cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A193231 Blue code for n: in binary coding of a polynomial over GF(2), substitute x+1 for x (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 2, 5, 4, 6, 7, 15, 14, 12, 13, 10, 11, 9, 8, 17, 16, 18, 19, 20, 21, 23, 22, 30, 31, 29, 28, 27, 26, 24, 25, 51, 50, 48, 49, 54, 55, 53, 52, 60, 61, 63, 62, 57, 56, 58, 59, 34, 35, 33, 32, 39, 38, 36, 37, 45, 44, 46, 47, 40, 41, 43, 42, 85, 84, 86
Offset: 0

Views

Author

Franklin T. Adams-Watters, Jul 18 2011

Keywords

Comments

This is a self-inverse permutation of the nonnegative integers.
The function "substitute x+1 for x" on polynomials over GF(2) is completely multiplicative.
What is the density of fixed points in this sequence? Do we get a different answer if we look only at irreducible polynomials?
From Antti Karttunen, Dec 27 2013: (Start)
As what comes to the above question, the number of fixed points in range [2^(n-1),(2^n)-1] of the sequence is given by A131575(n). In range [0,0] there is one fixed point: 0, in range [1,1] there is also one: 1, in range [2,3] there are no fixed points, in range [4,7] there are two fixed points: 6 and 7, and so on. (Cf. also the C-code given in A118666.)
Similarly, the number of cycles in such ranges begins as 1, 1, 1, 3, 4, 10, 16, 36, 64, 136, ... which is A051437 shifted two steps right (prepended with 1's): Because the sequence is a self-inverse permutation, the number of its cycles in range [2^(n-1),(2^n)-1] is computed as: cycles(n) = (A011782(n)-number_of_fixedpoints(n))/2 + number_of_fixedpoints(n), which matches with the identity: A051437(n-2) = (A011782(n)-A131575(n))/2 + A131575(n), for n>=2.
In OEIS terms, the above comment about multiplicativeness can be rephrased as: a(A048720(x,y)) = A048720(a(x),a(y)) for all integers x, y >= 0. Here A048720(x,y) gives the product of carryless binary multiplication of x and y.
The permutation conjugates between Gray code and its inverse: A003188(n) = a(A006068(a(n))) and A006068(n) = a(A003188(a(n))) [cf. the identity 1.19-9d: gB = Bg^{-1} given on page 53 of fxtbook].
Because of the multiplicativity, the subset of irreducible (and respectively: composite) polynomials over GF(2) is closed under this permutation. Cf. the following mappings: a(A014580(n)) = A234750(n) and a(A091242(n)) = A234745(n).
(End)

Examples

			11, binary 1011, corresponds to polynomial x^3+x+1, substituting: (x+1)^3+(x+1)+1 = x^3+x^2+x+1 + x+1 + 1 = x^3+x^2+1, binary 1101 = decimal 13, so a(11) = 13.
From _Tilman Piesk_, Jun 26 2025: (Start)
The binary exponents of 11 are {0, 1, 3}, because 11 = 2^0 + 2^1 + 2^3.
a(11) = A001317(0) XOR A001317(1) XOR A001317(3) = 1 XOR 3 XOR 15 = 13. (End)

Links

Crossrefs

Cf. A000069, A001969, A001317, A003987, A048720, A048724, A065621, A051437, A118666 (fixed points), A131575, A234022 (the number of 1-bits), A234023, A010060, A234745, A234750.
Similarly constructed permutation pairs: A003188/A006068, A135141/A227413, A232751/A232752, A233275/A233276, A233277/A233278, A233279/A233280.
Other permutations based on this (by conjugating, composing, etc): A234024, A234025/A234026, A234027, A234612, A234613, A234747, A234748, A244987, A245812, A245454.

Programs

  • Mathematica
    f[n_] := Which[0 <= # <= 1, #, EvenQ@ #, BitXor[2 #, #] &[f[#/2]], True, BitXor[#, 2 # + 1] &[f[(# - 1)/2]]] &@ Abs@ n; Table[f@ n, {n, 0, 66}] (* Michael De Vlieger, Feb 12 2016, after Robert G. Wilson v at A048724 and A065621 *)
  • PARI
    tox(n) = local(x=Mod(1,2)*X, xp=1, r); while(n>0,if(n%2,r+=xp);xp*=x;n\=2);r
a(n)=subst(lift(subst(tox(n),X,X+1)),X,2)
  • PARI
    a(n)={local(x='x);subst(lift(Mod(1,2)*subst(Pol(binary(n),x),x,1+x)),x,2)};
  • Python
    def a065621(n): return n^(2*(n - (n&-n)))
def a048724(n): return n^(2*n)
l=[0, 1]
for n in range(2, 101):
    if n%2==0: l.append(a048724(l[n//2]))
    else: l.append(a065621(1 + l[(n - 1)//2]))
print(l) # Indranil Ghosh, Jun 04 2017
  • Scheme
    ;; with memoizing macro definec available in Antti Karttunen's IntSeq-library:
(define (A193231 n) (let loop ((n n) (i 0) (s 0)) (cond ((zero? n) s) ((even? n) (loop (/ n 2) (+ 1 i) s)) (else (loop (/ (- n 1) 2) (+ 1 i) (A003987bi s (A001317 i))))))) ;; A003987bi implements binary XOR, A003987.
;; Antti Karttunen, Dec 27 2013
  • Scheme
    ;; With memoizing macro definec available in Antti Karttunen's IntSeq-library.
;; Alternative implementation, a recurrence based on entangling even & odd numbers with complementary pair A048724 and A065621:
(definec (A193231 n) (cond ((< n 2) n) ((even? n) (A048724 (A193231 (/ n 2)))) (else (A065621 (+ (A193231 (/ (- n 1) 2)) 1)))))
;; Antti Karttunen, Dec 27 2013

Formula

From Antti Karttunen, Dec 27 2013: (Start)
a(0) = 0, and for any n = 2^a + 2^b + ... + 2^c, a(n) = A001317(a) XOR A001317(b) XOR ... XOR A001317(c), where XOR is bitwise XOR (A003987) and all the exponents a, b, ..., c are distinct, that is, they are the indices of 1-bits in the binary representation of n.
From above it follows, because all terms of A001317 are odd, that A000035(a(n)) = A010060(n) = A000035(a(2n)). Conversely, we also have A010060(a(n)) = A000035(n). Thus the permutation maps any even number to some evil number, A001969 (and vice versa), like it maps any odd number to some odious number, A000069 (and vice versa).
a(0)=0, a(1)=1, and for n>1, a(2n) = A048724(a(n)), a(2n+1) = A065621(1+a(n)). [A recurrence based on entangling even & odd numbers with the complementary pair A048724/A065621]
For all n, abs(a(2n)-a(2n+1)) = 1.
a(A000079(n)) = A001317(n).
(End)
It follows from the first paragraph above that a(A003987(n,k)) = A003987(a(n), a(k)), that is a(n XOR k) = a(n) XOR a(k). - Peter Munn, Nov 27 2019

A005418 Number of (n-1)-bead black-white reversible strings; also binary grids; also row sums of Losanitsch's triangle A034851; also number of caterpillar graphs on n+2 vertices.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 36, 72, 136, 272, 528, 1056, 2080, 4160, 8256, 16512, 32896, 65792, 131328, 262656, 524800, 1049600, 2098176, 4196352, 8390656, 16781312, 33558528, 67117056, 134225920, 268451840, 536887296, 1073774592, 2147516416, 4295032832
Offset: 1

Views

Author

N. J. A. Sloane, R. K. Guy

Keywords

Comments

Equivalently, walks on triangle, visiting n+2 vertices, so length n+1, n "corners"; the symmetry group is S3, reversing a walk does not count as different. Walks are not self-avoiding. - Colin Mallows
Slavik V. Jablan observes that this is also the number of rational knots and links with n+2 crossings (cf. A018240). See reference. [Corrected by Andrey Zabolotskiy, Jun 18 2020]
Number of bit strings of length (n-1), not counting strings which are the end-for-end reversal or the 0-for-1 reversal of each other as different. - Carl Witty (cwitty(AT)newtonlabs.com), Oct 27 2001
The formula given in page 1095 of the Balasubramanian reference can be used to derive this sequence. - Parthasarathy Nambi, May 14 2007
Also number of compositions of n up to direction, where a composition is considered equivalent to its reversal, see example. - Franklin T. Adams-Watters, Oct 24 2009
Number of normally non-isomorphic realizations of the associahedron of type I starting with dimension 2 in Ceballos et al. - Tom Copeland, Oct 19 2011
Number of fibonacenes with n+2 hexagons. See the Balaban and the Dobrynin references. - Emeric Deutsch, Apr 21 2013
From the point of view of binary grids, it is a (1,n)-rectangular grid. A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11. - Yosu Yurramendi, May 19 2013
Number of n-vertex difference graphs (bipartite 2K_2-free graphs) [Peled & Sun, Thm. 9]. - Falk Hüffner, Jan 10 2016
The offset should be 0, since the first row of A034851 is row 0. The name would then be: "Number of n bead...". - Daniel Forgues, Jul 26 2018
a(n) is the number of non-isomorphic generalized rigid ladders with n cells. A generalized rigid ladder with n cells is a graph with vertex set is the union of {u_0, u_1, ..., u_n} and {v_0, v_1, ..., v_n}, and for every 0 <= i <= n-1, the edges are of the form {u_i,u_i+1}, {v_i, v_i+1}, {u_i,v_i} and either {u_i,v_i+1} or {u_i+1,v_i}. - Christian Barrientos, Jul 29 2018
Also number of non-isomorphic stairs with n+1 cells. A stair is a snake polyomino allowing only two directions for adjacent cells: east and north. - Christian Barrientos and Sarah Minion, Jul 29 2018
From Robert A. Russell, Oct 28 2018: (Start)
There are two different unoriented row colorings using two colors that give us very similar results here, a difference of one in the offset. In an unoriented row, chiral pairs are counted as one.
a(n) is the number of color patterns (set partitions) of an unoriented row of length n using two or fewer colors (subsets). Two color patterns are equivalent if the colors are permutable.
a(n+1) is the number of ways to color an unoriented row of length n using two noninterchangeable colors (one need not use both colors).
See the examples below of these two different colorings. (End)
Also arises from the enumeration of types of based polyhedra with exactly two triangular faces [Rademacher]. - N. J. A. Sloane, Apr 24 2020
a(n) is the number of (unlabeled) 2-paths with n+4 vertices. (A 2-path with order n at least 4 can be constructed from a 3-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to an existing 2-clique containing an existing 2-leaf.) - Allan Bickle, Apr 05 2022
a(n) is the number of caterpillars with a perfect matching and order 2n+2. - Christian Barrientos, Sep 12 2023
a(n) is also the number of distinct planar embeddings of the (n+2)-centipede graph (up to at least n=8 and likely for all larger n). - Eric W. Weisstein, May 21 2024
a(n) is also the number of distinct planar embeddings of the 2 X (n+2) grid graph i.e., the (n+2)-ladder graph. - Eric W. Weisstein, May 21 2024
Dimension of the homogeneous component of degree n of the free Jordan algebra on two generators (or, in this case, the free special Jordan algebra on two generators). It follows from (Shirshov 1956, Cohn 1959). - Vladimir Dotsenko, Mar 29 2025

Examples

			a(5) = 10 because there are 16 compositions of 5 (shown as <vectors>) but only 10 equivalence classes (shown as {sets}): {<5>}, {<4,1>,<1,4>}, {<3,2>,<2,3>}, {<3,1,1>,<1,1,3>}, {<1,3,1>},{<2,2,1>,<1,2,2>}, {<2,1,2>}, {<2,1,1,1>,<1,1,1,2>}, {<1,2,1,1>,<1,1,2,1>}, {<1,1,1,1,1>}. - _Geoffrey Critzer_, Nov 02 2012
G.f. = x + 2*x^2 + 3*x^3 + 6*x^4 + 10*x^5 + 20*x^6 + 36*x^7 + 72*x^8 + ... - _Michael Somos_, Jun 24 2018
From _Robert A. Russell_, Oct 28 2018: (Start)
For a(5)=10, the 4 achiral patterns (set partitions) are AAAAA, AABAA, ABABA, and ABBBA. The 6 chiral pairs are AAAAB-ABBBB, AAABA-ABAAA, AAABB-AABBB, AABAB-ABABB, AABBA-ABBAA, and ABAAB-ABBAB. The colors are permutable.
For n=4 and a(n+1)=10, the 4 achiral colorings are AAAA, ABBA, BAAB, and BBBB. The 6 achiral pairs are AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, and BABB-BBAB. The colors are not permutable. (End)

References

  • K. Balasubramanian, "Combinatorial Enumeration of Chemical Isomers", Indian J. Chem., (1978) vol. 16B, pp. 1094-1096. See page 1095.
  • Wayne M. Dymacek, Steinhaus graphs. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 399--412, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561065 (81f:05120)
  • Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007.
  • Joseph S. Madachy: Madachy's Mathematical Recreations. New York: Dover Publications, Inc., 1979, p. 46 (first publ. by Charles Scribner's Sons, New York, 1966, under the title: Mathematics on Vacation)
  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]
  • C. A. Pickover, Keys to Infinity, Wiley 1995, p. 75.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 2 of A320750 (set partitions).
Cf. A131577 (oriented), A122746(n-3) (chiral), A016116 (achiral), for set partitions with up to two subsets.
Column 2 of A277504, offset by one (colors not permutable).
Cf. A000079 (oriented), A122746(n-2) (chiral), and A060546 (achiral), for a(n+1).
Cf. A001998, A001444, A051436, A051437, A007582, A001445, A032085.

Programs

  • Haskell
    a005418 n = sum $ a034851_row (n - 1) -- Reinhard Zumkeller, Jan 14 2012
  • Maple
    A005418 := n->2^(n-2)+2^(floor(n/2)-1): seq(A005418(n), n=1..34);
  • Mathematica
    LinearRecurrence[{2,2,-4}, {1,2,3}, 40] (* or *) Table[2^(n-2)+2^(Floor[n/2]-1), {n,40}] (* Harvey P. Dale, Jan 18 2012 *)
  • PARI
    A005418(n)= 2^(n-2) + 2^(n\2-1); \\ Joerg Arndt, Sep 16 2013
  • Python
    def A005418(n): return 1 if n == 1 else 2**((m:= n//2)-1)*(2**(n-m-1)+1) # Chai Wah Wu, Feb 03 2022

Formula

a(n) = 2^(n-2) + 2^(floor(n/2) - 1).
G.f.: -x*(-1 + 3*x^2) / ( (2*x - 1)*(2*x^2 - 1) ). - Simon Plouffe in his 1992 dissertation
G.f.: x*(1+2*x)*(1-3*x^2)/((1-4*x^2)*(1-2*x^2)), not reduced. - Wolfdieter Lang, May 08 2001
a(n) = 6*a(n - 2) - 8*a(n - 4). a(2*n) = A063376(n - 1) = 2*a(2*n - 1); a(2*n + 1) = A007582(n). - Henry Bottomley, Jul 14 2001
a(n+2) = 2*a(n+1) - A077957(n) with a(1) = 1, a(2) = 2. - Yosu Yurramendi, Oct 24 2008
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Jaume Oliver Lafont, Dec 05 2008
Union of A007582 and A161168. Union of A007582 and A063376. - Jaroslav Krizek, Aug 14 2009
G.f.: G(0); G(k) = 1 + 2*x/(1 - x*(1+2^(k+1))/(x*(1+2^(k+1)) + (1+2^k)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 12 2011
a(2*n) = 2*a(2*n-1) and a(2*n+1) = a(2*n) + 4^(n-1) with a(1) = 1. - Johannes W. Meijer, Aug 26 2013
From Robert A. Russell, Oct 28 2018: (Start)
a(n) = (A131577(n) + A016116(n)) / 2 = A131577(n) - A122746(n-3) = A122746(n-3) + A016116(n), for set partitions with up to two subsets.
a(n+1) = (A000079(n) + A060546(n)) / 2 = A000079(n) - A122746(n-2) = A122746(n-2) + A060546(n), for two colors that do not permute.
a(n) = Sum_{j=0..k} (S2(n,j) + Ach(n,j)) / 2, where k=2 is the maximum number of colors, S2(n,k) is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n+1) = (k^n + k^ceiling(n/2)) / 2, where k=2 is number of colors we can use. (End)
E.g.f.: (cosh(2*x) + 2*cosh(sqrt(2)*x) + sinh(2*x) + sqrt(2)*sinh(sqrt(2)*x) - 3)/4. - Stefano Spezia, Jun 01 2022

A000238 Number of oriented trees with n nodes.

Original entry on oeis.org

1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, 492180, 2266502, 10598452, 50235931, 240872654, 1166732814, 5702001435, 28088787314, 139354922608, 695808554300, 3494390057212, 17641695461662, 89495023510876, 456009893224285, 2332997330210440
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 286.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 60, r(x).
  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000060, A000151, A051437 (linear oriented), A334827 (oriented star-like).
Diagonal of A335362.

Programs

  • Maple
    A:= proc(n) option remember; if n=0 then 0 else unapply(convert(series(x*exp(2* add(A(n-1)(x^k)/k, k=1..n-1)), x=0,n), polynom), x) fi end: a:= n-> coeff(series(A(n+1)(x) *(1-A(n+1)(x)), x=0, n+1), x,n): seq(a(n), n=1..26); # Alois P. Heinz, Aug 20 2008
  • Mathematica
    A[n_][y_] := A[n][y] = If[n == 0, 0, Normal[Series[x*Exp[2*Sum[A[n-1][x^k]/k, {k, 1, n-1}]], {x, 0, n}] /. x -> y]]; a[n_] := SeriesCoefficient[A[n+1][x]*(1-A[n+1][x]), {x, 0, n}]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Feb 12 2014, translated from Maple *)
  • PARI
    seq(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); Vec(Ser(A)-x*Ser(A)^2)} \\ Andrew Howroyd, May 13 2018

Formula

G.f. = x+x^2+3*x^3+8*x^4+27*x^5+... = R(x)-R(x)^2, where R(x) = g.f. for A000151.
a(n) ~ c * d^n / n^(5/2), where d = A245870 = 5.64654261623294971289271351621..., c = 0.22571615379282714232305... . - Vaclav Kotesovec, Dec 08 2014

Extensions

2 errors corrected by Paul Zimmermann, Mar 01 1996
More terms from N. J. A. Sloane, Mar 10 2007

A303864 Array read by antidiagonals: T(n,k) = number of noncrossing path sets on k*n nodes up to rotation with each path having exactly k nodes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 6, 2, 1, 1, 4, 36, 38, 3, 1, 1, 10, 210, 960, 384, 6, 1, 1, 16, 1176, 18680, 35956, 4425, 14, 1, 1, 36, 6328, 313664, 2280910, 1588192, 57976, 34, 1, 1, 64, 32896, 4683168, 111925464, 323840016, 77381016, 807318, 95, 1
Offset: 0

Views

Author

Andrew Howroyd, May 01 2018

Keywords

Examples

			Array begins:
=======================================================
n\k| 1  2     3        4           5              6
---+---------------------------------------------------
0  | 1  1     1        1           1              1 ...
1  | 1  1     1        3           4             10 ...
2  | 1  1     6       36         210           1176 ...
3  | 1  2    38      960       18680         313664 ...
4  | 1  3   384    35956     2280910      111925464 ...
5  | 1  6  4425  1588192   323840016    46552781760 ...
6  | 1 14 57976 77381016 50668922540 21346459738384 ...
...

Links

Crossrefs

Columns 2..4 are A002995(n+1), A303865, A303866.
Row n=1 is A051437(k-3).
Cf. A302828, A303844, A303869.
Cf. A295224 (polygon dissections), A303694 (sets of cycles instead of paths).

Programs

  • Mathematica
    nmax = 10; seq[n_, k_] := Module[{p, q, h}, p = 1 + InverseSeries[ x/(k*2^If[k == 1, 0, k - 3]*(1 + x)^k) + O[x]^n, x ]; h = p /. x -> x^2 + O[x]^n; q = x*D[p, x]/p; Integrate[((p - 1)/k + Sum[EulerPhi[d]*(q /. x -> x^d + O[x]^n), {d, 2, n}])/x, x] + If[OddQ[k], 0, 2^(k/2 - 2)*x*h^(k/2)] + 1];
Clear[col]; col[k_] := col[k] = CoefficientList[seq[nmax, k], x];
T[n_, k_] := col[k][[n + 1]];
Table[T[n - k, k], {n, 0, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 04 2018, after Andrew Howroyd *)
  • PARI
    seq(n,k)={ \\ gives gf of k'th column
my(p=1 + serreverse( x/(k*2^if(k==1, 0, k-3)*(1 + x)^k) + O(x*x^n) ));
my(h=subst(p,x,x^2+O(x*x^n)), q=x*deriv(p)/p);
intformal( ((p-1)/k + sum(d=2,n,eulerphi(d)*subst(q,x,x^d+O(x*x^n))))/x) + if(k%2, 0, 2^(k/2-2)*x*h^(k/2)) + 1;
}
Mat(vector(6, k, Col(seq(7, k))))

A303869 Triangle read by rows: T(n,k) = number of noncrossing path sets on n nodes up to rotation with k paths and isolated vertices allowed.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 4, 2, 1, 4, 11, 8, 2, 1, 10, 34, 39, 16, 3, 1, 16, 92, 144, 90, 25, 3, 1, 36, 256, 545, 473, 197, 40, 4, 1, 64, 672, 1878, 2184, 1246, 370, 56, 4, 1, 136, 1762, 6296, 9436, 7130, 2910, 658, 80, 5, 1, 256, 4480, 20100, 38025, 36690, 19698, 6090, 1080, 105, 5, 1
Offset: 1

Views

Author

Andrew Howroyd, May 01 2018

Keywords

Examples

			Triangle begins:
    1;
    1,    1;
    1,    1,    1;
    3,    4,    2,    1;
    4,   11,    8,    2,    1;
   10,   34,   39,   16,    3,    1;
   16,   92,  144,   90,   25,    3,   1;
   36,  256,  545,  473,  197,   40,   4,  1;
   64,  672, 1878, 2184, 1246,  370,  56,  4, 1;
  136, 1762, 6296, 9436, 7130, 2910, 658, 80, 5, 1;
  ...

Links

Crossrefs

Row sums are A303836.
Column 1 is A051437(n-3).
Cf. A303844, A303732, A303864, A303868.

Programs

  • PARI
    \\ See A303732 for NCPathSetsModCyclic
{ my(rows=Vec(NCPathSetsModCyclic(vector(10, k, y))-1));
for(n=1, #rows, for(k=1,n,print1(polcoeff(rows[n],k), ", ")); print;)}

A052957 Expansion of 2*(1-x-x^2)/((1-2*x)*(1-2*x^2)).

Original entry on oeis.org

2, 2, 6, 8, 20, 32, 72, 128, 272, 512, 1056, 2048, 4160, 8192, 16512, 32768, 65792, 131072, 262656, 524288, 1049600, 2097152, 4196352, 8388608, 16781312, 33554432, 67117056, 134217728, 268451840, 536870912, 1073774592, 2147483648
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Programs

  • GAP
    a:=[2,2,6];; for n in [4..30] do a[n]:=2*a[n-1]+2*a[n-2]-4*a[n-3]; od; a; # G. C. Greubel, Oct 22 2019
  • Magma
    [2] cat [Round(2^n +2^((n-1)/2)*(1+(-1)^n)/Sqrt(2)): n in [1..30]]; // G. C. Greubel, Oct 22 2019
  • Maple
    spec:= [S,{S=Union(Sequence(Prod(Union(Z,Z),Z)),Sequence(Union(Z,Z)))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series(2*(1-x-x^2)/((1-2*x)*(1-2*x^2)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 22 2019
  • Mathematica
    CoefficientList[Series[2*(1-x-x^2)/((1-2*x)*(1-2*x^2)), {x, 0, 31}], x] (* Michael De Vlieger, Sep 23 2016 *)
Join[{2}, Table[2^n +2^((n-1)/2)*(1+(-1)^n)/Sqrt[2], {n, 30}]] (* G. C. Greubel, Oct 22 2019 *)
LinearRecurrence[{2,2,-4},{2,2,6},40] (* Harvey P. Dale, Jul 19 2020 *)
  • PARI
    a(n)=2^n+if(n%2,,2^(n/2)) \\ Charles R Greathouse IV, Sep 23 2016
  • Sage
    [2]+[2^n +2^((n-1)/2)*(1+(-1)^n)/sqrt(2) for n in (1..30)] # G. C. Greubel, Oct 22 2019

Formula

G.f.: 2*(1-x-x^2)/((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 + Sum_{alpha=RootOf(-1+2*x^2)} alpha^(-n)/2.
a(n) = 2*A051437(n+1), n > 0. - R. J. Mathar, Nov 27 2011
From Colin Barker, Sep 23 2016: (Start)
a(n) = 2^(n/2) + 2^n for n even.
a(n) = 2^n for n odd.
(End)
E.g.f.: (1/2)*(2*exp(2*x) + exp(-sqrt(2)*x) + exp(sqrt(2)*x)). - Stefano Spezia, Oct 22 2019

Extensions

More terms from James Sellers, Jun 05 2000

A334827 The number of oriented star-like and star trees with n arcs.

Original entry on oeis.org

4, 17, 66, 221, 688, 2034, 5788, 15998, 43192, 114496, 298712, 769340, 1959064, 4940761, 12354210, 30660947, 75583868, 185208833, 451356846, 1094522547, 2642121008, 6351335083, 15208854510, 36288478177, 86295204732, 204571273167, 483532711338, 1139738858221
Offset: 3

Views

Author

R. J. Mathar, Jun 09 2020

Keywords

Examples

			a(6)=221 counts 132 oriented star-like trees with 3 rays and 6 arcs, 62 with 4 rays and 6 arcs, 20 with 5 rays and 6 arcs, and 7 star trees. In the illustrations in A000238 [Mathar] this is the same as 48 (shape 2) + 64 (shape 3) + 20 (shape 4) +32 (shape 7) + 30 (shape 8) +20 (shape 10) + 7 (shape 11).

Links

Crossrefs

Cf. A000238 (oriented trees), A051437 (linear oriented trees), A209406 (star-like oriented by number of arcs and rays), A004250 (undirected edges).

Formula

a(n) = A034899(n) -2^(n+1) = Sum_{k>=3} A209406(n,k).
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