqrt(4*n)).
Both results were proved (see MathOverflow link for details). (End)
From G. C. Greubel, Feb 17 2024: (Start)
T(n, floor(n^2/4)) = A000034(n).
Sum_{k=0..floor(n^2/4)} (-1)^k*T(n, k) = A016116(n+1).
Sum_{k=0..(n + 2) - ceiling(sqrt(4*n))} (-1)^k*T(n - k, k) = (-1)^n*A000025(n+1) = -A260460(n+1). (End)

Edited by R. J. Mathar, May 28 2009

New name using a comment from Geoffrey Critzer by Peter Luschny, Feb 17 2024

A233590 Decimal expansion of the continued fraction c(1) +c(1)/(c(2) +c(2)/(c(3) +c(3)/(c(4) +c(4)/....))), where c(i)=2^(i-1).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jan 06 2014

For more details about this type of continued fraction, see A233588.
This one corresponds to the powers of two sequence.
Corresponds to the regular continued fraction 1,2,2,4,4,8,8,16,16,... = A060546. - Jeffrey Shallit, Jun 14 2016

			1.408615979735005205132362590255795209484563373686888353703927022...

Stanislav Sykora, Table of n, a(n) for n = 1..20000 [a(1522) onward corrected by _Kevin Ryde_, Oct 26 2024]
Stanislav Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001
Stanislav Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki

Cf. A000079 (2^n), A096658, A060546.

Cf. Blazys's continued fractions: A233588, A233589, A233591 and Blazys' expansions: A233582, A233583, A233584, A233585, A233586, A233587

RealDigits[ Fold[(#2 + #2/#1) &, 1, Reverse@ (2^Range[0, 27])], 10, 111][[1]] (* Robert G. Wilson v, May 22 2014 *)

PARI
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See the link
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Equals 1+1/(2+2/(4+4/(8+8/(16+16/(32+...))))).

Equals Product_{k>=0} ((1 - 2^(5*k + 2))*(1 - 2^(5*k + 3)))/((1 - 2^(5*k + 1))*(1 - 2^(5*k + 4))). - Antonio Graciá Llorente, Mar 20 2024

A356185 The difference between number of even and number of odd Grassmannian permutations of size n.

Original entry on oeis.org

1, 1, 0, 1, 0, 3, 2, 9, 8, 23, 22, 53, 52, 115, 114, 241, 240, 495, 494, 1005, 1004, 2027, 2026, 4073, 4072, 8167, 8166, 16357, 16356, 32739, 32738, 65505, 65504, 131039, 131038, 262109, 262108, 524251, 524250, 1048537, 1048536, 2097111, 2097110, 4194261, 4194260

Offset: 0

Per W. Alexandersson, Jul 28 2022

A permutation is Grassmann if it has at most one descent. A closed-form formula was proved by J. B. Gil and J. A. Tomasko.

			For n=3, 123, 231, 312 are even Grassmann permutations, and 132, 213 are the odd ones. Hence a(3) = 1.

Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021.
Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, Enum. Combin. Appl. 2 (2022), no. 4, Article #S4PP6.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Cf. A000325, A001710, A032085, A060546, A122746, A233411.

Bisections give: A005803 (even part), A183155 (odd part).

Table[2^Floor[1 + (n - 1)/2] - n, {n, 1, 80}]

a(n) = 2^(1+floor((n-1)/2))-n.
From Alois P. Heinz, Jul 28 2022: (Start)
G.f.: -(4*x^3-3*x^2-x+1)/((2*x^2-1)*(x-1)^2).
a(n) = A000325(n) - A233411(n) = A060546(n) - n = 2^ceiling(n/2) - n.
a(n) = A000325(n) - 2*A032085(n) = A000325(n) - 2*A122746(n-2) for n>=2. (End)

A321391 Array read by antidiagonals: T(n,k) is the number of achiral rows of n colors using up to k colors.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 3, 4, 1, 0, 1, 5, 4, 9, 4, 1, 0, 1, 6, 5, 16, 9, 8, 1, 0, 1, 7, 6, 25, 16, 27, 8, 1, 0, 1, 8, 7, 36, 25, 64, 27, 16, 1, 0, 1, 9, 8, 49, 36, 125, 64, 81, 16, 1, 0, 1, 10, 9, 64, 49, 216, 125, 256, 81, 32, 1, 0

Offset: 0

Robert A. Russell, Nov 08 2018

The antidiagonals go from top-right to bottom-left.

			The array begins with T(0,0):
1 1  1   1    1     1     1      1      1      1       1       1 ...
0 1  2   3    4     5     6      7      8      9      10      11 ...
0 1  2   3    4     5     6      7      8      9      10      11 ...
0 1  4   9   16    25    36     49     64     81     100     121 ...
0 1  4   9   16    25    36     49     64     81     100     121 ...
0 1  8  27   64   125   216    343    512    729    1000    1331 ...
0 1  8  27   64   125   216    343    512    729    1000    1331 ...
0 1 16  81  256   625  1296   2401   4096   6561   10000   14641 ...
0 1 16  81  256   625  1296   2401   4096   6561   10000   14641 ...
0 1 32 243 1024  3125  7776  16807  32768  59049  100000  161051 ...
0 1 32 243 1024  3125  7776  16807  32768  59049  100000  161051 ...
0 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 ...
For T(3,3)=9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC.

Columns 0-6 are A000007, A000012, A060546, A056449, A056450, A056451, A056452.

Cf. A003992 (oriented), A277504 (unoriented), A293500 (chiral).

Table[If[n>0, (n-k)^Ceiling[k/2], 1], {n, 0, 12}, {k, 0, n}] // Flatten

T(n,k) = [n==0] + [n>0] * k^ceiling(n/2).

The generating function for column k is (1+k*x) / (1-k*x^2).

A274106 Triangle read by rows: T(n,k) = total number of configurations of k nonattacking bishops on the white squares of an n X n chessboard (0 <= k <= n-1+[n=0]).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 2, 1, 8, 14, 4, 1, 12, 38, 32, 4, 1, 18, 98, 184, 100, 8, 1, 24, 188, 576, 652, 208, 8, 1, 32, 356, 1704, 3532, 2816, 632, 16, 1, 40, 580, 3840, 12052, 16944, 9080, 1280, 16, 1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32, 1, 60, 1390, 16000, 98292, 322848, 540080, 412800, 116656, 7744, 32

Offset: 0

N. J. A. Sloane, Jun 14 2016

From Eder G. Santos, Dec 16 2024: (Start)
The sequence counts every possible nonattacking configuration of k bishops on the white squares of an n X n chess board.
It is assumed that the n X n chess board has a black square in the upper left corner.
(End)

			Triangle begins:
  1;
  1;
  1,  2;
  1,  4,    2;
  1,  8,   14,     4;
  1, 12,   38,    32,     4;
  1, 18,   98,   184,   100,      8;
  1, 24,  188,   576,   652,    208,      8;
  1, 32,  356,  1704,  3532,   2816,    632,     16;
  1, 40,  580,  3840, 12052,  16944,   9080,   1280,     16;
  1, 50,  940,  8480, 38932,  89256,  93800,  37600,   3856,   32;
  1, 60, 1390, 16000, 98292, 322848, 540080, 412800, 116656, 7744, 32;
  ...
From _Eder G. Santos_, Dec 16 2024: (Start)
For example, for n = 3, k = 2, the T(3,2) = 2 nonattacking configurations are:
  +---+---+---+   +---+---+---+
  |   | B |   |   |   |   |   |
  +---+---+---+   +---+---+---+
  |   |   |   | , | B |   | B |
  +---+---+---+   +---+---+---+
  |   | B |   |   |   |   |   |
  +---+---+---+   +---+---+---+
(End)

Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. See Section 9.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. See Section 9. [Annotated scanned copy]
J. Perott, Sur le problème des fous, Bulletin de la S. M. F., tome 11 (1883), pp. 173-186.
Eder G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas. arXiv:2411.16492 [math.CO], 2024. (considered as black board).
Eric Weisstein's World of Mathematics, White Bishop Graph.

Columns k=0-1 give: A000012, A007590.
Alternate rows give A088960.
Row sums are A216078(n+1).
T(2n,n) gives A191236.
T(2n+1,n) gives A217900(n+1).
T(n+1,n) gives A060546.
Cf. A274105 (black squares), A288182, A201862, A002465.

with(combinat): with(gfun):
T := n -> add(stirling2(n+1,n+1-k)*x^k, k=0..n):
# bishops on white squares
bish := proc(n) local m,k,i,j,t1,t2; global T;
    if n=0 then return [1] fi;
    if (n mod 2) = 0 then m:=n/2;
        t1:=add(binomial(m,k)*T(2*m-1-k)*x^k, k=0..m);
    else
        m:=(n-1)/2;
        t1:=add(binomial(m,k)*T(2*m-k)*x^k, k=0..m+1);
    fi;
    seriestolist(series(t1,x,2*n+1));
end:
for n from 0 to 12 do lprint(bish(n)); od:

T[n_] := Sum[StirlingS2[n+1, n+1-k]*x^k, {k, 0, n}];
bish[n_] := Module[{m, t1, t2}, If[Mod[n, 2] == 0,
   m = n/2;     t1 = Sum[Binomial[m, k]*T[2*m-1-k]*x^k, {k, 0, m}],
   m = (n-1)/2; t1 = Sum[Binomial[m, k]*T[2*m - k]*x^k, {k, 0, m+1}]];
CoefficientList[t1 + O[x]^(2*n+1), x]];
Table[bish[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 25 2022, after Maple code *)

def stirling2_negativek(n, k):
  if k < 0: return 0
  else: return stirling_number2(n, k)
print([sum([binomial(floor(n/2), j)*stirling2_negativek(n-j, n-k) for j in [0..k]]) for n in [0..10] for k in [0..n-1+kronecker_delta(n,0)]]) # Eder G. Santos, Dec 01 2024

From Eder G. Santos, Dec 01 2024: (Start)
T(n,k) = Sum_{j=0..k} binomial(floor(n/2),j) * Stirling2(n-j,n-k).
T(n,k) = T(n-1,k) + (n-k+1-A000035(n)) * T(n-1,k-1), T(n,0) = 1, T(0,k) = delta(k,0). (End)

T(0,0) prepended by Eder G. Santos, Dec 01 2024

A060552 a(n) is the number of distinct (modulo geometric D3-operations) nonsymmetric (no reflective nor rotational symmetry) patterns which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 14, 35, 70, 154, 310, 650, 1300, 2666, 5332, 10788, 21588, 43428, 86856, 174244, 348488, 697992, 1396040, 2794120, 5588240, 11180680, 22361360, 44730896, 89462032, 178940432, 357880864, 715794960

Offset: 1

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Harry J. Smith, Table of n, a(n) for n=1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-4,-4,10,-4,-4,4,8,8,-16).

Cf. A000079, A060547, A060548, A060546, A060552, A008611, A008615, A008619.

a(n) = { (2^(n-1)-2^(floor(n/3)+(n%3)%2-1))/3+2^(floor((n+3)/6)+(n%6==1)-1)-2^floor((n-1)/2) } \\ Harry J. Smith, Jul 07 2009

a(n) = (2^(n-1) - 2^(floor(n/3) + (n mod 3)mod 2 - 1))/3 + 2^(floor((n+3)/6) + d(n) - 1) - 2^floor((n-1)/2), with d(n)=1 if n mod 6=1 else d(n)=0.
a(n) = (A000079(n-1) - A060547(n)/2)/3 + A060548(n)/2 -A060546(n)/2.
a(n) = (A000079(n-1) - 2^(A008611(n-1) - 1))/3 + 2^(A008615(n+1) - 1) - 2^(A008619(n-1) - 1), n >= 1.
From R. J. Mathar, Aug 03 2009: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - 4*a(n-4) - 4*a(n-5) + 10*a(n-6) - 4*a(n-7) - 4*a(n-8) + 4*a(n-9) + 8*a(n-10) + 8*a(n-11) - 16*a(n-12).
G.f.: -x^4*(-1 - x^2 - x^4 + 2*x^3 + 2*x^5 + 2*x^6)/((2*x-1)*(2*x^2-1)*(2*x^3-1)*(2*x^6-1)). (End)

cp's OEIS Frontend

A320770 a(n) = (-1)^floor(n/4) * 2^floor(n/2).

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A343177 a(0)=4; if n > 0 is even then a(n) = 2^(n/2+1)+3, otherwise a(n) = 3*(2^((n-1)/2)+1).

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A354785 Numbers of the form 3*2^k or 9*2^k.

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A036991 Numbers k with the property that in the binary expansion of k, reading from right to left, the number of 0's never exceeds the number of 1's.

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A083906 Table read by rows: T(n, k) is the number of length n binary words with exactly k inversions.

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A233590 Decimal expansion of the continued fraction c(1) +c(1)/(c(2) +c(2)/(c(3) +c(3)/(c(4) +c(4)/....))), where c(i)=2^(i-1).

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A354785 Numbers of the form 32^k or 92^k.