A000243 -id:A000243 - cp's OEIS frontend

A004050 Numbers of the form 2^j + 3^k, for j and k >= 0.

2, 3, 4, 5, 7, 9, 10, 11, 13, 17, 19, 25, 28, 29, 31, 33, 35, 41, 43, 59, 65, 67, 73, 82, 83, 85, 89, 91, 97, 113, 129, 131, 137, 145, 155, 209, 244, 245, 247, 251, 257, 259, 265, 275, 283, 307, 337, 371, 499, 513, 515, 521, 539, 593, 730, 731, 733, 737, 745, 755

Offset: 1

N. J. A. Sloane

nonn

Donovan Johnson, Table of n, a(n) for n = 1..10000
Douglas Edward Iannucci, On duplicate representations as 2^x+3^y for nonnegative integers x and y, arXiv:1907.03347 [math.NT], 2019.

Cf. A085634, A219835.
Cf. A226806-A226832 (cases to 8^j + 9^k).
Cf. A004051 (primes), A000079, A000243.

import Data.Set (singleton, deleteFindMin, insert)
a004050 n = a004050_list !! (n-1)
a004050_list = f 1 $ singleton (2, 1, 1) where
   f x s = if y /= x then y : f y s'' else f x s''
           where s'' = insert (u * 2 + v, u * 2, v) $
                       insert (u + 3 * v, u, 3 * v) s'
                 ((y, u, v), s') = deleteFindMin s
-- Reinhard Zumkeller, May 20 2015

lincom:=proc(a,b,n) local i,j,s,m; s:={}; for i from 0 to n do for j from 0 to n do m:=a^i+b^j; if m<=n then s:={op(s),m} fi od; od; lprint(sort([op(s)])); end: lincom(2,3,760); # Zerinvary Lajos, Feb 24 2007

mx = 760; s = Union@ Flatten@ Table[2^i + 3^j, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx - 2^i]}] (* Robert G. Wilson v, Sep 19 2012 *)

ispow2(n)=n>>valuation(N,2)==1
is(n)=my(k); if(n%2, if(n<3, return(0)); for(k=0,logint(n-2,3), if(ispow2(n-3^k), return(1))); 0, ispower(n-1,,&k); k==3 || n==2 || n==4) \\ Charles R Greathouse IV, Aug 29 2016

def aupto(lim):
    s, pow3 = set(), 1
    while pow3 < lim:
        for j in range((lim-pow3).bit_length()):
            s.add(2**j + pow3)
        pow3 *= 3
    return sorted(set(s))
print(aupto(756)) # Michael S. Branicky, Jul 29 2021

There are log^2 x/(log 2 log 3) + O(log x) terms up to x. Bounds on the error term can be made explicit. - Charles R Greathouse IV, Oct 28 2022

More terms from Sascha Kurz, Jan 02 2003

A034799 Triangle read by rows: T(n,k) is the number of partially labeled trees with n nodes, k of which are labeled, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 4, 9, 16, 16, 3, 9, 26, 67, 125, 125, 6, 20, 75, 251, 680, 1296, 1296, 11, 48, 214, 888, 3135, 8716, 16807, 16807, 23, 115, 612, 3023, 13155, 47787, 134960, 262144, 262144, 47, 286, 1747, 10038, 51873, 232154, 858578, 2450309, 4782969, 4782969

Offset: 0

N. J. A. Sloane

			Triangle begins:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 3, 3;
  2, 4, 9, 16, 16;
  ...

J. Riordan, An Introduction to Combinatorial Analysis, p. 138.

Index entries for sequences related to trees

Columns are A000055, A000081, A000243, A000269, A000485, A000526, A064785.

Cf. A008295, A034800.

Reference gives generating function.

E.g.f.: r(x,y) - (1/2)*r(x,y)^2 + (1/2)*r(x^2) where r(x,y) is the e.g.f. for A008295 and r(x) is the o.g.f. for A000081. - Sean A. Irvine, Sep 04 2020

More terms from Sean A. Irvine, Sep 04 2020

Name edited by Andrew Howroyd, Mar 23 2023

A303833 Birooted trees: number of unlabeled trees with n nodes rooted at 2 indistinguishable roots.

Original entry on oeis.org

0, 1, 2, 6, 15, 43, 116, 329, 918, 2609, 7391, 21099, 60248, 172683, 495509, 1424937, 4102693, 11830006, 34148859, 98686001, 285459772, 826473782, 2394774727, 6944343654, 20151175658, 58513084011, 170007600051, 494230862633

Offset: 1

R. J. Mathar, Brendan McKay, May 01 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Subsets of graphs in A303831. Cf. A000243 (distinguishable roots), A000055 (not rooted).
Third column of A294783.
Cf. A000081, A000107, A027852, A303840, A304067, A339303.

a000081 := [1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, 87811, 235381, 634847, 1721159, 4688676, 12826228, 35221832, 97055181, 268282855, 743724984, 2067174645, 5759636510, 16083734329, 45007066269, 126186554308, 354426847597] ;
g81 := add( op(i,a000081)*x^i,i=1..nops(a000081) ) ;
g := 0 ;
nmax := nops(a000081) ;
for m from 0 to nmax do
    mhalf := floor(m/2) ;
    ghalf := g81^(mhalf+1) ;
    gcyc := (ghalf^2+subs(x=x^2,ghalf))/2 ;
    if type(m,odd) then
        gcyc := gcyc*g81 ;
    end if;
    g := g+gcyc ;
end do:
taylor(g,x=0,nmax) ;
gfun[seriestolist](%) ; # R. J. Mathar, May 01 2018

TreeGf is A000081 as g.f.
TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n), t2=subst(t,x,x^2)+O(x*x^n)); Vec((2*t^2-1)/(1-t) + (1+t)/(1-t2), -n)/2} \\ Andrew Howroyd, Dec 04 2020

G.f.: [g81(x)^2/{1-g81(x)} +(1+g81(x))*g81(x^2)/{1-g81(x^2)}] /2 = [ g243(x) +(1+g81(x))*g107(x^2)]/2, where g81 is the g.f. of A000081, g243 the g.f. of A000243 and g107 the g.f. of A000107. - R. J. Mathar, May 02 2018
a(n) = A027852(n) + A304067(n). - Brendan McKay, May 05 2018
a(n) = A303840(n+2) - A000081(n). - Andrew Howroyd, Dec 04 2020

A255884 Expansion of exp( Sum_{n >= 1} A002438(n)*x^n/n ).

Original entry on oeis.org

1, 5, 115, 7955, 1179715, 304888655, 121350927565, 68751844662605, 52528700295424915, 52031089992310711055, 64835758857480094584265, 99249388572274155967996505, 183075972804988649078529524365, 400493686169423616676960341062705, 1025151296160300228944197705742007715

Offset: 0

Peter Bala, Mar 09 2015

A002438(n+1) =(-1)^n*6^(2*n)*E(2*n,1/6), where E(n,x) denotes the n-th Euler polynomial. In general it appears that when k is a nonzero integer, the expansion of exp( Sum_{n >= 1} k^(2*n)*E(2*n,1/k)*(-x)^n/n ) has (positive) integer coefficients. See A255881 (k = 2), A255882(k = 3) and A255883 (k = 4).

G. C. Greubel, Table of n, a(n) for n = 0..200
E. W. Weisstein, Euler Polynomial

Cf. A002438, A188514, A255881, A255882, A255883.

#A255884
k := 6:
exp(add(k^(2*n)*euler(2*n, 1/k)*(-x)^n/n, n = 1 .. 14)): seq(coeftayl(%, x = 0, n), n = 0 .. 14);

A000243[n_]:= (1 + 9^(n - 1))*Abs[EulerE[2*(n - 1)]]/2; a:= With[{nmax = 75}, CoefficientList[Series[Exp[Sum[A000243[k + 1]*x^(k)/(k), {k, 1, 85}]], {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Aug 26 2018 *)

O.g.f.: exp( 5*x + 205*x^2/2 + 22265*x^3/3 + 4544185 *x^4/4 + ... ) = 1 + 5*x + 115*x^2 + 7955*x^3 + 1179715*x^4 + ....
a(0) = 1 and for n >= 1, n*a(n) = Sum_{k = 1..n} (-1)^k*6^(2*k)*E(2*k,1/6)*a(n-k).
a(n) ~ 2^(4*n + 2) * 3^(2*n) * n^(2*n - 1/2) / (exp(2*n) * Pi^(2*n + 1/2)). - Vaclav Kotesovec, Jun 08 2019

A304489 Triangle read by rows: T(n,k) = number of rooted signed trees with n nodes and k positive edges (0 <= k < n).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 9, 9, 4, 9, 26, 37, 26, 9, 20, 75, 134, 134, 75, 20, 48, 214, 469, 596, 469, 214, 48, 115, 612, 1577, 2445, 2445, 1577, 612, 115, 286, 1747, 5204, 9480, 11513, 9480, 5204, 1747, 286, 719, 4995, 16865, 35357, 50363, 50363, 35357, 16865, 4995, 719

Offset: 1

Andrew Howroyd, May 13 2018

Equivalently, the number of rooted trees with 2-colored non-root nodes, n nodes and k nodes of the first color.

			Triangle begins:
    1;
    1,    1;
    2,    3,    2;
    4,    9,    9,    4;
    9,   26,   37,   26,     9;
   20,   75,  134,  134,    75,   20;
   48,  214,  469,  596,   469,  214,   48;
  115,  612, 1577, 2445,  2445, 1577,  612,  115;
  286, 1747, 5204, 9480, 11513, 9480, 5204, 1747, 286;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A000151.
Columns k=0..1 are A000081, A000243.
Cf. A294783, A302939, A331114.

R(n, y)={my(v=vector(n)); v[1]=1; for(k=1, n-1, my(p=(1+y)*v[k]); my(q=Vec(prod(j=0, poldegree(p, y), (1/(1-x*y^j) + O(x*x^(n\k)))^polcoeff(p, j)))); v=vector(n, j, v[j] + sum(i=1, (j-1)\k, v[j-i*k] * q[i+1]))); v; }
{ my(A=R(10,y)); for(n=1, #A, print(Vecrev(A[n]))) }

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
R(n, y)={my(v=[1]); for(k=2,n,v=concat([1], EulerMT(v*(1+y)))); v}
{ my(A=R(10,y)); for(n=1, #A, print(Vecrev(A[n]))) }

A000269 Number of trees with n nodes, 3 of which are labeled.

Original entry on oeis.org

3, 16, 67, 251, 888, 3023, 10038, 32722, 105228, 334836, 1056611, 3311784, 10322791, 32026810, 98974177, 304835956, 936147219, 2867586542, 8764280567, 26733395986, 81399821915, 247459136331, 751211286356, 2277496842016

Offset: 3

N. J. A. Sloane

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).

T. D. Noe, Table of n, a(n) for n=3..200
Index entries for sequences related to trees

Column k=3 of A034799.

b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n+1 - j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[ b[k]*x^k, {k, 1, n}]; a[n_] := SeriesCoefficient[ B[n-1]^3 * (2*B[n-1]-3) / (B[n-1]-1)^3, {x, 0, n}]; Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Jan 27 2015 *)

G.f.: A(x) = B(x)^3*(3-2*B(x))/(1-B(x))^3, where B(x) is g.f. for rooted trees with n nodes, cf. A000081. - Vladeta Jovovic, Oct 19 2001

a(n) = A000524(n) - 2*A000243(n).

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

A304068 Number of trees on n vertices rooted at an oriented non-edge.

Original entry on oeis.org

0, 0, 1, 4, 14, 45, 140, 424, 1269, 3760, 11080, 32517, 95190, 278154, 811887, 2367973, 6903453, 20120905, 58639016, 170894228, 498084608, 1451899005, 4232957241, 12343454790, 36001675800, 105028397290, 306472665459, 894497511566

Offset: 1

Brendan McKay, May 05 2018

nonn

			a(4)=4 is based on the same examples as in A304067, but the oriented edge that spans a leaf with the node at distance 2 in the linear graph may have 2 orientations, so a(4) is one larger than A304067(4).

Cf. A000055 (not rooted), A000106 (rooted at oriented edge)

A000106(n) + a(n) = A000243(n).

A280784 Triangle read by rows: numbers of nonintersecting circles with one of them marked.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 13, 9, 3, 1, 35, 26, 10, 3, 1, 95, 75, 30, 10, 3, 1, 262, 214, 91, 31, 10, 3, 1, 727, 612, 268, 95, 31, 10, 3, 1, 2033, 1747, 790, 284, 96, 31, 10, 3, 1, 5714, 4995, 2308, 848, 288, 96, 31, 10, 3, 1, 16136, 14294, 6737, 2506, 864, 289, 96, 31, 10, 3, 1

Offset: 1

N. J. A. Sloane, Jan 20 2017

			Triangle begins:
1,
2,1,
5,3,1,
13,9,3,1,
35,26,10,3,1,
95,75,30,10,3,1,
262,214,91,31,10,3,1,
727,612,268,95,31,10,3,1,
2033,1747,790,284,96,31,10,3,1,
...

R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.

Rows sums are A000243.

cp's OEIS Frontend

A004050 Numbers of the form 2^j + 3^k, for j and k >= 0.

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A034799 Triangle read by rows: T(n,k) is the number of partially labeled trees with n nodes, k of which are labeled, 0 <= k <= n.

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A303833 Birooted trees: number of unlabeled trees with n nodes rooted at 2 indistinguishable roots.

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A255884 Expansion of exp( Sum_{n >= 1} A002438(n)*x^n/n ).

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A304489 Triangle read by rows: T(n,k) = number of rooted signed trees with n nodes and k positive edges (0 <= k < n).

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A000269 Number of trees with n nodes, 3 of which are labeled.

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A304068 Number of trees on n vertices rooted at an oriented non-edge.

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A280784 Triangle read by rows: numbers of nonintersecting circles with one of them marked.

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