A029744 -id:A029744 - cp's OEIS frontend

A183978 1/4 the number of (n+1) X 2 binary arrays with all 2 X 2 subblock sums the same.

4, 6, 9, 15, 25, 45, 81, 153, 289, 561, 1089, 2145, 4225, 8385, 16641, 33153, 66049, 131841, 263169, 525825, 1050625, 2100225, 4198401, 8394753, 16785409, 33566721, 67125249, 134242305, 268468225, 536920065, 1073807361, 2147581953, 4295098369

Offset: 1

R. H. Hardin, Jan 08 2011

Column 1 of A183986

Based on the conjectured recursion formula, it is also the number of notches in a sheet of paper when you fold it n times and cut off the four corners (see A274230). - Philippe Gibone, Jul 06 2016

			Some solutions for 5X2
..0..1....1..0....1..0....1..1....0..1....1..0....1..0....0..1....0..1....0..1
..0..0....1..0....1..0....1..0....1..0....1..0....1..0....0..1....1..0....0..1
..1..0....1..0....0..1....1..1....0..1....0..1....0..1....1..0....0..1....1..0
..0..0....1..0....1..0....0..1....1..0....1..0....0..1....1..0....0..1....0..1
..1..0....1..0....1..0....1..1....1..0....0..1....0..1....1..0....1..0....0..1

R. H. Hardin, Table of n, a(n) for n = 1..46
Robert Israel, Proof of empirical formulas for A183978
Index entries for linear recurrences with constant coefficients, signature (3, 0, -6, 4).

Cf. A274230.

Conjectured to be the main diagonal of A274636.

seq((1+2^floor((n-1)/2))*(1+2^ceil((n-1)/2)), n=1..20); # Robert Israel, May 21 2019

Empirical: a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4)
Based on the conjectured recursion formula, we may prove (by a tedious induction) that a(n) = (2^ceiling(n/2) + 1) * (2^floor(n/2) + 1) = A051032(n) * A051032(n-1) for n >= 1. - Philippe Gibone, Jul 06 2016, corrected by Robert Israel, May 21 2019
Empirical: G.f.: -x*(4-6*x-9*x^2+12*x^3) / ( (x-1)*(2*x-1)*(2*x^2-1) ). - R. J. Mathar, Jul 15 2016
Empirical formulas verified (see link): Robert Israel, May 21 2019.
2*a(n) = 2+2^n+A029744(n+3). - R. J. Mathar, Jul 19 2024

A257569 Triangular array read by rows: T(h,k) = number of steps from (h,k) to (0,0), where one step is (x,y) -> (x-1, y) if x is odd or (x,y) -> (y, x/2) if x is even.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 6, 7, 6, 7, 7, 8, 7, 7, 7, 7, 7, 8, 8, 8, 8, 7, 8, 7, 8, 7, 9, 8, 9, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 8, 9, 8, 9, 8, 9, 8, 10, 9, 10, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 8, 10, 9, 10, 8

Offset: 1

Clark Kimberling, May 01 2015

The number of pairs (h,k) satisfying T(h,k) = n is F(n), where F = A000045, the Fibonacci numbers.  The number of such pairs having odd h is F(n-2), and the number having even h is F(n-1).
Let c(n,k) be the number of pairs (h,k) satisfying T(h,k) = n; in particular, c(n,0) is the number of integers (pairs of the form (h,0)) satisfying T(h,0) = n.  Let p(n) = A000931(n).  Then c(n,0) = p(n+3) for n >= 0.  More generally, for fixed k >=0, the sequence satisfies the recurrence r(n) = r(n-2) + r(n-3) except for initial terms.
The greatest h for which some (h,k) is n steps from (0,0) is H = A029744(n-1) for n >= 2, and the only such pair is (H,0).
T(n,k) is also the number of steps from h + k*sqrt(2) to 0, where one step is x + y*sqrt(2) -> x-1 + y*sqrt(2) if x is odd, and x + sqrt(y) -> y + (x/2)*sqrt(2) if x is even.

			First ten rows:
0
1   2
3   3   4
4   4   5   5
5   5   5   6   6
6   6   6   6   7   7
6   7   6   7   7   8   7
7   7   7   7   8   8   8   8
7   8   7   8   7   9   8   9   8
8   8   8   8   8   8   9   9   9   9
Row 3 counts the pairs (2,0), (1,1), (0,2), for which the paths to (0,0) are as shown here:
(2,0) -> (0,1) -> (1,0) -> (0,0)  (3 steps);
(1,1) -> (0,1) -> (1,0) -> (0,0)  (3 steps);
(0,2) -> (2,0) -> (0,1) -> (1,0) -> (0,0) (4 steps).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A257570, A257571, A257572, A000045, A000931, A029744.

f[{x_, y_}] := If[EvenQ[x], {y, x/2}, {x - 1, y}];
g[{x_, y_}] := Drop[FixedPointList[f, {x, y}], -1];
h[{x_, y_}] := -1 + Length[g[{x, y}]];
t = Table[h[{n - k, k}], {n, 0, 16}, {k, 0, n}];
TableForm[t] (* A257569 array *)
Flatten[t]   (* A257569 sequence *)

A364295 Numbers k such that A292943(k) = A292944(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 45, 48, 64, 72, 90, 96, 128, 144, 165, 180, 189, 192, 256, 288, 330, 360, 378, 384, 512, 576, 660, 720, 756, 768, 1024, 1152, 1320, 1440, 1512, 1536, 2048, 2304, 2640, 2880, 3024, 3072, 4096, 4608, 5280, 5760, 6048, 6144, 8192, 9216, 10560, 11520, 12096, 12288, 16384

Offset: 1

Antti Karttunen, Jul 26 2023

nonn

If n is present, then 2*n is also present, and vice versa.

A007283 is included as a subsequence, because it gives the known fixed points of map n -> A163511(n).

Cf. A163511, A243071, A292943, A292944.
Subsequences: A000079, A007283, A029744, A364296 (odd terms).
Cf. also A364494, A364496.

A004754(n) = (n+(1<<(#binary(n)-1)));
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); };
A292272(n) = (n - bitand(n,n\2));
A292944(n) = (A292272(A004754(n)) - 2*A053644(n));
A054429(n) = ((3<<#binary(n\2))-n-1);
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A243071(n) = if(n<=2, n-1, A054429(A156552(n)));
A292943(n) = A292944(A243071(n));
isA364295(n) = (A292943(n)==A292944(n));

A123760 Numbers whose binary expansion is 1xy100...0 where x,y = 0 or 1.

Original entry on oeis.org

9, 11, 13, 15, 18, 22, 26, 30, 36, 44, 52, 60, 72, 88, 104, 120, 144, 176, 208, 240, 288, 352, 416, 480, 576, 704, 832, 960, 1152, 1408, 1664, 1920, 2304, 2816, 3328, 3840, 4608, 5632, 6656, 7680, 9216, 11264, 13312, 15360, 18432, 22528, 26624

Offset: 1

Reinhard Zumkeller, Oct 13 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A029744, A070875, A070876, A093873.

Union[Join @@ (Table[FromDigits[Join[#, Table[0, {n}]], 2], {n, 0, 11}] & /@ {{1, 0, 0, 1}, {1, 0, 1, 1}, {1, 1, 0, 1}, {1, 1, 1, 1}})] (* Amiram Eldar, Mar 28 2022 *)

a(n) = 2*a(n-4) for n > 4.
A093873(a(n)) = 3.
Sum_{n>=1} 1/a(n) = 4448/6435. - Amiram Eldar, Mar 28 2022

A180249 a(n) is the total number of k-reverses of n.

Original entry on oeis.org

1, 2, 4, 8, 16, 26, 50, 80, 130, 212, 342, 518, 820, 1276, 1864, 2960, 4336, 6704, 9710, 15068, 21368, 33420, 47082, 72950, 102316, 158888, 220882, 342616, 475108, 734816, 1015778, 1569680, 2161944, 3337952, 4587200, 7069748, 9699292, 14932444, 20445520

Offset: 1

John P. McSorley, Aug 19 2010

nonn

See sequence A180171 for the definition of a k-reverse of n.
Briefly, a k-reverse of n is a k-composition of n whose reverse is cyclically equivalent to itself.
This sequence is the total number of k-reverses of n for k=1,2,...,n.
It is the row sums of the 'R(n,k)' triangle from sequence A180171.
For example a(6)=26 because there are 26 k-reverses of n=6 for k=1,2,3,4,5, or 6.
They are, in cyclically equivalent, classes: {6}, {15,51}, {24,42},{33},{114,411,141},{222} {1113,3111,1311,1131}, {1122,2112,2211,1221}, {1212,2121}, {11112,21111,12111,11211,11121}, {111111}.

John P. McSorley: Counting k-compositions with palindromic and related structures. Preprint, 2010.

Andrew Howroyd, Table of n, a(n) for n = 1..200

If we ask for the number of cyclically equivalent classes we get sequence A052955.
For example the 6th term of A052955 is 11, corresponding to the 11 classes in the example above.
Row sums of A180171.
Cf. A056493, A180322.

f[n_Integer] := Block[{c = 0, k = 1, ip = IntegerPartitions@ n, lmt = 1 + PartitionsP@ n, ipk}, While[k < lmt, c += g[ ip[[k]]]; k++ ]; c]; g[lst_List] := Block[{c = 0, len = Length@ lst, per = Permutations@ lst}, While[ Length@ per > 0, rl = Union[ RotateLeft[ per[[1]], # ] & /@ Range@ len]; If[ MemberQ[rl, Reverse@ per[[1]]], c += Length@ rl]; per = Complement[ per, rl]]; c]; Array[f, 24] (* Robert G. Wilson v, Aug 25 2010 *)
b[n_] := Sum[MoebiusMu[n/d] * If[OddQ[d], 2, 3] * 2^Quotient[d-1, 2], {d, Divisors[n]}]; a[n_] := Sum[d*b[d], {d, Divisors[n]}] / 2; Array[a, 39] (* Jean-François Alcover, Nov 04 2017, after Andrew Howroyd *)

\\ here b(n) is A056493
b(n) = sumdiv(n, d, moebius(n/d) * if(d%2,2,3) * 2^((d-1)\2));
a(n) = sumdiv(n, d, d*b(d)) / 2; \\ Andrew Howroyd, Oct 07 2017

a(n) = Sum_{d|n} d*A056493(d)/2. - Andrew Howroyd, Oct 07 2017
From Petros Hadjicostas, Oct 15 2017: (Start)
a(n) = (n/2)*Sum_{d|n} (phi^(-1)(d)/d)*b(n/d), where phi^(-1)(n) = A023900(n) is the Dirichlet inverse of the Euler totient function and b(n) = A029744(n+1) (= 3*2^((n/2)-1), if n is even, and = 2^((n+1)/2), if n is odd).
G.f.: Sum_{n>=1} phi^(-1)(n)*g(x^n), where phi^(-1)(n) = A023900(n) and g(x) = x*(x+1)*(2*x+1)/(1-2*x^2)^2.
(End)

a(11) - a(24) from Robert G. Wilson v, Aug 25 2010
a(25) - a(27) from Robert G. Wilson v, Aug 29 2010
Terms a(28) and beyond from Andrew Howroyd, Oct 07 2017

A247283 Positions of subrecords in A048673.

Original entry on oeis.org

5, 7, 9, 15, 18, 27, 36, 54, 72, 108, 144, 216, 288, 432, 576, 864, 1152, 1728, 2304, 3456, 4608, 6912, 9216, 13824, 18432, 27648, 36864, 55296, 73728, 110592, 147456, 221184, 294912, 442368, 589824, 884736, 1179648, 1769472, 2359296, 3538944, 4718592, 7077888

Offset: 1

Antti Karttunen, Sep 11 2014

nonn

Odd bisection seems to be A116453 (i.e. A005010, 9*2^n from a(3)=9 onward).

After terms 7 and 15, even bisection from a(6)=27 onward seems to be A175806 (27*2^n).

			The fourth (A246360(4) = 5) and the fifth (A246360(5) = 8) record of A048673 (1, 2, 3, 5, 4, 8, ...) occur at A029744(4) = 4 and A029744(5) = 6 respectively. In range between, the maximum must occur at 5, thus a(4-3) = a(1) = 5. (All the previous records of A048673 are in consecutive positions, 1, 2, 3, 4, thus there are no previous subrecords).
The ninth (A246360(9) = 68) and the tenth (A246360(10) = 122) record of A048673 occur at A029744(9) = 24 and A029744(10) = 32 respectively. For n in range 25 .. 31 the values of A048673 are: 25, 26, 63, 50, 16, 53, 19, of which 63 is the maximum, and because it occurs at n = 27, we have a(9-3) = a(6) = 27.

Antti Karttunen, Table of n, a(n) for n = 1..60

A247284 gives the corresponding values.

Cf. A005010 (A116453), A175806, A029744, A048673, A064216, A246360, A245449.

\\ Compute A245449, A246360, A247283 and A247284 at the same time:
default(primelimit,(2^31)+(2^30));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A048673(n) = (A003961(n)+1)/2;
n = 0; i2 = 0; i3 = 0; ir = 0; prevmax = 0; submax = 0; while(n < 2^32, n++; a_n = A048673(n); if((A048673(a_n) == n), i2++; write("b245449.txt", i2, " ", n)); if((a_n > prevmax), if(submax > 0, i3++; write("b247283.txt", i3, " ", submaxpt); write("b247284.txt", i3, " ", submax)); prevmax = a_n; submax = 0; ir++; write("b029744_empirical.txt", ir, " ", n); write("b246360_empirical.txt", ir, " ", a_n), if((a_n > submax), submax = a_n; submaxpt = n)));

(definec (A247283 n) (max_pt_in_range A048673 (+ (A029744 (+ n 3)) 1) (- (A029744 (+ n 4)) 1)))
(define (max_pt_in_range intfun lowlim uplim) (let loop ((i (+ 1 lowlim)) (maxnow (intfun lowlim)) (maxpt lowlim)) (cond ((> i uplim) maxpt) (else (let ((v (intfun i))) (if (> v maxnow) (loop (+ 1 i) v i) (loop (+ 1 i) maxnow maxpt)))))))

a(n) = A064216(A247284(n)).
Conjectures from Chai Wah Wu, Jul 30 2020: (Start)
a(n) = 2*a(n-2) for n > 6.
G.f.: x*(3*x^5 - x^3 + x^2 - 7*x - 5)/(2*x^2 - 1). (End)

A247284 Subrecords in A048673: maximum value between two consecutive records in A048673.

Original entry on oeis.org

4, 6, 13, 18, 38, 63, 113, 188, 338, 563, 1013, 1688, 3038, 5063, 9113, 15188, 27338, 45563, 82013, 136688, 246038, 410063, 738113, 1230188, 2214338, 3690563, 6643013, 11071688, 19929038, 33215063, 59787113, 99645188, 179361338, 298935563, 538084013, 896806688

Offset: 1

Antti Karttunen, Sep 11 2014

nonn

			The fourth (A246360(4) = 5) and the fifth (A246360(5) = 8) record of A048673 (1, 2, 3, 5, 4, 8, ...) occur at A029744(4) = 4 and A029744(5) = 6 respectively. In range between, the maximum must occur at 5, where A048673(5) = 4, thus a(4-3) = a(1) = 4. (All the previous records of A048673 are in consecutive positions, 1, 2, 3, 4, thus there are no previous subrecords).
The ninth (A246360(9) = 68) and the tenth (A246360(10) = 122) record of A048673 occur at A029744(9) = 24 and A029744(10) = 32 respectively. For n in range 25 .. 31 the values of A048673 are: 25, 26, 63, 50, 16, 53, 19, of which 63 is the maximum, thus a(9-3) = a(6) = 63.

Antti Karttunen, Table of n, a(n) for n = 1..60

A247283 gives the corresponding positions.

Cf. A029744, A048673, A246360.

PARI
```
\\ See program in A247283.
```

(define (A247284 n) (A048673 (A247283 n)))

a(n) = A048673(A247283(n)).
Conjectures from Chai Wah Wu, Jul 30 2020: (Start)
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) for n > 7.
G.f.: x*(-10*x^6 + 10*x^5 - x^4 - x^3 - 5*x^2 + 2*x + 4)/((x - 1)*(3*x^2 - 1)). (End)

A372817 Table read by antidiagonals: T(m,n) = number of 1-metered (m,n)-parking functions.

Original entry on oeis.org

1, 0, 2, 0, 3, 3, 0, 4, 8, 4, 0, 6, 21, 15, 5, 0, 8, 55, 56, 24, 6, 0, 12, 145, 209, 115, 35, 7, 0, 16, 380, 780, 551, 204, 48, 8, 0, 24, 1000, 2912, 2640, 1189, 329, 63, 9, 0, 32, 2625, 10868, 12649, 6930, 2255, 496, 80, 10, 0, 48, 6900, 40569, 60606, 40391, 15456, 3905, 711, 99, 11

Offset: 1

Spencer Daugherty, May 13 2024

			For T(3,2) the 1-metered (3,2)-parking functions are 111, 121, 211, 212.
Table begins:
  1,  2,    3,     4,     5,      6,      7, ...
  0,  3,    8,    15,    24,     35,     48, ...
  0,  4,   21,    56,   115,    204,    329, ...
  0,  6,   55,   209,   551,   1189,   2255, ...
  0,  8,  145,   780,  2640,   6930,  15456, ...
  0, 12,  380,  2912, 12649,  40391, 105937, ...
  0, 16, 1000, 10868, 60606, 235416, 726103, ...
  ...

Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.

Main diagonal is A097690 and first row of A372816.
First, second, and third diagonals above main are A097691, A342167, A342168.
Second column A029744. Second row A005563. Third row A242135.
Cf. A001353, A004254, A001109, A004187, A372818, A372821.

T(m,n) = (n*(n+sqrt(n^2 - 4))-2)/(n*(n+sqrt(n^2 - 4))-4)*((n+sqrt(n^2-4))/2)^m + (n*(n-sqrt(n^2 - 4))-2)/(n*(n-sqrt(n^2 - 4))-4)*((n-sqrt(n^2-4))/2)^m.

T(m,n) = n*T(m-1,n) - T(m-2,n) with T(0,n) = 1.

A029746 Numbers of the form 2^k or 7*2^k.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 128, 224, 256, 448, 512, 896, 1024, 1792, 2048, 3584, 4096, 7168, 8192, 14336, 16384, 28672, 32768, 57344, 65536, 114688, 131072, 229376, 262144, 458752, 524288, 917504, 1048576, 1835008, 2097152

Offset: 0

N. J. A. Sloane

nonn

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

Cf. A000079, A005009, A029744, A029748.

Sort[Flatten[{#,7#}&/@(2^Range[0,21])]] (* or *) CoefficientList[Series[ (1+2x+2x^2+3x^3)/(1-2x^2),{x,0,50}],x]  (* Harvey P. Dale, Apr 23 2011 *)

a(n)=if(n<0,0,if(n<2,2^n,if(n%2,7/2,2)*2^(n\2)))

G.f.: (1+2x+2x^2+3x^3)/(1-2x^2). - Michael Somos, Nov 05 2002

Sum_{n>=0} 1/a(n) = 16/7. - Amiram Eldar, Jan 17 2022

A075825 a(0) = 1, a(1) = 2; for n>0, a(2n) = |a(n)-a(n-1)|, a(2n+1) = a(n)+a(n-1).

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 2, 4, 2, 4, 2, 4, 1, 5, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 3, 5, 4, 6, 3, 7, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 3, 9, 2, 8, 1, 9, 2, 10, 3, 9, 4, 10, 3, 11, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4

Offset: 0

John W. Layman, Oct 14 2002

For 2*2^k-2 <= n <= 3*2^k-1, a(n) alternates: 2^floor(k/2) if n is even, A029744(k+2) if n is odd. - Robert Israel, Nov 08 2016

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A029744.

A[0]:= 1: A[1]:= 2:
for n from 1 to 100 do
  A[2*n]:= abs(A[n]-A[n-1]);
  A[2*n+1]:= A[n]+A[n-1];
od:
seq(A[n],n=0..201); # Robert Israel, Nov 08 2016

a[0]=1; a[1]=2; a[n_]:=If[EvenQ[n],Abs[a[n/2]-a[n/2-1]],a[(n-1)/2]+a[(n-3)/2]]; Array[a,95,0] (* Stefano Spezia, Apr 04 2024 *)

cp's OEIS Frontend

A183978 1/4 the number of (n+1) X 2 binary arrays with all 2 X 2 subblock sums the same.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A257569 Triangular array read by rows: T(h,k) = number of steps from (h,k) to (0,0), where one step is (x,y) -> (x-1, y) if x is odd or (x,y) -> (y, x/2) if x is even.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A364295 Numbers k such that A292943(k) = A292944(k).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A123760 Numbers whose binary expansion is 1xy100...0 where x,y = 0 or 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A180249 a(n) is the total number of k-reverses of n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A247283 Positions of subrecords in A048673.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Scheme

Formula

A247284 Subrecords in A048673: maximum value between two consecutive records in A048673.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Scheme

Formula

A372817 Table read by antidiagonals: T(m,n) = number of 1-metered (m,n)-parking functions.

Views

Author

Keywords

Examples