A090666 For each possible representation of n as n = 2*nb + 3*nu + K with nb, nu, K nonnegative integers, add numbers K, K+1, ..., 2*K except for 2*K-1 into a multiset. a(n) is the size of the resulting multiset minus the number of distinct numbers in it.

0, 0, 0, 0, 2, 3, 7, 11, 18, 26, 36, 48, 63, 79, 99, 121, 146, 174, 206, 240, 279, 321, 367, 417, 472, 530, 594, 662, 735, 813, 897, 985, 1080, 1180, 1286, 1398, 1517, 1641, 1773, 1911, 2056, 2208, 2368, 2534, 2709, 2891, 3081, 3279, 3486, 3700, 3924, 4156

Offset

0,5

Comments

Number of repetitions (defined as the number of appearances minus one) of L quantum number for a given value of N=2*nb+tau=0,1,2,... principal quantum number for the 5 dimensional harmonic oscillator (connected to the solution of Bohr equation in 5 dimensional). For each tau, nu=0,1,..,[tau/3] and K=tau-3*nu. Finally L=K,K+1,K+2,...,2*K-2,2*K (or alternatively from K to 2*K with the exception of 2*K-1).

Nuclear Physics, collective model: classification of states based on quantum numbers and group theory.

Links

Bill McEachen, Table of n, a(n) for n = 0..671

Lorenzo Fortunato, Solutions of the Bohr Hamiltonian, a compendium, The European Physical Journal A - Hadrons and Nuclei, Vol. 26, Suppl. 1 (2005), pp. 1-30; arXiv preprint, arXiv:nucl-th/0411087, 2004.

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Formula

N = 2*nb+tau = 0, 1, 2, ..... nb=0, 1, 2, .... tau=0, 1, 2, .... nu=0, 1, .., [tau/3] K=tau-3*nu L=K, K+1, K+2, ..., 2*K-2, 2*K

G.f.: x^4*(x^2-x+1)*(x^5-x^4-2x^3+x+2)/((x-1)^4*(x+1)*(x^2+x+1)). - Jean-François Alcover, Feb 18 2019

Example

a(N=0)=0 because N=0 implies nb=0 and tau=0. Hence nu=0, K=0 and L=0. There are no repetitions.
a(N=4)=2 because N=4 implies (nb,tau)= (0,4),(1,2),(2,0). At the end L=0,2,4,2,4,5,6,8, so that only 2 repetitions are found.

Prog

(Fortran) implicit integer(a-z) dimension mrep(0:100) do N=0, 30 do L=0, 100 mrep(L)=0 enddo do nb=0, nint(real(N)/2.-0.01) tau=N-2*nb numax=int(tau/3) do nu=0, numax K=tau-3*nu do L=K, 2*K if(L.eq.(2*K-1)) goto 100 mrep(L)=mrep(L)+1 100 enddo enddo enddo sum=0 do L=0, 100 if(mrep(L).gt.0) then sum=sum+mrep(L)-1 endif enddo print *, N, sum enddo end
(PARI) a(n)={my(v=vector(2*n+1)); for(nb=0, n\2, for(nu=0, (n-2*nb)\3, my(K=n-2*nb-3*nu); for(L=K, 2*K-2, v[1+L]++); v[1+2*K]++)); sum(i=1, #v, if(v[i], v[i]-1))} \\ Andrew Howroyd, Sep 16 2025

Crossrefs

Cf. A212254.

Sequence in context: A060341 A114345 A077165 * A140409 A201011 A108541

Adjacent sequences: A090663 A090664 A090665 * A090667 A090668 A090669

Keyword

nonn,easy

Author

Lorenzo Fortunato (fortunat(AT)pd.infn.it), Dec 16 2003

Extensions

Extended by Bill McEachen, Sep 16 2025

Edited by Andrei Zabolotskii, Sep 21 2025

Status

approved