Susanna Cuyler - cp's OEIS frontend

A265046 Coordination sequence for a 4.6.6 point in the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6} of the plane by squares and dominoes (hexagons).

1, 3, 5, 8, 13, 18, 23, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset: 0

N. J. A. Sloane and Susanna Cuyler, Dec 27 2015

This tiling is 3-transitive but not 3-uniform since the polygons are not regular. It is a common floor-tiling.

The coordination sequences with respect to the points of types 4.6.6 (labeled "C" in the illustration), 6.6.6 ("B"), 6.6.6.6 ("A") are A265046, A265045, and A008574, respectively. The present sequence is for a "C" point.

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, A portion of the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6}
N. J. A. Sloane, A portion of the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6} showing the three types of point
N. J. A. Sloane, Hand-drawn illustration showing a(0) to a(8)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008574, A265045.

Vec((1+x)*(1+x^3+x^4-x^5+x^6-x^7)/(1-x)^2+ O(x^100)) \\ Colin Barker, Jan 01 2016

For n >= 7 all three sequences equal 4n. (For n >= 7 the n-th shell contains n-1 points in the interior of each quadrant plus 4 points on the axes.)
From Colin Barker, Jan 01 2016: (Start)
a(n) = 2*a(n-1)-a(n-2) for n>8.
a(n) = 4*n for n>6.
G.f.: (1+x)*(1+x^3+x^4-x^5+x^6-x^7) / (1-x)^2.
(End)

A265045 Coordination sequence for a 6.6.6 point in the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6} of the plane by squares and dominoes (hexagons).

Original entry on oeis.org

1, 3, 7, 11, 14, 18, 23, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232

Offset: 0

N. J. A. Sloane and Susanna Cuyler, Dec 27 2015

This tiling is 3-transitive but not 3-uniform since the polygons are not regular. It is a common floor-tiling.

The coordination sequences with respect to the points of types 4.6.6 (labeled "C" in the illustration), 6.6.6 ("B"), 6.6.6.6 ("A") are A265046, A265045, and A008574, respectively. The present sequence is for a "B" point.

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, A portion of the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6}
N. J. A. Sloane, A portion of the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6} showing the three types of point
N. J. A. Sloane, Hand-drawn illustration showing a(0) to a(10)
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008574, A265046.

LinearRecurrence[{2,-1},{1,3,7,11,14,18,23,28,32},60] (* Harvey P. Dale, Sep 23 2017 *)

Vec((1+x)*(1+2*x^2-2*x^3+x^4+x^6-x^7)/(1-x)^2 + O(x^100)) \\ Colin Barker, Jan 01 2016

For n >= 7 all three sequences equal 4n. (For n >= 7 the n-th shell contains n-1 points in the interior of each quadrant plus 4 points on the axes.)
From Colin Barker, Jan 01 2016: (Start)
a(n) = 2*a(n-1)-a(n-2) for n>8.
a(n) = 4*n for n>6.
G.f.: (1+x)*(1+2*x^2-2*x^3+x^4+x^6-x^7) / (1-x)^2.
(End)

A181753 Universal sequence of period 56 which contains every 3-subset of {1,2,...,8} exactly once.

Original entry on oeis.org

1, 3, 5, 6, 7, 2, 5, 6, 8, 2, 3, 4, 7, 2, 3, 5, 7, 8, 1, 4, 7, 8, 2, 4, 5, 6, 1, 4, 5, 7, 1, 2, 3, 6, 1, 2, 4, 6, 7, 8, 3, 6, 7, 1, 3, 4, 5, 8, 3, 4, 6, 8, 1, 2, 5, 8, 1, 3, 5, 6, 7, 2, 5, 6, 8, 2, 3, 4, 7, 2, 3, 5, 7, 8, 1, 4, 7, 8, 2, 4, 5, 6, 1, 4, 5, 7, 1, 2, 3, 6, 1, 2, 4, 6, 7, 8, 3, 6, 7, 1, 3, 4, 5, 8, 3, 4, 6, 8, 1, 2, 5, 8, 1, 3, 5, 6, 7, 2, 5, 6, 8, 2, 3, 4, 7, 2, 3, 5, 7, 8, 1, 4, 7, 8, 2, 4, 5, 6, 1, 4, 5, 7, 1, 2, 3, 6, 1, 2, 4, 6, 7, 8, 3, 6, 7, 1, 3, 4, 5, 8, 3, 4, 6, 8, 1, 2, 5, 8

Offset: 1

Susanna Cuyler, Nov 14 2010

Each successive block of length 7 is obtained by adding 5 mod 8 to the previous block.

			The period is 1356725 6823472 3578147 8245614 5712361 2467836 7134583 4681258.

G. Hurlbert, On universal cycles for k-subsets of an n-element set, SIAM J. Discrete Math., 7 (1994), 598-604.
A. Leitner and A. Godbole, Universal cycles of classes of restricted words, Discrete Math., 310 (2010) 3303-3309.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1,0,0,0,0,0,-1, 1).

Cf. A010887.

a181753 n = a181753_list !! (n-1)
a181753_list = concat $ iterate
               (map ((+ 1) . flip mod 8 . (+ 4))) [1,3,5,6,7,2,5]
-- Reinhard Zumkeller, Nov 09 2014

From Chai Wah Wu, Jun 13 2020: (Start)
a(n) = a(n-1) - a(n-7) + a(n-8) - a(n-14) + a(n-15) - a(n-21) + a(n-22) - a(n-28) + a(n-29) - a(n-35) + a(n-36) - a(n-42) + a(n-43) - a(n-49) + a(n-50) for n > 50.
G.f.: x*(-8*x^49 + 3*x^48 + 3*x^47 + x^46 - 7*x^45 + 2*x^44 + 2*x^43 - 7*x^42 - 2*x^41 + 6*x^40 + 2*x^39 - 6*x^38 + 4*x^37 - 4*x^36 - 6*x^35 + x^34 + x^33 + 3*x^32 - 5*x^31 + 6*x^30 - 2*x^29 - 5*x^28 - 4*x^27 + 4*x^26 + 4*x^25 - 4*x^24 - 4*x^21 - x^20 - x^19 + 5*x^18 - 3*x^17 + 2*x^16 - 6*x^15 - 3*x^14 + 2*x^13 + 2*x^12 - 2*x^11 - 2*x^10 + 4*x^9 - 4*x^8 - 2*x^7 - 3*x^6 + 5*x^5 - x^4 - x^3 - 2*x^2 - 2*x - 1)/(x^50 - x^49 + x^43 - x^42 + x^36 - x^35 + x^29 - x^28 + x^22 - x^21 + x^15 - x^14 + x^8 - x^7 + x - 1). (End)

A176998 a(n) = binary "k-number" for n-rowed Kayles.

Original entry on oeis.org

1, 10, 11, 1, 100, 11, 10, 1, 100, 10, 110, 100, 1, 10, 111, 1, 100, 11, 10, 1, 100, 110, 111, 100, 1, 10, 1000, 101, 100, 111, 10, 1, 1000, 110, 111, 100, 1, 10, 11, 1, 100, 111, 10, 1, 1000, 10, 111, 100, 1, 10, 1000, 1, 100, 111, 10, 1, 100, 10, 111, 100, 1, 10, 1000, 1, 100, 111, 10, 1, 1000, 110

Offset: 1

Susanna Cuyler, Dec 19 2010

nonn

See Gardner for precise definition.

M. Gardner, Mathematical Carnival, Random House, NY, 1975; page 215.

A074742 a(n) = (n^3 + 6n^2 - n + 12)/6.

Original entry on oeis.org

2, 3, 7, 15, 28, 47, 73, 107, 150, 203, 267, 343, 432, 535, 653, 787, 938, 1107, 1295, 1503, 1732, 1983, 2257, 2555, 2878, 3227, 3603, 4007, 4440, 4903, 5397, 5923, 6482, 7075, 7703, 8367, 9068, 9807, 10585, 11403, 12262, 13163, 14107, 15095, 16128, 17207, 18333

Offset: 0

Susanna Cuyler, Sep 06 2002

A. Schultze, Advanced Algebra, Macmillan, London, 1910; p. 552.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A027965.

[(n^3 + 6*n^2 - n + 12)/6: n in [0..50]]; // Vincenzo Librandi, Jan 13 2012

Table[(n^3 + 6n^2 - n + 12)/6, {n, 0, 49}] (* Alonso del Arte, Jan 13 2012 *)
CoefficientList[Series[(2-5x+7x^2-3x^3)/(1-x)^4,{x,0,50}],x] (* or *) LinearRecurrence[ {4,-6,4,-1},{2,3,7,15},50] (* Harvey P. Dale, Aug 05 2022 *)

a(n)=n*(n^2+6*n-1)/6+2 \\ Charles R Greathouse IV, Jan 13 2012

From R. J. Mathar, Sep 23 2008: (Start)
G.f.: (2 - 5*x + 7*x^2 - 3*x^3)/(1-x)^4.
a(n) = A027965(n+1), n > 0. (End)
E.g.f.: exp(x)*(12 + 6*x + 9*x^2 + x^3)/6. - Stefano Spezia, Jul 12 2023

A020767 Product_{k=1..n} b(k), where b(k) = binary expansion of k (A007088) but read as if it were a decimal number.

Original entry on oeis.org

1, 1, 10, 110, 11000, 1111000, 122210000, 13565310000, 13565310000000, 13578875310000000, 13714664063100000000, 13865525367794100000000, 15252077904573510000000000, 16792537772935434510000000000, 18639716927958332306100000000000, 20708725506961707192077100000000000

Offset: 0

Susanna Cuyler, May 23 2003

			a(4) = 1*10*11*100 = 11000.

Partial products of A007088.

A005354 Number of asymmetric planar trees with n nodes.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 3, 9, 28, 85, 262, 827, 2651, 8626, 28507, 95393, 322938, 1104525, 3812367, 13266366, 46504495, 164098390, 582521687, 2079133141, 7457788295, 26872946466, 97238824018, 353218128299, 1287657977946, 4709784136316

Offset: 0

N. J. A. Sloane, Simon Plouffe, Susanna Cuyler

a(13) in the Labelle table is a typographical error. - R. J. Mathar, Feb 03 2010

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)
Gilbert Labelle, Counting asymmetric enriched trees, J. Symbolic Comput. 14 (1992), no. 2-3, 211-242.
Torsten Mütze and Franziska Weber, Construction of 2-factors in the middle layer of the discrete cube, arXiv preprint arXiv:1111.2413 [math.CO], 2011.
T. Mütze and F. Weber, Construction of 2-factors in the middle layer of the discrete cube, Journal of Combinatorial Theory, Series A, 119(8) (2012), 1832-1855.
Index entries for sequences related to trees

Cf. A000108, A002995, A022553.

From R. J. Mathar, Feb 03 2010: (Start)
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A007727 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do a := a+binomial(2*d,d)*numtheory[mobius](n/d) ; end do ; a ; end proc;
A022553 := proc(n) A007727(n)/2/n ; end proc:
A005354 := proc(n) local a; if n <=1 then 1; else a := A022553(n-1) ; a := a-A000108(n-1)/2 ; if type(n,'even') then a := a-A000108(n/2-1)/2 ; end if; a ; end if; end proc: seq(A005354(n),n=0..20) ; (End)

a[0] = a[1] = 1; a[n_] := DivisorSum[n-1, MoebiusMu[(n-1)/#]*Binomial[2#, #]&]/(2(n-1)) - CatalanNumber[n-1]/2 - Boole[EvenQ[n]]*CatalanNumber[n/2 - 1]/2; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, May 09 2012, after R. J. Mathar, updated Jan 31 2018 *)

From Christian G. Bower, Dec 15 1999: (Start)
G.f.: 1+B(x)+(C(x^2)-C(x)^2)/2 where B is g.f. of A022553(n-1) and C is g.f. of A000108(n-1).
a(n) = A022553(n-1) - A000108(n-2)/2 - (if n is even) A000108(n/2-1)/2. (End)

More terms from Christian G. Bower, Dec 15 1999

A005355 Number of asymmetric permutation rooted trees with n nodes.

Original entry on oeis.org

0, 1, 1, 1, 3, 7, 21, 61, 187, 577, 1825, 5831, 18883, 61699, 203429, 675545, 2258291, 7592249, 25656477, 87096661, 296891287, 1015797379, 3487272317, 12008898531, 41471260883, 143588078449, 498343911529, 1733410858955, 6041795275027, 21098924740155

Offset: 0

N. J. A. Sloane, Simon Plouffe, Susanna Cuyler

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Gilbert Labelle, Counting asymmetric enriched trees, J. Symbolic Comput. 14 (1992), no. 2-3, 211-242.
Index entries for sequences related to rooted trees

Cf. A004111, A050383.

a[0] = 1; a[1] = 1; a[n_] := a[n] = Sum[a[k]*a[n-k], {k, 1, n-1}] - If[EvenQ[n-1], a[(n-1)/2], 0]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jan 04 2016 *)

Shifts left under transform T where Ta has g.f. (1-A(x^2))/(1-A(x)).

More terms, formula from Christian G. Bower, Nov 15 1999

cp's OEIS Frontend

User: Susanna Cuyler

A265046 Coordination sequence for a 4.6.6 point in the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6} of the plane by squares and dominoes (hexagons).

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A265045 Coordination sequence for a 6.6.6 point in the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6} of the plane by squares and dominoes (hexagons).

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A181753 Universal sequence of period 56 which contains every 3-subset of {1,2,...,8} exactly once.

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A176998 a(n) = binary "k-number" for n-rowed Kayles.

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A074742 a(n) = (n^3 + 6n^2 - n + 12)/6.

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Magma

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A020767 Product_{k=1..n} b(k), where b(k) = binary expansion of k (A007088) but read as if it were a decimal number.

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A005354 Number of asymmetric planar trees with n nodes.

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Maple

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A005355 Number of asymmetric permutation rooted trees with n nodes.

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