A053730 -id:A053730 - cp's OEIS frontend

A104270 a(n) = 2^(n - 2)*(binomial(n,2) + 2).

1, 3, 10, 32, 96, 272, 736, 1920, 4864, 12032, 29184, 69632, 163840, 380928, 876544, 1998848, 4521984, 10158080, 22675456, 50331648, 111149056, 244318208, 534773760, 1166016512, 2533359616, 5486149632, 11844714496, 25501368320, 54760833024, 117306294272, 250718715904

Offset: 1

Ralf Stephan, Apr 17 2005

Cardinality of set of crossing-similarity classes.
Total number of hexagonal systems with n hexagons that cannot be placed in a cage of size n-1. - Parthasarathy Nambi, Sep 06 2006
a(n+1) is equal to n! times the determinant of the n X n matrix whose (i,j)-entry is KroneckerDelta[i,j](((i+2)/(i)) - 1) + 1. - John M. Campbell, May 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..237
M. Klazar, On identities concerning the numbers of crossings and nestings of two edges in matchings
Tosic R., Masulovic D., Stojmenovic I., Brunvoll J., Cyvin B. N. and Cyvin S. J., Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., 1995, 35, 181-187, Table 1 (with an error at h=16).
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Equals (1/2) A053730. Partial sums of A084264.

[2^(n-2)*(Binomial(n,2)+2): n in [1..30]]; // Vincenzo Librandi, May 24 2011

Table[n!*Det[Array[KroneckerDelta[#1,#2](((#1+2)/(#1))-1)+1 &, {n,n}]], {n, 1, 10}] (* John M. Campbell, May 20 2011 *)
LinearRecurrence[{6,-12,8},{1,3,10},30] (* Harvey P. Dale, Jul 03 2017 *)

a(n)=(binomial(n,2)+2)<<(n-2) \\ Charles R Greathouse IV, May 24 2011

G.f.: x*(1 - 3*x + 4*x^2)/(1-2*x)^3. - Colin Barker, Apr 01 2012

A300451 a(n) = (3n^2 - 3n + 8)*2^(n - 3).

Original entry on oeis.org

1, 2, 7, 26, 88, 272, 784, 2144, 5632, 14336, 35584, 86528, 206848, 487424, 1134592, 2613248, 5963776, 13500416, 30343168, 67764224, 150470656, 332398592, 730857472, 1600126976, 3489660928, 7583301632, 16424894464, 35467034624, 76369887232, 164014063616

Offset: 0

Franck Maminirina Ramaharo, Mar 06 2018

First difference yields A295288.
1 and 7 are the only odd terms.
a(n) gives the number of words of length n defined over the alphabet {a,b,c,d} such that letters from {a,b} are only used in pairs of at most one, and consist of (a,a), (a,b) and (b,a).

			a(4) = 88. The corresponding words are cccc, cccd, ccdc, ccdd, cdcc, cdcd, cddc, cddd, dccc, dccd, dcdc, dcdd, ddcc, ddcd, dddc, dddd, caac, caca, ccaa, caad, cada, caad, cabc, cacb, ccab, cabd, cadb, cabd, cbac, cbca, ccba, cbad, cbda, cbad, daac, daca, dcaa, daad, dada, daad, dabc, dacb, dcab, dabd, dadb, dabd, dbac, dbca, dcba, dbad, dbda, dbad, aacc, acac, acca, aacd, acad, acda, aadc, adac, adca, aadd, adad, adda, abcc, acbc, accb, abcd, acbd, acdb, abdc, adbc, adcb, abdd, adbd, addb, bacc, bcac, bcca, bacd, bcad, bcda, badc, bdac, bdca, badd, bdad, bdda.

Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015.
Ian F. Blake, The Mathematical Theory of Coding, Academic Press, 1975.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Hermann Gruber, Jonathan Lee and Jeffrey Shallit, Enumerating regular expressions and their languages, arXiv preprint arXiv:1204.4982 [cs.FL], 2012.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Cf. A000079, A000292, A001788, A005448, A006003, A045943, A052481, A053730, A081908, A295288, A300184.

List([0..30],n->(3*n^2-3*n+8)*2^(n-3)); # Muniru A Asiru, Mar 09 2018

[(3*n^2-3*n+8)*2^(n-3): n in [0..30]]; // Vincenzo Librandi, Mar 10 2018

A := n -> (3*n^2 - 3*n + 8)*2^(n - 3);
seq(A(n), n = 0 .. 70);

Table[(3 n^2 - 3 n + 8) 2^(n - 3), {n, 0, 70}]
CoefficientList[Series[(1 - 4x + 7x^2)/(1 - 2x)^3, {x, 0, 30}], x] (* or *)
LinearRecurrence[{6, -12, 8}, {1, 2, 7}, 30] (* Robert G. Wilson v, Mar 07 2018 *)

makelist((3*n^2 - 3*n + 8)*2^(n - 3), n, 0, 70);

a(n) = (3*n^2-3*n+8)*2^(n-3); \\ Altug Alkan, Mar 09 2018

G.f.:  (1 - 4*x + 7*x^2)/(1 - 6*x + 12*x^2 - 8*x^3).
E.g.f: (1/2)*(3*x^2 + 2)*exp(2*x).
a(n) = ((3/4)*binomial(n, 2) + 1)*2^n.
a(n) = 2*a(n-1) + 3*(n - 1)*2^(n - 2), with a(0) = 1.
a(n) = 3*A001788(n) + A000079(n).
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), for n >= 3, with a(0) = 1, a(1) = 2 and a(2) = 7.
a(n) = A300184(n,2).

cp's OEIS Frontend

A104270 a(n) = 2^(n - 2)*(binomial(n,2) + 2).

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A300451 a(n) = (3n^2 - 3n + 8)*2^(n - 3).

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cp's OEIS Frontend

A104270 a(n) = 2^(n - 2)*(binomial(n,2) + 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A300451 a(n) = (3*n^2 - 3*n + 8)*2^(n - 3).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

Maxima

PARI

Formula

A300451 a(n) = (3n^2 - 3n + 8)*2^(n - 3).