A130517 -id:A130517 - cp's OEIS frontend

A213371 Triangle read by rows in which row n lists the number of pairs of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.

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

Offset: 1

Omar E. Pol, Jul 16 2012

First differs from A212121 at a(14). The illustration of initial terms is essentially the same as the illustration of initial terms of A213372, the main entry for this sequence.

			Written as an irregular triangle in which row n represents the n-th shell of nucleus. Note that row 4 has only one term. Triangle begins:
1;
2, 1;
3, 1, 2;
4;
2, 3, 1, 5;
4, 3, 6, 2, 1;
4, 5, 7, 2, 3, 1;

MIT OpenCourseWare, Energy Levels of Nucleons in a Smoothly-Varying Potential Well

Partial sums give A213373. Other version are A004736, A130517, A212121, A213361.

Cf. A213372, A213374.

a(n) = A213372(n)/2.

A072785 Differences between A072781 and A072738.

Original entry on oeis.org

0, 0, -1, 0, 0, 1, 0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Antti Karttunen, Jun 12 2002

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, 2012, arXiv:1212.2732 [math.CO], 2012.

Cf. A130517, A162630.

(define (A072785 n) (- (A072781 n) (A072738 n)))

From Boris Putievskiy, Jan 16 2013: (Start)
a(n) =  floor((2*A000027(n)-A003056(n)^2-A003056(n))/(A003056(n)+3))*(-1)^A003056(n).
a(n) =  floor((2*n-t*t-t)/(t+3))*(-1)^t where t=floor((-1+sqrt(8*n-7))/2).
(End)

A162518 Characteristic function of magic numbers A018226: 1 if n is a magic number else 0.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jul 07 2009

nonn

Sequence related to atomic nuclei.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Cf. A018226, A130517, A130556, A130598, A130602, A162519, A162521, A162522, A162523, A162524, A162525.

A162518(n) = if(n<1, 0, polcoeff( x^126+x^82+x^50+x^28+x^20+x^8+x^2, n)); \\ Antti Karttunen, Dec 24 2018, after code in A089011

Data section extended up to a(126) by Antti Karttunen, Dec 24 2018

A377137 Array read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n contains 3n/2 elements if n is even, and (n+1)/2 elements if n is odd; ; see Comments.

Original entry on oeis.org

1, 4, 2, 3, 6, 5, 12, 10, 8, 7, 9, 11, 15, 13, 14, 24, 22, 20, 18, 16, 17, 19, 21, 23, 28, 26, 25, 27, 40, 38, 36, 34, 32, 30, 29, 31, 33, 35, 37, 39, 45, 43, 41, 42, 44, 60, 58, 56, 54, 52, 50, 48, 46, 47, 49, 51, 53, 55, 57, 59, 66, 64, 62, 61, 63, 65, 84, 82, 80, 78, 76, 74, 72, 70, 68, 67, 69, 71, 73, 75, 77, 79, 81, 83, 91, 89, 87, 85, 86

Offset: 1

Boris Putievskiy, Oct 17 2024

Row n has length A064455(n). The sequence A064455 is non-monotonic.
The array consists of two triangular arrays alternating row by row.
For odd n, row n consists of permutations of the integers from A001844((n-1)/2) to A265225(n-1). For even n, row n consists of permutations of the integers from A130883(n/2) to A265225(n-1).
These permutations are generated by the algorithm described A130517.
The sequence is an intra-block permutation of the positive integers.

			Array begins:
     k =  1   2   3   4   5   6
  n=1:    1;
  n=2:    4,  2,  3;
  n=3:    6,  5;
  n=4:   12, 10,  8,  7,  9, 11;
The triangular arrays alternate by row: n=1 and n=3 comprise one, and n=2 and n=4 comprise the other.
Subtracting (n^2 - 1)/2 if n is odd from each term in row n produces a permutation of 1 .. (n+1)/2. Subtracting (n^2 - n)/2 if n is even from each term in row n produces a permutation of 1 .. 3n/2:
  1,
  3, 1, 2,
  2, 1,
  6, 4, 2, 1, 3, 5,
  ...

Boris Putievskiy, Table of n, a(n) for n = 1..9940
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A064455, A001844, A130517, A130883, A162630, A265225, A375602, A376180.

a[n_]:=Module[{L,R, P,Result},L=Ceiling[Max[x/.NSolve[x*(2*(x-1)-Cos[Pi*(x-1)]+5)-4*n==0,x,Reals]]]; R=n-If[EvenQ[L],(L^2-L)/2,(L^2-1)/2]; P[(L+1)*(2*L-(-1)^L+5)/4]=If[EvenQ[L],3L/2,(L+1)/2]; P[3]=2; P= Abs[2*R-If[EvenQ[L],3L/2,(L+1)/2]-If[2*R<=If[EvenQ[L],3L/2,(L+1)/2]+1,2,1]]; Res=P+If[EvenQ[L],(L^2-L)/2,(L^2-1)/2]; Result=Res; Result] Nmax= 12; Table[a[n],{n,1,Nmax}]

Linear sequence:
a(n) = P(n) + B(L(n)-1), where L(n) = ceiling(x(n)), x(n) is largest real root of the equation B(x) - n = 0. B(n) = (n+1)*(2*n-(-1)^n+5)/4 = A265225(n). P(n) = A162630(n)/2.
Array T(n,k) (see Example):
T(n, k) = P(n, k) + (n^2 - n)/2 if n is even, T(n, k) = P(n, k) + (n^2 - 1)/2 if n is odd, T(n, k) = P(n, k) +  A265225(n-1). P(n, k) = |2k - 3n / 2 - 2| if n is even and if 2k <= 3n / 2 + 1, P(n, k) = |2k - 3n / 2 - 1| if n is even and if 2k > 3n / 2 + 1. P(n, k) = |2k - (n + 1) / 2 - 2| if n is odd and if 2k <=  (n + 1) / 2 + 1, P(n, k) = |2k - (n + 1) / 2 - 1| if n is odd and if 2k > (n + 1) / 2 + 1. There are several special cases: P(n, 1) = 3n/2 if n is even, P(n, 1) = (n+1)/2 if n is odd. P(2, 2) = 1. P(n, n) = n/2 - 1 if n is even, P(n, n) = (n-3)/2 if n is odd.

A162519 Characteristic function of magic numbers A018226: 0 if n is a magic number else 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jul 07 2009

Also, successive digits of A130598.

This sequence is related to atomic nuclei. See also A162518.

Index entries for characteristic functions

Cf. A018226, A130517, A130556, A130598, A130602, A162518, A162521, A162522, A162523, A162524, A162525.

A210842 Number of states in the n-th shell of the nuclear shell model.

Original entry on oeis.org

2, 6, 12, 8, 22, 32, 44, 58

Offset: 1

Omar E. Pol, Jun 04 2012

Partial sums of the first seven terms give the "magic numbers" A018226.

Also the partial sums of the first eight terms give the positive first eight terms of A162626 (and possibly more).

M. Goeppert Mayer and J. Hans D. Jensen, Elementary Theory of Nuclear Shell Structure, J. Wiley and Sons, Inc. (1955).
I. Talmi, Simple Models of Complex Nuclei, Hardwood Academic Publishers (1993).

Row sums of triangle A212122.

Cf. A018226, A130517, A162522, A162626, A162630, A212121, A212123, A212124, A213361-A213364.

A141110 Number of cycles and fixed points in the permutation (n, n-2, n-4, ..., 1, ..., n-3, n-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 4, 3, 1, 3, 2, 3, 5, 1, 2, 5, 1, 3, 4, 1, 1, 7, 6, 1, 3, 1, 4, 5, 3, 1, 4, 1, 7, 3, 4, 5, 7, 3, 2, 7, 1, 1, 8, 1, 3, 3, 4, 3, 7, 5, 2, 5, 3, 9, 10, 1, 5, 7, 2, 1, 3, 3, 6, 5, 1, 5, 8, 7, 3, 3, 4, 1, 9, 1, 2, 11

Offset: 1

Ramasamy Chandramouli, Jun 05 2008

The above permutation (see A130517) can be generated by taking S_n: (1, 2, ..., n) and reversing the first two, first three and so on till first n, elements in sequence. Interestingly this permutation orbit has length given by A003558.

			a(20) = 2, since (20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19) has two cycles (1, 20, 19, 17, 13, 5, 12, 3, 16, 11) and (2, 18, 15, 9, 4, 14, 7, 8, 6, 10).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A130517 (permutations), A003558 (order).

from sympy.combinatorics import Permutation
def a(n):
    p = list(range(n, 0, -2)) + list(range(1+(n%2), n, 2))
    return Permutation([pi-1 for pi in p]).cycles
print([a(n) for n in range(1, 86)]) # Michael S. Branicky, Dec 27 2021

A377280 Given n cards, each time you reversing the order of the top 1, 2, 3, ..., n-1, n cards, then repeat reversing 1, 2, 3, ... cards. Do reversing at least once. the minimum number of steps required to return all the cards to their original position.

Original entry on oeis.org

1, 4, 9, 12, 25, 36, 28, 32, 81, 60, 121, 120, 117, 196, 75, 80, 204, 324, 228, 200, 147, 264, 529, 504, 200, 676, 540, 252, 841, 900, 186, 192, 1089, 748, 1225, 324, 740, 1140, 1521, 1080, 1681, 336, 1204, 484, 540, 460, 1692, 1152, 735, 2500, 2601, 624, 2809, 972, 1980, 784, 2508, 696, 1416, 3300

Offset: 1

Youhua Li, Oct 22 2024

nonn

The sequence is only restored after one of the full length n reversal, so a(n) >= n.

The process of reversing blocks from 1 to n corresponds to the order transformation of numbers in sequence A130517.

			For example, using "abc" to represent three cards, the card positions at the end of each step are: abc, bac, cab, cab, acb, bca, bca, cba, abc. Therefore, it takes 9 steps. If there are 4 cards "abcd", the sequence of changes is: abcd, bacd, cabd, dbac, dbac, bdac, adbc, cbda, cbda, bcda, dcba, abcd, so it takes 12 steps

Cf. A003558. A130517.

A377280(n)=my(M=Mod(2,2*n+1),o=znorder(M));if(o%2==0&&M^(o/2)==-1,n*o/2,o*n) \\ Kevin Ryde

from sympy.ntheory import n_order
def A377280(n):
    modular = 2*n + 1
    order = n_order(2, 2*n+1)
    if order % 2 == 0 and pow(2, order//2, modular) == modular - 1:
        return (order//2) * n
    else:
        return order * n # after Kevin Ryde

a(n) = n * A003558(n).

cp's OEIS Frontend

A213371 Triangle read by rows in which row n lists the number of pairs of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.

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A072785 Differences between A072781 and A072738.

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A162518 Characteristic function of magic numbers A018226: 1 if n is a magic number else 0.

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A377137 Array read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n contains 3n/2 elements if n is even, and (n+1)/2 elements if n is odd; ; see Comments.

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A162519 Characteristic function of magic numbers A018226: 0 if n is a magic number else 1.

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A210842 Number of states in the n-th shell of the nuclear shell model.

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A141110 Number of cycles and fixed points in the permutation (n, n-2, n-4, ..., 1, ..., n-3, n-1).

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A377280 Given n cards, each time you reversing the order of the top 1, 2, 3, ..., n-1, n cards, then repeat reversing 1, 2, 3, ... cards. Do reversing at least once. the minimum number of steps required to return all the cards to their original position.

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