A034877 -id:A034877 - cp's OEIS frontend

A032085 Number of reversible strings with n beads of 2 colors. If more than 1 bead, not palindromic.

2, 1, 2, 6, 12, 28, 56, 120, 240, 496, 992, 2016, 4032, 8128, 16256, 32640, 65280, 130816, 261632, 523776, 1047552, 2096128, 4192256, 8386560, 16773120, 33550336, 67100672, 134209536, 268419072, 536854528

Offset: 1

Christian G. Bower

a(n) is also the number of induced subgraphs with odd number of edges in the path graph P(n) if n>0. - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 06 2009
A common recurrence of the bisections A020522 and A006516 means a(n+4) = 6*a(n+2) - 8*a(n), n>1. - Yosu Yurramendi, Aug 07 2008
Also, the decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 05 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
C. G. Bower, Transforms (2)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1022
S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A005418, A016116. Essentially the same as A122746.
Row sums of triangle A034877.
Cf. A289404, A289405, A052551.

[2] cat [2^(n-1)-2^Floor((n-1)/2) : n in [2..40]]; // Wesley Ivan Hurt, Jul 03 2020

Join[{2}, LinearRecurrence[{2, 2, -4}, {1, 2, 6}, 29]] (* Jean-François Alcover, Oct 11 2017 *)

a(n)=([0,1,0; 0,0,1; -4,2,2]^(n-1)*[2;1;2])[1,1] \\ Charles R Greathouse IV, Oct 21 2022

"BHK" (reversible, identity, unlabeled) transform of 2, 0, 0, 0, ...
a(n) = 2^(n-1)-2^floor((n-1)/2), n > 1. - Vladeta Jovovic, Nov 11 2001
G.f.: 2*x+x^2/((1-2*x)*(1-2*x^2)). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 25 2004
a(n) = A005418(n+1)-A016116(n+2), n>1. - Yosu Yurramendi, Aug 07 2008
a(n+1) = A077957(n) + 2*a(n), n>1. a(n+2) = A000079(n+1) + 2*a(n), n>1. - Yosu Yurramendi, Aug 10 2008
First differences: a(n+1)-a(n) = A007179(n) = A156232(n+2)/4, n>1. - Paul Curtz, Nov 16 2009
a(n) = 2*(a(n-1) bitwiseOR a(n-2)), n>3. - Pierre Charland, Dec 12 2010
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Wesley Ivan Hurt, Jul 03 2020

A006584 If n mod 2 = 0 then n(n^2-4)/12 else n(n^2-1)/12.

Original entry on oeis.org

0, 0, 0, 2, 4, 10, 16, 28, 40, 60, 80, 110, 140, 182, 224, 280, 336, 408, 480, 570, 660, 770, 880, 1012, 1144, 1300, 1456, 1638, 1820, 2030, 2240, 2480, 2720, 2992, 3264, 3570, 3876, 4218, 4560, 4940, 5320

Offset: 0

N. J. A. Sloane

Graded dimension of L''/[L',L''] for the free Lie algebra on 2 generators. Let L be a free Lie algebra with 2 generators graded by the total degree. Set L'=[L,L] and L''=[L',L']. Then a(n) is equal to the dimension of the homogeneous subspace of degree n+2 in the quotient L''/[L',L'']. - Sergei Duzhin, Mar 15 2004
Also the 2nd Witt transform of A000027. - R. J. Mathar, Nov 08 2008
Also the number of 3-element subsets of {1..n+1} whose elements sum up to an odd integer, i.e., the third column of A159916: e.g. a(3)=2 corresponds to the two subsets {1,2,4} and {2,3,4} of {1..4}. - M. F. Hasler, May 01 2009
The set of magic numbers for an idealized harmonic oscillator nucleus with a biaxially deformed prolate ellipsoid shape and an oscillator ratio of 2:1. - Jess Tauber, May 13 2013
Quasipolynomial of order 2. - Charles R Greathouse IV, May 14 2013

W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.

Moussa Benoumhani and Messaoud Kolli, Finite topologies and partitions, JIS 13 (2010), Article 10.3.5, Lemma 6 6th line.
Marc Le Brun, Email to N. J. A. Sloane, Jul 1991.
Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Pieter Moree, Convoluted convolved Fibonacci numbers, arXiv:math/0311205 [math.CO], 2003.
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000027, A003451, A006918, A027656, A034877.

Partial sums of A110660.

A006584:=n->`if`(n mod 2 = 0, n*(n^2-4)/12, n*(n^2-1)/12): seq(A006584(n), n=0..100); # Wesley Ivan Hurt, May 08 2016

If[EvenQ@ #, #*(#^2 - 4)/12, #*(#^2 - 1)/12] & /@ Range[0, 40] (* or *) Table[n*(2*n^2 - 5 - 3*(-1)^n)/24, {n, 0, 40}] (* Michael De Vlieger, Apr 03 2015 *)

A006584(n)=n*(n^2-if(n%2,1,4))\12 \\ M. F. Hasler, May 01 2009

a(n)=n*if(n%2, n^2-1, n^2-4)/12 \\ Charles R Greathouse IV, Aug 11 2017

a(n+3) = A003451(n) + A027656(n). - Yosu Yurramendi, Aug 07 2008
G.f.: 2*x^3/((1-x)^4*(1+x)^2). a(n) = 2*A006918(n-2). - R. J. Mathar, Nov 08 2008
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Jaume Oliver Lafont, Dec 05 2008
a(n) = n*(2*n^2-5-3*(-1)^n)/24. - Luce ETIENNE, Apr 03 2015
a(n) = Sum_{i=1..n} floor(i*(n-i)/2). - Wesley Ivan Hurt, May 07 2016
E.g.f.: x*(x*(x + 3)*exp(x) - 3*sinh(x))/12. - Ilya Gutkovskiy, May 08 2016
Sum_{n>=3} 1/a(n) = 75/8 - 12*log(2). - Amiram Eldar, Sep 17 2022

A034852 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 2, 4, 4, 2, 0, 0, 3, 6, 10, 6, 3, 0, 0, 3, 9, 16, 16, 9, 3, 0, 0, 4, 12, 28, 32, 28, 12, 4, 0, 0, 4, 16, 40, 60, 60, 40, 16, 4, 0, 0, 5, 20, 60, 100, 126, 100, 60, 20, 5, 0, 0, 5, 25, 80, 160, 226, 226, 160, 80, 25, 5, 0, 0, 6, 30, 110, 240

Offset: 0

N. J. A. Sloane

Also number of linear unbranched n-4-catafusenes of C_{2v} symmetry.
Number of n-bead black-white reversible strings with k black beads; also binary grids; string is not palindromic. - Yosu Yurramendi, Aug 08 2008
The first seven columns are A004526, A002620, A006584, A032091, A032092, A032093, A032094. Row sums give essentially A032085. - Yosu Yurramendi, Aug 08 2008
From Álvar Ibeas, Jun 01 2020: (Start)
T(n, k) is the sum of odd-degree coefficients of the Gaussian polynomial [n, k]_q. The area below a NE lattice path between (0,0) and (k, n-k) is even for A034851(n, k) paths and odd for T(n, k) of them.
For a (non-reversible) string of k black and n-k white beads, consider the minimum number of bead transpositions needed to place the black ones to the left and the white ones to the right (in other words, the number of inversions of the permutation obtained by labeling the black beads by integers 1,...,k and the white ones by k+1,...,n, in the same order they take on the string). It is even for A034851(n, k) strings and odd for T(n, k) cases.
(End)

			Triangle begins:
  0;
  0 0;
  0 1 0;
  0 1 1 0;
  0 2 2 2 0;
  0 2 4 4 2 0;
  ...

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
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

Cf. A007318, A034851, A051159.

Essentially the same as A034877.

a034852 n k = a034852_tabl !! n !! k
a034852_row n = a034852_tabl !! n
a034852_tabl = zipWith (zipWith (-)) a007318_tabl a034851_tabl
-- Reinhard Zumkeller, Mar 24 2012

nmax = 12; t[n_?EvenQ, k_?EvenQ] := (Binomial[n, k] - Binomial[n/2, k/2])/ 2; t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_?OddQ, k_?EvenQ] := (Binomial[n, k] - Binomial[(n-1)/2, k/2])/2; t[n_?OddQ, k_?OddQ] := (Binomial[n, k] - Binomial[(n-1)/2, (k-1)/2])/2; Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 15 2011, after Yosu Yurramendi *)

Equals (A007318-A051159)/2. - Yosu Yurramendi, Aug 08 2008

T(n, k) = T(n - 1, k - 1) + T(n - 1, k); except when n is even and k odd, in which case T(n, k) = A034851(n, k) = T(n - 1, k - 1) + A034841(n - 1, k) = A034841(n - 1, k - 1) + T(n - 1, k) = C(n, k) / 2. - Álvar Ibeas, Jun 01 2020

More terms from James Sellers, May 04 2000

A032095 Number of reversible strings with n-1 black beads and n-1 white beads. String is not palindromic.

Original entry on oeis.org

1, 1, 2, 10, 32, 126, 452, 1716, 6400, 24310, 92252, 352716, 1351616, 5200300, 20056584, 77558760, 300533760, 1166803110, 4537543340, 17672631900, 68923172032, 269128937220, 1052049129144, 4116715363800

Offset: 1

Christian G. Bower

nonn

C. G. Bower, Transforms (2)

Central column of triangle A034877.

Join[{1},Table[If[EvenQ[n],Binomial[2n-2,n-1]/2,(Binomial[2n-2,n-1]- Binomial[ n-1,(n-1)/2])/2],{n,2,30}]] (* Harvey P. Dale, Sep 12 2013 *)

"BHK[ n ](2n-1)" (reversible, identity, unlabeled, n parts, 2n-1 elements) transform of 1, 1, 1, 1...
for n>1 a(n)=(1/2)*C(2n-2, n-1) if n is even, a(n)=(1/2)[C(2n-2, n-1)-C(n-1, (n-1)/2)] if n is odd. (Emeric Deutsch Jul 25 2002)
E.g.f. : 1+exp(x)*sinh(x)*BesselI(0,2x). [From Paul Barry, Aug 11 2009]

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A032085 Number of reversible strings with n beads of 2 colors. If more than 1 bead, not palindromic.

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A006584 If n mod 2 = 0 then n*(n^2-4)/12 else n*(n^2-1)/12.

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A034852 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).

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A032095 Number of reversible strings with n-1 black beads and n-1 white beads. String is not palindromic.

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A006584 If n mod 2 = 0 then n(n^2-4)/12 else n(n^2-1)/12.