A004146 -id:A004146 - cp's OEIS frontend

A002985 Number of trees in an n-node wheel.

1, 1, 1, 2, 3, 6, 11, 20, 36, 64, 108, 179, 292, 464, 727, 1124, 1714, 2585, 3866, 5724, 8418, 12290, 17830, 25713, 36898, 52664, 74837, 105873, 149178, 209364, 292793, 407990, 566668, 784521, 1082848, 1490197, 2045093, 2798895, 3820629, 5202085

Offset: 1

N. J. A. Sloane

nonn

This is the number of nonequivalent spanning trees of the n-wheel graph up to isomorphism of the trees.

			All trees that span a wheel on 5 nodes are equivalent to one of the following:
      o         o         o
      |         | \     /   \
   o--o--o   o--o  o   o--o  o
      |         |           /
      o         o         o

F. Harary, P. E. O'Neil, R. C. Read and A. J. Schwenk, The number of trees in a wheel, in D. J. A. Welsh and D. R. Woodall, editors, Combinatorics. Institute of Mathematics and Its Applications. Southend-on-Sea, England, 1972, pp. 155-163.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..200
Andrew Howroyd, Derivation of formula.
Eric Weisstein's World of Mathematics, Wheel Graph.
Index entries for sequences related to trees

Cf. A003293, A004146, A008804.

terms = 40;
A003293[n_] := SeriesCoefficient[Product[(1-x^k)^(-Ceiling[k/2]), {k, 1, terms}], {x, 0, n}];
A008804[n_] := SeriesCoefficient[1/((1-x)^4 (1+x)^2 (1+x^2)), {x, 0, n}];
a[n_] := A003293[n-1] - A008804[n-3];
Array[a, terms] (* Jean-François Alcover, Sep 02 2019 *)

\\ here b(n) is A003293 and d(n) is A008804.
b(n)={polcoeff( prod(k=1, n, (1-x^k+x*O(x^n))^-ceil(k/2)), n)}
d(n)={(84+12*(-1)^n+6*I*((-I)^n-I^n)+(85+3*(-1)^n)*n+24*n^2+2*n^3)/96}
a(n)=b(n-1)-d(n-3); \\ Andrew Howroyd, Oct 09 2017

a(n) = A003293(n-1) - A008804(n-3). - Andrew Howroyd, Oct 09 2017

Terms a(32) and beyond from Andrew Howroyd, Oct 09 2017

A122679 Invariant number of internal vertices of n-circum-C_5 H_5 systems.

Original entry on oeis.org

0, 5, 25, 80, 225, 605, 1600, 4205, 11025, 28880, 75625, 198005, 518400, 1357205, 3553225, 9302480, 24354225, 63760205, 166926400, 437019005, 1144130625, 2995372880, 7841988025, 20530591205, 53749785600, 140718765605, 368406511225, 964500768080

Offset: 1

N. J. A. Sloane, Sep 23 2006

Colin Barker, Table of n, a(n) for n = 1..1000
J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Azulenoids, MATCH, No. 34, 1996, 91-108.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

LinearRecurrence[{4,-4,1},{0,5,25},40] (* Harvey P. Dale, Apr 21 2015 *)

concat(0, Vec(-5*x^2*(1+x)/((x-1)*(x^2-3*x+1)) + O(x^40))) \\ Colin Barker, Nov 03 2016

a(n) = 15*Fibonacci(2*k-1)-5*Fibonacci(2*k)-10 = 5*A004146(n-1).
G.f.: -5*x^2*(1+x) / ( (x-1)*(x^2-3*x+1) ). - R. J. Mathar, Nov 23 2014
a(1)=0, a(2)=5, a(3)=25, a(n) = 4*a(n-1)-4*a(n-2)+a(n-3). - Harvey P. Dale, Apr 21 2015
a(n) = -5*2^(-1-n)*(2^(2+n)-(3-sqrt(5))^n*(3+sqrt(5))+(-3+sqrt(5))*(3+sqrt(5))^n). - Colin Barker, Nov 03 2016

More terms from Harvey P. Dale, Apr 21 2015

A364754 Smallest nonnegative integer not expressible by the addition and subtraction of fewer than n Lucas numbers.

Original entry on oeis.org

0, 1, 5, 23, 99, 421, 1785, 7563, 32039, 135721, 574925, 2435423, 10316619, 43701901, 185124225, 784198803, 3321919439, 14071876561, 59609425685, 252509579303, 1069647742899, 4531100550901, 19194049946505, 81307300336923, 344423251294199, 1459000305513721, 6180424473349085

Offset: 0

Mike Speciner, Oct 20 2023

			a(0) = 0, since 0 is expressible as the sum of 0 Lucas numbers.
a(1) = 1, since 1 is a Lucas number.
a(2) = 5, since 2, 3, and 4 are all Lucas numbers; while 5=1+4, the sum of 2 Lucas numbers.
a(3) = 23, since integers less than 23 are expressible with 2 or fewer Lucas numbers, while 23 = 1+4+18 requires 3 terms.

Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Cf. A000032, A004146 (adding positive Lucas numbers), A365907 (adding any Lucas numbers).

Cf. A001076 (with Fibonacci numbers).

a[n_] := (LucasL[3*n - 1] - 1)/2; a[0] = 0; Array[a, 27, 0] (* Amiram Eldar, Oct 21 2023 *)

from sympy import lucas
a = lambda n: n and (lucas(3*n-1)-1)//2

a(0) = 0.
a(n) = (A000032(3*n-1)-1)/2, for n > 0.
a(n) = 1 + Sum_{i=1..n-1} A000032(3*i), for n > 0.
G.f.: x*(1 + x^2)/((1 - x)*(1 - 4*x - x^2)). -  Stefano Spezia, Oct 21 2023

A111091 Successive generations of a Kolakoski(3,1) rule starting with 1 (see A066983).

Original entry on oeis.org

1, 3, 111, 313, 1113111, 313111313, 11131113131113111, 3131113131113111313111313, 1113111313111311131311131311131113131113111

Offset: 1

Benoit Cloitre, Oct 12 2005

nonn

Terms are palindromic. If b_3(n) denotes the number of 3's in a(n) then b(n) satisfies the recursion: b_3(1)=0, b_3(2)=1 and b_3(n) = b_3(n-1) + b_3(n-2) + (-1)^n so that b_3(2n)=A055588(n) and b_3(2n+1)=A027941(n). If b_1(n) denotes the number of 1's: b_1(1)=1, b_1(2)=0 and b_1(n) = b_1(n-1) + b_1(n-2) - 2*(-1)^n so that b_1(2n)=A004146(n) and b_1(2n+1) = A000032(n-2) - 2.

			1 --> 3 --> 111 --> 313 --> 1113111 --> 313111313

Cf. A111081.

As n grows, a(2n-1) converges toward A095345 (read as a word) and a(2n) converges toward A095346.

A206723 a(n) = 7*( ((3 + sqrt(5))/2)^n + ((3 - sqrt(5))/2)^n - 2 ).

Original entry on oeis.org

7, 35, 112, 315, 847, 2240, 5887, 15435, 40432, 105875, 277207, 725760, 1900087, 4974515, 13023472, 34095915, 89264287, 233696960, 611826607, 1601782875, 4193522032, 10978783235, 28742827687, 75249699840, 197006271847, 515769115715, 1350301075312, 3535134110235, 9255101255407, 24230169656000, 63435407712607, 166076053481835, 434792752732912

Offset: 1

N. J. A. Sloane, Feb 11 2012

Hang Gu and Robert M. Ziff, Crossing on hyperbolic lattices, arXiv preprint arXiv:1111.5626, 2011
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Equals 7*A004146 for n >= 1.

LinearRecurrence[{4, -4, 1}, {7, 35, 112}, 33]  (* Jean-François Alcover, Sep 21 2017 *)

G.f.: -7*x*(1+x) / ( (x-1)*(x^2-3*x+1) ). - R. J. Mathar, Nov 15 2013

A328986 The sequence C(n) defined in the comments (A and B smallest missing numbers, offset 1).

Original entry on oeis.org

4, 10, 16, 21, 28, 33, 39, 45, 51, 57, 62, 68, 74, 80, 86, 91, 98, 103, 109, 115, 120, 127, 132, 138, 144, 150, 156, 161, 168, 173, 179, 185, 190, 197, 202, 208, 214, 220, 226, 231, 237, 243, 249, 255, 260, 267, 272, 278, 284, 290, 296, 301, 307, 313, 319

Offset: 1

N. J. A. Sloane, Nov 07 2019

nonn

Define a triple of sequences A,B,C by A[1]=1, B[1]=2, C[1]=4; for n>=2, A[n] = smallest missing number from the terms of A,B,C defined so far; B[n] = = smallest missing number from the terms of A,B,C defined so far; C[n] = n+A[n]+B[n].
Then A = A286660, B = A080652, C = the present sequence.
Inspired by the triples [A003144, A003145, A004146] and [A298468, A298469, A047218].

			The initial terms are:
n: 1, 2, 3, 4,  5,  6,  7,  8.  9. 10. 11, 12, ...
A: 1, 3, 6, 8, 11, 13, 15, 18, 20, 23, 25, 27, ...
B: 2, 5, 7, 9, 12, 14, 17, 19, 22, 24, 26, 29, ...
C: 4, 10, 16, 21, 28, 33, 39, 45, 51, 57, 62, 68, ...

Cf. A286660, A080652; A003144, A003145, A003146; A298468, A298469, A047218.

cp's OEIS Frontend

A002985 Number of trees in an n-node wheel.

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A122679 Invariant number of internal vertices of n-circum-C_5 H_5 systems.

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A364754 Smallest nonnegative integer not expressible by the addition and subtraction of fewer than n Lucas numbers.

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A111091 Successive generations of a Kolakoski(3,1) rule starting with 1 (see A066983).

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A206723 a(n) = 7*( ((3 + sqrt(5))/2)^n + ((3 - sqrt(5))/2)^n - 2 ).

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A328986 The sequence C(n) defined in the comments (A and B smallest missing numbers, offset 1).

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