A171386 -id:A171386 - cp's OEIS frontend

A103221 Number of partitions of n into parts 2 and 3.

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 13, 14, 14, 14, 14, 15, 14, 15, 15

Offset: 0

Michael Somos, Jan 25 2005

Essentially the same as A008615.
Poincaré series [or Poincare series] for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).
Dimension of the space of weight 2n+12 cusp forms for Gamma_0( 1 ).
Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
a(n) is the number of partitions of n into two nonnegative parts congruent modulo 3. - Andrew Baxter, Jun 28 2006
Also number of equivalence classes of period 2n billiards on an equilateral triangle. - Andrew Baxter, Jun 06 2008
a(n) is also the number of 2-regular multigraphs on n vertices, where each component is either a pair of parallel edges, or a triangle. - Jason Kimberley, Oct 14 2011
For n>1, a(n) is the number of partitions of 2n into positive parts x,y, and z such that x>=y and y=z. This sequence is used in calculating the probability of the need for a run-off election when n voters randomly cast ballots for two of three candidates running for two empty slots on a county commission. - Dennis P. Walsh, Apr 25 2013
Also, Molien series for invariants of finite Coxeter group A_2. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k, not the alternating group Alt_k. - N. J. A. Sloane, Jan 11 2016
The coefficient of x^(2*n+1) in the power series expansion of the Weierstrass sigma function is a polynomial in the invariants g2 and g3 with a(n) terms. - Michael Somos, Jun 14 2016
a(n) is also the dimension of the complex vector space of modular forms M_{2*n} of weight 2*n and level 1 (full modular group). See Apostol p. 119, eq. (9) for k=2*n, and Ash and Gross, p. 178, Table 13.1. For a(6*k+1) = a(6*k+j)-1 for j = 0,2,3,4,5 and k >= 0 see A016921 (so-called dips, cf. Ash and Gross, p. 178.). - Wolfdieter Lang, Sep 16 2016
In an hexagonal tiling of the plane where the base tile is (0,0)--(2,1)--(3,3)--(1,4)--(-1,3)--(-2,1)--(0,0), a(n) is the number of vertices on the (n,0)--(n,n) closed line segment. - Luc Rousseau, Mar 22 2018

			For n=8, a(n)=2 since there are two partitions of 16 into 3 positive parts x, y, and z such that x >= y and y=z, namely, 16 = 8+4+4 and 16 = 6+5+5. - _Dennis P. Walsh_, Apr 25 2013
G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + 2*x^11 + ...

T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 119.
Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.
J.-M. Kantor, Ou en sont les mathématiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 26. - N. J. A. Sloane, Aug 28 2010.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, (2024). See p. 5.
Andrew M. Baxter and Ron Umble, Periodic Orbits of Billiards on an Equilateral Triangle, Amer. Math. Monthly, 115 (No. 6, 2008), 479-491.
Ansgar Freyer, Monika Ludwig, and Martin Rubey, Unimodular Valuations beyond Ehrhart, arXiv:2407.07691 [math.MG], 2024. See p. 8.
Jun-Ichi Igusa, On Siegel modular forms of genus 2 (II), Amer. J. Math., 86 (1964), 392-412, esp. p. 402.
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 15.
Luc Rousseau, a(n) in an hexagonal tiling
Tetsuji Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89, 1022-1046, 1967.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for Molien series
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1).

Cf. A008615, A001399 (partial sums), A128115, A171386, A081753.
Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776, A266777, A266778, A266779, A266780, A266781.
Cf. A004009, A013973, A016921, A079978.

[Floor((n+2)/2)-Floor((n+2)/3): n in [0..100]]; // Vincenzo Librandi, Sep 18 2016

A103221:=n->floor((n+2)/2)-floor((n+2)/3): # Andrew Baxter, Jun 06 2008

a=b=c=d=0;Table[e=a+b-d+1;a=b;b=c;c=d;d=e,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
LinearRecurrence[{0, 1, 1, 0, -1},{1, 0, 1, 1, 1},88] (* Ray Chandler, Sep 23 2015 *)
a[ n_] := With[{m = Max[-5 - n, n]}, (-1)^Boole[n < 0] SeriesCoefficient[ 1 / ((1 - x^2) (1 - x^3)), {x, 0, m}]]; (* Michael Somos, Jun 02 2019 *)

{a(n) = if( n<-4, -a(-5-n), polcoeff( 1 / ((1 - x^2) * (1 - x^3)) + x * O(x^n), n))};

a(n)=n+=2; n\2 - n\3 \\ Charles R Greathouse IV, Jul 31 2017

def A103221(n): return (n>>1)+1-(n+2)//3 # Chai Wah Wu, Apr 15 2025

def a(n) : return( len( CuspForms( Gamma0( 1), 2*n + 12, prec=1). basis())); # Michael Somos, May 29 2013

Euler transform of finite sequence [0, 1, 1] with offset 1, which is A171386.
a(n) = A008615(n+2). First differences of A001399.
a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - Henry Bottomley, Sep 02 2000
G.f.: 1/((1-x^2)*(1-x^3)).
a(n) = floor((n+2)/2) - floor((n+2)/3). - Andrew Baxter, Jun 06 2008
For odd n, a(n)=floor((n+3)/6). For even n, a(n)=floor((n+6)/6). - Dennis P. Walsh, Apr 25 2013
a(n) = floor(n/6)+1 unless n == 1 (mod 6); if n == 1 (mod 6), a(n) = floor(n/6). - Bob Selcoe, Sep 27 2014
a(n) = A081753(2*n); see the Dennis P. Walsh formula. - Wolfdieter Lang, Sep 16 2016
a(n)-a(n-2) = A079978(n). - R. J. Mathar, Jun 23 2021
E.g.f.: (3*(4 + x)*cosh(x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2)) + 3*(1 + x)*sinh(x))/18. - Stefano Spezia, Mar 05 2023
a(n) = A008615(n-1)+A059841(n). - R. J. Mathar, May 03 2023

Name changed by Wolfdieter Lang, Sep 16 2016

A018226 Magic numbers of nucleons: nuclei with one of these numbers of either protons or neutrons are more stable against nuclear decay.

Original entry on oeis.org

2, 8, 20, 28, 50, 82, 126

Offset: 1

John Raithel (raithel(AT)rahul.net)

In the shell model for the nucleus, magic numbers are the numbers of either protons or neutrons at which a shell is filled.
First seven positive terms of A162626. - Omar E. Pol, Jul 07 2009
Steppenbeck: "The results of the experiment indicate that 54Ca's first excited state lies at a relatively high energy, which is characteristic of a large nuclear shell gap, thus indicating that N = 34 in 54Ca is a new magic number, as predicted theoretically by the University of Tokyo group in 2001. By conducting a more detailed comparison to nuclear theory the researchers were able to show that the N = 34 magic number is equally as significant as some other nuclear shell gaps."

Dictionary of Science (Simon and Schuster), see the entry for "Magic number".

S. Bjornholm, Clusters, condensed matter in embryonic form, Contemp. Phys. 31 1990 pp. 309-324 (p. 312).
Encyclopedia Britannica, magic number
J. Fridmann et al., 'Magic' nucleus 42-Si, Nature, 435 (2005), 922-924 and 897-898.
J. Glanz, Uut and Uup Add Their Atomic Mass to Periodic Table, New York Times, Feb 01, 2003, pages 1 and 26.
R. V. F. Janssens, Unexpected doubly magic nucleus, Nature, 459 (Jun 25 2009), 1069-1070. [_Added by N. J. A. Sloane, Jul 05 2009]
Radoslav Jovanovic, Magic Numbers and the Pascal Triangle
Lutvo Kuric, Digital nuclear shell model, International Letters of Chemistry, Physics and Astronomy, 13(2) (2014) 160-173; ISSN 2299-3843.
V. Ladma, Magic Numbers
NAPC Isotope Hydrology Section, Chapter 2, Atomic Systematics and Nuclear Structure [Broken link?]
R. Nave, Shell Model of Nucleus
R. Nave, Enhanced Abundance of Magic Number Nuclei
Rachele Nerattini, Johann S. Brauchart, and Michael K.-H. Kiessling, Magic numbers in Smale's 7th problem, arXiv:1307.2834v1 [math-ph], July 10, 2013.
Phys.org, Evidence for a new nuclear 'magic number', Oct 9, 2013.
D. Steppenbeck et al., Evidence for a new nuclear 'magic number' from the level structure of 54Ca, Nature, 2013 DOI: 10.1038/nature12522.
D. Warner, Not-so-magic numbers, Nature, 430 (Jul 29 2004), 517-519.
D. Weise, The Pythagorean Approach to Problems of Periodicity in Chemistry and Nuclear Physics
Wikipedia, Magic number (physics)

Cf. A018227 Number of electrons (which equals number of protons) such that they are arranged into complete shells within the atom.

Cf. A033547, A110856, A130598, A162626, A046939, A046940.

If 1 <= n <= 3 then a(n)=n*(n+1)*(n+2)/3, else if 4 <= n <= 7 then a(n)=n(n^2+5)/3. - Omar E. Pol, Jul 07 2009 [This needs to be clarified. - Joerg Arndt, May 03 2011]
From Daniel Forgues, May 03 2011: (Start)
If 1 <= n <= 3 then a(n) = 2 T_n, else
if 4 <= n <= 7 then a(n) = 2 (T_n - t_{n-1}),
where T_n is the n-th tetrahedral number, t_n the n-th triangular number.
G.f.: (2*x*(1 - 6*x^3 + 14*x^4 - 11*x^5 + 3*x^6))/(1 - x)^4, 1 <= n <= 7.
Using those formulas for n >= 0 gives A162626. (End)
a(n) = n*(n^2+5)/3 + (4*n-6)*A171386(n). - Omar E. Pol, Aug 14 2013

A181354 Number of n-digit perfect cubes.

Original entry on oeis.org

2, 2, 5, 12, 25, 53, 116, 249, 535, 1155, 2487, 5358, 11545, 24871, 53584, 115444, 248715, 535841, 1154435, 2487154, 5358411, 11544347, 24871542, 53584111, 115443470, 248715414, 535841116, 1154434691, 2487154143, 5358411166

Offset: 1

Martin Renner, Jan 28 2011

a(n) is also the total number of n-digit numbers requiring 1 positive cube in their representation as sum of cubes.
a(n) + A181376(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + A171386(n) = A052268(n).
Differs from A062941 only at n=1, because 0 is considered a 0-digit, not a 1-digit number here. - R. J. Mathar, Jul 09 2011

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A000578

a:=n->ceil(10^(n/3))-ceil(10^((n-1)/3));

With[{c = Range[4650000]^3}, Length[#]&/@Table[Select[c, IntegerLength[#] == n &], {n, 20}]] (* Harvey P. Dale, Feb 01 2011 *)
Differences[Ceiling[10^(Range[0, 30]/3)]]

a(n) = A061439(n) - A061439(n-1).

More terms from T. D. Noe, Feb 01 2011

A185012 Characteristic function of two.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Jason Kimberley, Oct 11 2011

The Euler transformation of this sequence is A059841.

This sequence is the first differences of A157928.

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

Cf. A171386.

Characteristic function of g: A000007 (g=0), A063524 (g=1), this sequence (g=2), A185013 (g=3), A185014 (g=4), A185015 (g=5), A185016 (g=6), A185017 (g=7).

def A185012(n): return int(n==2) # Chai Wah Wu, Feb 04 2022

A181376 Total number of n-digit numbers requiring 2 positive cubes in their representation as a sum of cubes.

Original entry on oeis.org

2, 7, 32, 161, 736, 3416, 15976, 74295, 345334, 1605089, 7455698, 34623338, 160759047, 746318897, 3464508951, 16081935250, 74648713406

Offset: 1

Martin Renner, Jan 28 2011

A181354(n) + a(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + A171386(n) = A052268(n).

			a(1) = 2 from 1+1=2, 1+8=9.
a(2) = 7 from 8+8=16, 1+27=28, 35, 54, 65, 72, 91.

Eric W. Weisstein, MathWorld -- Waring's Problem.

Cf. A003325.

Table[Length[c = Table[j^3, {j, (10^n - 1)^(1/3)}];
  Select[Union[Flatten[Outer[Plus, c, c]]],
IntervalMemberQ[Interval[{10^(n - 1), 10^n - 1}], #] &]], {n, 10}] (* Robert Price, Apr 18 2019 *)

a(n)=my(N=10^n, Nn=N/10, v=List(), x3, t); sum(x=sqrtnint(Nn\2,3), sqrtnint(N-1, 3), x3=x^3; sum(y=1, min(sqrtnint(N-x3, 3), x), t=x3+y^3; t>=Nn && !ispower(t, 3) && listput(v, t))); #vecsort(v, , 8) \\ Charles R Greathouse IV, Oct 16 2013

a(n) = A181375(n)-A181375(n-1).

a(6)-a(11) from Charles R Greathouse IV, Oct 16 2013
a(12) from Lars Blomberg, Jan 15 2014
a(13)-a(17) from Hiroaki Yamanouchi, Jul 13 2014

cp's OEIS Frontend

A103221 Number of partitions of n into parts 2 and 3.

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A018226 Magic numbers of nucleons: nuclei with one of these numbers of either protons or neutrons are more stable against nuclear decay.

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A181354 Number of n-digit perfect cubes.

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A185012 Characteristic function of two.

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A181376 Total number of n-digit numbers requiring 2 positive cubes in their representation as a sum of cubes.

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A181378 Total number of n-digit numbers requiring 3 positive cubes in their representation as sum of cubes.

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A181380 Total number of n-digit numbers requiring 4 positive cubes in their representation as sum of cubes.

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