A077957 -id:A077957 - cp's OEIS frontend

A367301 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

1, 3, 3, 10, 16, 8, 33, 75, 63, 21, 109, 320, 380, 220, 55, 360, 1296, 1980, 1620, 720, 144, 1189, 5070, 9459, 9940, 6255, 2262, 377, 3927, 19353, 42615, 54561, 44085, 22635, 6909, 987, 12970, 72532, 184034, 277480, 272854, 179972, 78230, 20672, 2584

Offset: 1

Clark Kimberling, Dec 23 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
     1
     3      3
    10     16      8
    33     75     63     21
   109    320    380    220     55
   360   1296   1980   1620    720    144
  1189   5070   9459   9940   6255   2262   377
  3927  19353  42615  54561  44085  22635  6909  987
Row 4 represents the polynomial p(4,x) = 33 + 75*x + 63*x^2 + 21*x^3, so (T(4,k)) = (33,75,63,21), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A006190 (column 1); A001906 (p(n,n-1)); A154244 (row sums, p(n,1)); A077957 (alternating row sums, p(n,-1)); A190984 (p(n,2)); A006190 (signed, p(n,-2)); A154244 (p(n,-3)); A190984 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300.

p[1, x_] := 1; p[2, x_] := 3 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 3 + 3*x, u = p(2,x), and v = 1 - 2*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(13 + 10*x + 5*x^2)), b = (1/2) (3*x + 3 + 1/k), c = (1/2) (3*x + 3 - 1/k).

A007179 Dual pairs of integrals arising from reflection coefficients.

Original entry on oeis.org

0, 1, 1, 4, 6, 16, 28, 64, 120, 256, 496, 1024, 2016, 4096, 8128, 16384, 32640, 65536, 130816, 262144, 523776, 1048576, 2096128, 4194304, 8386560, 16777216, 33550336, 67108864, 134209536, 268435456, 536854528, 1073741824, 2147450880, 4294967296, 8589869056

Offset: 0

N. J. A. Sloane, Simon Plouffe

			From _Gus Wiseman_, Feb 26 2022: (Start)
Also the number of integer compositions of n with at least one odd part. For example, the a(1) = 1 through a(5) = 16 compositions are:
  (1)  (1,1)  (3)      (1,3)      (5)
              (1,2)    (3,1)      (1,4)
              (2,1)    (1,1,2)    (2,3)
              (1,1,1)  (1,2,1)    (3,2)
                       (2,1,1)    (4,1)
                       (1,1,1,1)  (1,1,3)
                                  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
J. Heading, Theorem relating to the development of a reflection coefficient in terms of a small parameter, J. Phys. A 14 (1981), 357-367.
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Column k=2 of A309748.
Odd bisection is A000302.
Even bisection is A006516 = 2^(n-1)*(2^n - 1).
The complement is counted by A077957, internal version A027383.
The internal case is A274230, even bisection A134057.
A000045(n-1) counts compositions without odd parts, non-singleton A077896.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A034871, A097805, and A345197 count compositions by alternating sum.
A052952 (or A074331) counts non-singleton compositions without even parts.
Cf. A000918, A033484, A052955, A060867, A116406, A138364.

[Floor(2^n/2-2^(n/2)*(1+(-1)^n)/4): n in [0..40]]; // Vincenzo Librandi, Aug 20 2011

f := n-> if n mod 2 = 0 then 2^(n-1)-2^((n-2)/2) else 2^(n-1); fi;

LinearRecurrence[{2,2,-4},{0,1,1},30] (* Harvey P. Dale, Nov 30 2015 *)
Table[2^(n-1)-If[EvenQ[n],2^(n/2-1),0],{n,0,15}] (* Gus Wiseman, Feb 26 2022 *)

Vec(x*(1-x)/((1-2*x)*(1-2*x^2)) + O(x^50)) \\ Michel Marcus, Jan 28 2016

From Paul Barry, Apr 28 2004: (Start)
  Binomial transform is (A000244(n)+A001333(n))/2.
  G.f.: x*(1-x)/((1-2*x)*(1-2*x^2)).
  a(n) = 2*a(n-1)+2*a(n-2)-4*a(n-3).
  a(n) = 2^n/2-2^(n/2)*(1+(-1)^n)/4. (End)
G.f.: (1+x*Q(0))*x/(1-x), where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
a(n) = A011782(n+2) - A077957(n) - Gus Wiseman, Feb 26 2022

A051437 Number of undirected walks of length n+1 on an oriented triangle, visiting n+2 vertices, with n "corners"; the symmetry group is C3. Walks are not self-avoiding.

Original entry on oeis.org

1, 3, 4, 10, 16, 36, 64, 136, 256, 528, 1024, 2080, 4096, 8256, 16384, 32896, 65536, 131328, 262144, 524800, 1048576, 2098176, 4194304, 8390656, 16777216, 33558528, 67108864, 134225920, 268435456, 536887296, 1073741824, 2147516416, 4294967296, 8590000128

Offset: 0

Colin Mallows

For a way to obtain this sequence from symmetry in quilts, see the Tom Young web page.
Also arises from the enumeration of based polyhedra with exactly two triangular faces [Rademacher]. - N. J. A. Sloane, Apr 24 2020
a(n-1) is the number of linear oriented trees with n arcs (n+1 nodes). - R. J. Mathar, Jun 09 2020

			For n=3 the walks visit vertices 1212, 1213, 1232, 1231.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hans Rademacher, On the number of certain types of polyhedra, Illinois Journal of Mathematics 9.3 (1965): 361-380. Reprinted in Coll. Papers, Vol II, MIT Press, 1974, pp. 544-564. See Theorem 7, Eq. 14.1.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
Joseph D. Yelk, Molecular Dynamics Investigations of Duplex Columnar Liquid Crystal Phases of Nucleoside Triphosphates, Ph. D. thesis, Northwestern University (2008).
Tom Young, Math Research Quilt Pattern Symmetry [Broken link]
Tom Young, Unique symmetrical triangle quilt patterns along the diagonal of an nxn square (An archived copy from the above page)
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A000079, A000129, A000244, A000302, A005418, A007582, A077957, A193231.

LinearRecurrence[{2,2,-4},{1,3,4},50] (* or *) CoefficientList[ Series[ (1+x-4x^2)/((1-2x)(1-2x^2)),{x,0,50}],x] (* Harvey P. Dale, Jun 06 2011 *)

(define (A051437 n) (if (zero? n) 1 (+ (A000079 n) (A077957 (- n 1))))) ;; Antti Karttunen, Dec 29 2013

a(2n+1) = A007582(n+1). a(2n) = A000302(n).
a(n) = A000079(n) + A077957(n-1). - Antti Karttunen, Dec 29 2013
From Paul Barry, Apr 28 2004: (Start)
Binomial transform is 3^n + Pell(n) = (A000244(n) + A000129(n)).
G.f.: (1+x-4*x^2)/((1-2*x)(1-2*x^2));
a(n) = 2^n + 2^(n/2)*(1-(-1)^n)/(2*sqrt(2)). (End)
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3); a(0)=1, a(1)=3, a(2)=4. - Harvey P. Dale, Jun 06 2011
a(n) = 2*a(n-2) + 2^(n-1), a(0)=1, a(1)=3. - Yuchun Ji, Aug 12 2020
E.g.f.: cosh(2*x) + sinh(2*x) + sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, Jun 03 2022

More terms from Harvey P. Dale, Jun 06 2011

A077966 Expansion of 1/(1+2*x^2).

Original entry on oeis.org

1, 0, -2, 0, 4, 0, -8, 0, 16, 0, -32, 0, 64, 0, -128, 0, 256, 0, -512, 0, 1024, 0, -2048, 0, 4096, 0, -8192, 0, 16384, 0, -32768, 0, 65536, 0, -131072, 0, 262144, 0, -524288, 0, 1048576, 0, -2097152, 0, 4194304, 0, -8388608, 0, 16777216, 0, -33554432, 0, 67108864, 0, -134217728, 0, 268435456

Offset: 0

N. J. A. Sloane, Nov 17 2002

Normally sequences like this are not included, since with the alternating 0's deleted it is already in the database.
Inverse binomial transform of A087455. - Philippe Deléham, Dec 02 2008
Pisano period lengths: 1, 1, 2, 1, 8, 2, 12, 1, 6, 8, 10, 2, 24, 12, 8, 1, 16, 6, 18, 8,... - R. J. Mathar, Aug 10 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-2).

Cf. A000079, A077957.

a:=[1,0];; for n in [3..60] do a[n]:=-2*a[n-2]; od; a; # G. C. Greubel, Jun 24 2019

I:=[1,0]; [n le 2 select I[n] else -2*Self(n-2): n in [1..60]]; // G. C. Greubel, Jun 24 2019

A077966:=n->(1+(-1)^n)*(-2)^(n/2)/2; seq(A077966(n), n=0..50); # Wesley Ivan Hurt, Apr 02 2014

CoefficientList[Series[1/(1 + 2*x^2), {x,0,60}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
LinearRecurrence[{0,-2}, {1,0}, 60] (* G. C. Greubel, Jun 24 2019 *)

Vec(1/(1+2*x^2)+O(x^60)) \\ Charles R Greathouse IV, Sep 24 2012

for(n=0, 51, print1(imag(quadgen(-8)^(n+1)), ", ")) \\ Arkadiusz Wesolowski, Dec 26 2012

[lucas_number1(n,0,2) for n in range(1,60)] # Zerinvary Lajos, Jul 16 2008

a(n) = (1+(-1)^n)*(-2)^(n/2)/2. - R. J. Mathar, Apr 23 2009
a(n) = ((n+1) mod 2 )*(-2)^floor((n+1)/2). - Wesley Ivan Hurt, Apr 06 2014
E.g.f.: cos(sqrt(2)*x). - G. C. Greubel, Jun 24 2019

A158474 Triangle read by rows generated from (x-1)(x-2)(x-4)*...

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -7, 14, -8, 1, -15, 70, -120, 64, 1, -31, 310, -1240, 1984, -1024, 1, -63, 1302, -11160, 41664, -64512, 32768, 1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152, 1, -255, 21590, -777240, 12850368, -99486720, 353730560

Offset: 0

Gary W. Adamson, Mar 20 2009

Row sum of the unsigned triangle = A028361: (1, 2, 6, 30, 270, 4590, ...).
Right border of the unsigned triangle = A006125: (1, 1, 2, 8, 64, 1024, ...).
From Philippe Deléham, Mar 20 2009: (Start)
Unsigned triangle: A077957(n) DELTA A007179(n+1) = [1,0,2,0,4,0,8,0,16,0,32,0,...]DELTA[1,1,4,6,16,28,64,120,256,496,...], where DELTA is the operator defined in A084938.
Signed triangle: [1,0,2,0,4,0,8,0,16,0,...]DELTA[-1,-1,-4,-6,-16,-28,-64,...]. (End)

			First few rows of the triangle =
1;
1,  -1;
1,  -3,     2;
1,  -7,    14,     -8;
1, -15,    70,   -120,       64;
1, -31,   310,  -1240,     1984,    -1024;
1, -63,  1302, -11160,    41664,   -64512,     32768;
1,-127,  5334, -94488,   755904, -2731008,   4161536,  -2097152;
1,-255, 21590,-777240, 12850368,-99486720, 353730560,-534773760, 268435456;
...
Example: row 3 = x^3 - 7x^2 + 14x - 8 = (x-1)*(x-2)*(x-4).

Cf. A028361, A006125.

Cf. A157963, A135950. - R. J. Mathar, Mar 20 2009

A158474 := proc(n,k) mul(x-2^j,j=0..n-1) ; expand(%); coeftayl(%,x=0,n-k) ; end proc: # R. J. Mathar, Aug 27 2011

{{1}}~Join~Table[Reverse@ CoefficientList[Fold[#1 (x - #2) &, 1, 2^Range[0, n]], x], {n, 0, 7}] // Flatten (* Michael De Vlieger, Dec 22 2016 *)

T(n,k) = coefficient [x^(n-k)] of (x-1)*(x-2)*(x-4)*...*(x-2^(n-1)).
T(n,k) = (-1)^k*(Sum_{j=0..k} T(k,j)*2^((k-j)*n))/(Product_{i=1..k} (2^i-1)) for n >= 0 and k > 0, i.e., e.g.f. of col k > 0 is: (-1)^k*(Sum_{j=0..k} T(k,j)* exp(2^(k-j)*t))/(Product_{i=1..k} (2^i-1)). - Werner Schulte, Dec 18 2016
T(n,k)/T(k,k) = A022166(n,k) for 0 <= k <= n. - Werner Schulte, Dec 21 2016

A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

Original entry on oeis.org

0, 3, 10, 15, 36, 43, 58, 63, 136, 147, 170, 175, 228, 235, 250, 255, 528, 547, 586, 591, 676, 683, 698, 703, 904, 915, 938, 943, 996, 1003, 1018, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2362, 2367, 2696, 2707, 2730, 2735, 2788, 2795, 2810, 2815, 3600, 3619

Offset: 1

Gus Wiseman, Feb 01 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and the corresponding compositions begin:
    0:         0  ()
    3:        11  (1,1)
   10:      1010  (2,2)
   15:      1111  (1,1,1,1)
   36:    100100  (3,3)
   43:    101011  (2,2,1,1)
   58:    111010  (1,1,2,2)
   63:    111111  (1,1,1,1,1,1)
  136:  10001000  (4,4)
  147:  10010011  (3,3,1,1)
  170:  10101010  (2,2,2,2)
  175:  10101111  (2,2,1,1,1,1)
  228:  11100100  (1,1,3,3)
  235:  11101011  (1,1,2,2,1,1)
  250:  11111010  (1,1,1,1,2,2)
  255:  11111111  (1,1,1,1,1,1,1,1)

The case of twins (binary weight 2) is A000120.
The Heinz numbers of these compositions are given by A000290.
All terms are evil numbers A001969.
Partitions of this type are counted by A035363, any length A351004.
These compositions are counted by A077957(n-2), see also A016116.
The strict case (distinct twins) is A351009, counted by A032020 with 0's.
The anti-run case is A351011, counted by A003242 interspersed with 0's.
A011782 counts integer compositions.
A085207/A085208 represent concatenation of standard compositions.
A333489 ranks anti-runs, complement A348612.
A345167/A350355/A350356 rank alternating compositions.
A351014 counts distinct runs in standard compositions.
Cf. A018819, A025047, A027383, A035457, A053738, A088218, A106356, A238279, A344604, A351012, A351015.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],And@@EvenQ/@Length/@Split[stc[#]]&]

A204293 Pascal's triangle interspersed with rows of zeros, and the rows of Pascal's triangle are interspersed with zeros.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 0, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 0, 15, 0, 20

Offset: 0

Reinhard Zumkeller, Jan 14 2012

Auxiliary array for computing Losanitsch's triangle A034851;

T(n, k) + T(n, k + 2) = T(n + 2, k + 2) for k < n - 1.

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
N. J. A. Sloane, Classic Sequences: Losanitsch's Triangle
Eric Weisstein's World of Mathematics, Losanitsch's Triangle
Index entries for triangles and arrays related to Pascal's triangle

Cf. A077957 (row sums), A126869 (central terms); A108044, A007318.

a204293 n k = a204293_tabl !! n !! k
a204293_row n = a204293_tabl !! n
a204293_tabl = [1] : [0,0] : f [1] [0,0] where
   f xs ys = xs' : f ys xs' where
     xs' = zipWith (+) ([0,0] ++ xs) (xs ++ [0,0])

t[n_?EvenQ, k_?EvenQ] := Binomial[n/2, k/2]; t[, ] = 0; Flatten[Table[t[n, k], {n, 0, 12}, {k, 0, n}]] (* Jean-François Alcover, Feb 07 2012 *)

T(n, k) = (1 - n mod 2) * (1 - k mod 2) * binomial(floor(n/2),floor(k/2)).

Formula for T(n,k) corrected by Peter Bala, Jul 06 2015

A251732 a(n) = 3^n*A123335(n). Rational parts of the integers in Q(sqrt(2)) giving the length of a Lévy C-curve variant at iteration step n.

Original entry on oeis.org

1, -3, 27, -189, 1377, -9963, 72171, -522693, 3785697, -27418419, 198581787, -1438256493, 10416775041, -75444958683, 546420727467, -3957528992949, 28662960504897, -207595523965923, 1503539788339611, -10889598445730973, 78869448769442337, -571223078628232779

Offset: 0

Kival Ngaokrajang, Dec 07 2014

The irrational part is given in A251733.
Inspired by the Lévy C-curve, and generated using different construction rules as shown in the links.
The length of this variant Lévy C-curve is an integer in the real quadratic number field Q(sqrt(2)), namely L(n) = A(n) + B(n)*sqrt(2) with A(n) = a(n) = 3^n*A123335(n) and B(n) = A251733(n) = 3^n*A077985(n-1), with A077985(-1) =  0. See the construction rule and the illustration in the links.
The total length of the Lévy C-curve after n iterations is sqrt(2)^n, also an integer in Q(sqrt(2)) (see a comment on A077957). The fractal dimension of the Lévy C-curve is 2, but for this modified case it is log(3)/log(1+sqrt(2)) = 1.2464774357... .

			The first lengths a(n) + A251733(n)*sqrt(2) are:
1, -3 + 3*sqrt(2), 27 - 18*sqrt(2), -189 + 135*sqrt(2), 1377 - 972*sqrt(2), -9963 + 7047*sqrt(2), 72171 - 51030*sqrt(2), -522693 + 369603*sqrt(2), 3785697 - 2676888*sqrt(2), -27418419 + 19387755*sqrt(2), 198581787 - 140418522*sqrt(2), ... - _Wolfdieter Lang_, Dec 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
MathImages, Lévy's C-curve
Kival Ngaokrajang, Construction rule, Illustration of modified Lévy C curve
Wikipedia, Lévy C curve
Index entries for linear recurrences with constant coefficients, signature (-6,9).

Cf. A017910, A077985, A123335, A251733.

[Round(((3*(-1+Sqrt(2)))^n + (-3*(1+Sqrt(2)))^n)/2): n in [0..30]]; // G. C. Greubel, Nov 18 2017

LinearRecurrence[{-6,9}, {1,-3}, 30] (* G. C. Greubel, Nov 18 2017 *)

Vec(-(3*x+1) / (9*x^2-6*x-1) + O(x^100)) \\ Colin Barker, Dec 07 2014

a(n) = 3^n*A123335(n).
a(n) = -6*a(n-1) + 9*a(n-2). - Colin Barker, Dec 07 2014
G.f.: -(3*x+1)/(9*x^2-6*x-1). - Colin Barker, Dec 07 2014
a(n) = ((3*(-1+sqrt(2)))^n + (-3*(1+sqrt(2)))^n) / 2. - Colin Barker, Jan 21 2017
E.g.f.: exp(-3*x)*cosh(3*sqrt(2)*x). - Stefano Spezia, Feb 01 2023

More terms from Colin Barker, Dec 07 2014

Edited: Name specified, Q(sqrt(2))remarks given earlier in a comment to a first version, MathImages link added. - Wolfdieter Lang, Dec 07 2014

A221408 T(n,k)=Number of nXk arrays of occupancy after each element stays put or moves to some horizontal or vertical neighbor, with every occupancy equal to zero or two.

Original entry on oeis.org

0, 2, 2, 0, 6, 0, 4, 16, 16, 4, 0, 44, 0, 44, 0, 8, 120, 436, 436, 120, 8, 0, 328, 0, 4518, 0, 328, 0, 16, 896, 12668, 47156, 47156, 12668, 896, 16, 0, 2448, 0, 492102, 0, 492102, 0, 2448, 0, 32, 6688, 368268, 5133044, 18852560, 18852560, 5133044, 368268, 6688, 32

Offset: 1

R. H. Hardin Jan 15 2013

Table starts
..0.....2........0.........4..........0...........8...........0............16
..2.....6.......16........44........120.........328.........896..........2448
..0....16........0.......436..........0.......12668...........0........368268
..4....44......436......4518......47156......492102.....5133044......53529806
..0...120........0.....47156..........0....18852560...........0....7505597654
..8...328....12668....492102...18852560...722442370.27665242848.1058237377670
..0...896........0...5133044..........0.27665242848
.16..2448...368268..53529806.7505597654
..0..6688........0.558164472
.32.18272.10705360
..0.49920
.64

			Some solutions for n=3 k=4
..2..2..0..2....2..0..2..0....2..0..2..2....2..0..2..0....0..2..2..0
..0..0..0..0....0..2..2..0....0..0..2..0....2..2..2..0....2..0..0..0
..2..2..2..0....2..0..0..2....0..2..2..0....0..0..2..0....2..2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..84

Column 1 is A077957

Column 2 is A002605(n+1)

A351011 Numbers k such that the k-th composition in standard order has even length and alternately equal and unequal parts, i.e., all run-lengths equal to 2.

Original entry on oeis.org

0, 3, 10, 36, 43, 58, 136, 147, 228, 235, 528, 547, 586, 676, 698, 904, 915, 2080, 2115, 2186, 2347, 2362, 2696, 2707, 2788, 2795, 3600, 3619, 3658, 3748, 3770, 8256, 8323, 8458, 8740, 8747, 8762, 9352, 9444, 9451, 10768, 10787, 10826, 11144, 11155, 14368

Offset: 1

Gus Wiseman, Feb 03 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and standard compositions begin:
    0:           0  ()
    3:          11  (1,1)
   10:        1010  (2,2)
   36:      100100  (3,3)
   43:      101011  (2,2,1,1)
   58:      111010  (1,1,2,2)
  136:    10001000  (4,4)
  147:    10010011  (3,3,1,1)
  228:    11100100  (1,1,3,3)
  235:    11101011  (1,1,2,2,1,1)
  528:  1000010000  (5,5)
  547:  1000100011  (4,4,1,1)
  586:  1001001010  (3,3,2,2)
  676:  1010100100  (2,2,3,3)
  698:  1010111010  (2,2,1,1,2,2)
  904:  1110001000  (1,1,4,4)
  915:  1110010011  (1,1,3,3,1,1)

The case of twins (binary weight 2) is A000120.
All terms are evil numbers A001969.
These compositions are counted by A003242 interspersed with 0's.
Partitions of this type are counted by A035457, any length A351005.
The Heinz numbers of these compositions are A062503.
Taking singles instead of twins gives A333489, complement A348612.
This is the anti-run case of A351010.
The strict case (distinct twins) is A351009, counted by A077957(n-2).
A011782 counts compositions.
A085207/A085208 represent concatenation of standard compositions.
A345167 ranks alternating compositions, counted by A025047.
A350355 ranks up/down compositions, counted by A025048.
A350356 ranks down/up compositions, counted by A025049.
A351014 counts distinct runs in standard compositions.
Cf. A008965, A018819, A027383, A032020, A035363, A088218, A106356, A122129, A122134, A238279, A351007.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],And@@(#==2&)/@Length/@Split[stc[#]]&]

cp's OEIS Frontend

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