A056951 -id:A056951 - cp's OEIS frontend

A130517 Triangle read by rows: row n counts down from n in steps of 2, then counts up the remaining elements in the set {1,2,...,n}, again in steps of 2.

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

Offset: 1

Omar E. Pol, Aug 08 2007

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.
Row n lists a permutation of the first n positive integers.
If n is odd then row n lists the first (n+1)/2 odd numbers in decreasing order together with the first (n-1)/2 positive even numbers.
If n is even then row n lists the first n/2 even numbers in decreasing order together with the first n/2 odd numbers.
Row n >= 2, with its floor(n/2) last numbers taken as negative, lists the n different eigenvalues (in decreasing order) of the odd graph O(n). The odd graph O(n) has the (n-1)-subsets of a (2*n-1)-set as vertices, with two (n-1)-subsets adjacent if and only if they are disjoint. For example, O(3) is isomorphic to the Petersen graph. - Miquel A. Fiol, Apr 07 2024

			A geometric model of the atomic nucleus:
......-------------------------------------------------
......|...-----------------------------------------...|
......|...|...---------------------------------...|...|
......|...|...|...-------------------------...|...|...|
......|...|...|...|...-----------------...|...|...|...|
......|...|...|...|...|...---------...|...|...|...|...|
......|...|...|...|...|...|...-...|...|...|...|...|...|
......i...h...g...f...d...p...s...p...d...f...g...h...i
......|...|...|...|...|...|.......|...|...|...|...|...|
......|...|...|...|...|.......1.......|...|...|...|...|
......|...|...|...|.......2.......1.......|...|...|...|
......|...|...|.......3.......1.......2.......|...|...|
......|...|.......4.......2.......1.......3.......|...|
......|.......5.......3.......1.......2.......4.......|
..........6.......4.......2.......1.......3.......5....
......7.......5.......3.......1.......2.......4.......6
.......................................................
...13/2.11/2.9/2.7/2.5/2.3/2.1/2.1/2.3/2.5/2.7/2.9/2.11/2
......|...|...|...|...|...|...|...|...|...|...|...|...|
......|...|...|...|...|...|...-----...|...|...|...|...|
......|...|...|...|...|...-------------...|...|...|...|
......|...|...|...|...---------------------...|...|...|
......|...|...|...-----------------------------...|...|
......|...|...-------------------------------------...|
......|...---------------------------------------------
.
Triangle begins:
   1;
   2, 1;
   3, 1, 2;
   4, 2, 1, 3;
   5, 3, 1, 2, 4;
   6, 4, 2, 1, 3, 5;
   7, 5, 3, 1, 2, 4, 6;
   8, 6, 4, 2, 1, 3, 5, 7;
   9, 7, 5, 3, 1, 2, 4, 6, 8;
  10, 8, 6, 4, 2, 1, 3, 5, 7, 9;
  ...
Also:
                     1;
                   2,  1;
                 3,  1,  2;
               4,  2,  1,  3;
             5,  3,  1,  2,  4;
           6,  4,  2,  1,  3,  5;
         7,  5,  3,  1,  2,  4,  6;
       8,  6,  4,  2,  1,  3,  5,  7;
     9,  7,  5,  3,  1,  2,  4,  6,  8;
  10,  8,  6,  4,  2,  1,  3,  5,  7,  9;
  ...
In this view each column contains the same numbers.
From _Miquel A. Fiol_, Apr 07 2024: (Start)
Eigenvalues of the odd graphs O(n) for n=2..10:
   2, -1;
   3,  1, -2;
   4,  2, -1, -3;
   5,  3,  1, -2, -4;
   6,  4,  2, -1, -3, -5;
   7,  5,  3,  1, -2, -4, -6;
   8,  6,  4,  2, -1, -3, -5, -7;
   9,  7,  5,  3,  1, -2, -4, -6, -8;
  10,  8,  6,  4,  2, -1, -3, -5, -7, -9;
... (End)

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
N. Bigss, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, 2012, arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Odd graph

Absolute values of A056951. Column 1 is A000027. Row sums are in A000217.
Cf. A130556, A130598, A130602.
Other versions are A004736, A212121, A213361, A213371.
Cf. A028310 (right edge), A000012 (central terms), A220073 (mirrored), A220053 (partial sums in rows), A375303.

a130517 n k = a130517_tabl !! (n-1) !! (k-1)
a130517_row n = a130517_tabl !! (n-1)
a130517_tabl = iterate (\row -> (head row + 1) : reverse row) [1]
-- Reinhard Zumkeller, Dec 03 2012

A130517 := proc(n,k)
     if k <= (n+1)/2 then
        n-2*(k-1) ;
    else
        1-n+2*(k-1) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2012

t[n_, 1] := n; t[n_, n_] := n-1; t[n_, k_] := Abs[2*k-n - If[2*k <= n+1, 2, 1]]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 03 2013, from abs(A056951) *)

a130517_row(n) = my(v=vector(n), s=1, n1=0, n2=n+1); forstep(k=n, 1,-1, s=-s; if(s>0, n2--; v[n2]=k, n1++; v[n1]=k)); v \\ Hugo Pfoertner, Aug 26 2024

a(n) = A162630(n)/2. - Omar E. Pol, Sep 02 2012
T(1,1) = 1; for n > 1: T(n,1) = T(n-1,1)+1 and T(n,k) = T(n-1,n-k+1), 1 < k <= n. - Reinhard Zumkeller, Dec 03 2012
From Boris Putievskiy, Jan 16 2013: (Start)
a(n) = |2*A000027(n) - A003056(n)^2 - 2*A003056(n) - 3| + floor((2*A000027(n) - A003056(n)^2 - A003056(n))/(A003056(n)+3)).
a(n) = |2*n - t^2 - 2*t - 3| + floor((2*n - t^2 - t)/(t+3)) where t = floor((-1+sqrt(8*n-7))/2). (End)

A130598 A shell geometric model of the nucleus. The location of the magic numbers. A triangle.

Original entry on oeis.org

10, 1111, 10, 111111, 11, 1110, 11111110, 1111, 11, 111111, 1111111110, 111111, 11, 1111, 11111111, 111111111110, 11111111, 1111, 11, 111111, 1111111111, 11111111111110, 1111111111, 111111, 11, 1111, 11111111, 111111111111

Offset: 1

Omar E. Pol, Aug 10 2007

The magic numbers of the atomic nucleus: 2, 8, 20, 28, 50, 82, 126. 0 is the location of a magic number. 10 or 11 is equal to 2 protons (or neutrons). 1110 or 1111 is equal to 2+2 protons (or neutrons). 111111 is equal to 2+2+2 protons (or neutrons)... The 2D model is a triangle and a spiral. The 3D model is a double tetrahedron and a double spiral.

			......|----------------------- h -------------------|.....
......|.....|----------------- g --------------|....|.....
......|.....|.....|----------- f ---------|....|....|.....
......|.....|.....|....|------ d -----|...|....|....|.....
......|.....|.....|....|...|-- p -|...|...|....|....|.....
......|.....|.....|....|...|.. s .|...|...|....|....|.....
......|.....|.....|....|...|......|...|...|....|....|.....
......|.....|.....|....|......10......|...|....|....|.....
......|.....|.....|......1111....10.......|....|....|.....
......|.....|.......111111....11....1110.......|....|.....
......|........11111110..1111....11....111111.......|.....
.......1111111110...111111....11....1111...11111111.|.....
111111111110...11111111..1111....11....111111...1111111111
......|.....|.....|....|...|..|...|...|...|....|....|.....
......|.....|.....|....|...|..|1/2|...|...|....|....|.....
......|.....|.....|....|...|-- 3/2 ---|...|....|....|.....
......|.....|.....|....|------ 5/2 -------|....|....|.....
......|.....|.....|----------- 7/2 ------------|....|.....
......|.....|----------------- 9/2 -----------------|.....
......|---------------------- 11/2 -----------------------

Cf. A056951, A130517, A130556, A130602.

A130556 A model of the atomic nucleus (Shell model of nucleus). A triangle.

Original entry on oeis.org

1, 11, 1, 111, 1, 11, 1111, 11, 1, 111, 11111, 111, 1, 11, 1111, 111111, 1111, 11, 1, 111, 11111, 1111111, 11111, 111, 1, 11, 1111, 111111, 11111111, 111111, 1111, 11, 1, 111, 11111, 1111111

Offset: 1

Omar E. Pol, Aug 09 2007, Aug 12 2007

1 is equal to 2 protons, 11 is equal to 2+2 protons, 111 is equal to 2+2+2 protons...

Repunit numbers represent the subshells.

			A geometric model of the shell structure of nucleus:
...|----------------------.i.----------------------|
...|...|------------------.h.------------------|...|
...|...|...|--------------.g.--------------|...|...|
...|...|...|...|----------.f.----------|...|...|...|
...|...|...|...|...|------.d.------|...|...|...|...|
...|...|...|...|...|...|--.p.--|...|...|...|...|...|
...|...|...|...|...|...|...s...|...|...|...|...|...|
...|...|...|...|...|...|...|...|...|...|...|...|...|
...|...|...|...|...|...|.......|...|...|...|...|...|
...|...|...|...|...|.......1.......|...|...|...|...|
...|...|...|...|......11.......1.......|...|...|...|
...|...|...|......111......1......11.......|...|...|
...|...|.....1111.....11.......1......111......|...|
...|.....11111....111......1......11.....1111......|
....111111...1111.....11.......1......111....11111....
1111111..11111....111......1......11.....1111...111111
......................................................
...|...|...|...|...|...|...|...|...|...|...|...|...|
...|...|...|...|...|...|...|1/2|...|...|...|...|...|
...|...|...|...|...|...|----3/2----|...|...|...|...|
...|...|...|...|...|--------5/2--------|...|...|...|
...|...|...|...|------------7/2------------|...|...|
...|...|...|----------------9/2----------------|...|
...|...|-------------------11/2--------------------|
...|----------------------.13/2.--------------------

Cf. A056951, A130517, A130598, A130602.

A130602 A shell geometric model of the atomic nucleus.

Original entry on oeis.org

11, 1111, 11, 111111, 11, 1111, 11111111, 1111, 11, 111111, 1111111111, 111111, 11, 1111, 11111111, 111111111111, 11111111, 1111, 11, 111111, 1111111111, 11111111111111, 1111111111, 111111, 11, 1111, 11111111, 111111111111

Offset: 1

Omar E. Pol, Aug 10 2007, Aug 12 2007

nonn

11 is equal to 2 protons. 1111 is equal 2+2 protons. 111111 is equal 2+2+2 protons...

Repunit numbers represent the subshells.

			See the model in the entry: A130517, A130556.

Cf. A056951, A130517, A130556.

A089645 Penny Flipping, or Flipping Coins: Given a stack of n coins, flip the top coin, then the stack of the top two coins, then the stack of the top three etc... starting again with the top coin after flipping all n coins. A flip of m coins reverses their order and inverts their state. This is the number of flips required to restore the stack to its original configuration.

Original entry on oeis.org

2, 3, 9, 11, 24, 35, 28, 31, 80, 60, 121, 119, 116, 195, 75, 79, 204, 323, 228, 199, 146, 264, 529, 504, 200, 675, 540, 251, 840, 899, 186, 191, 1088, 748, 1225, 324, 740, 1140, 1521, 1079, 1680, 336, 1204, 484, 540, 460, 1692, 1151, 734, 2499

Offset: 1

Richard Forster (gbrl01(AT)yahoo.co.uk), Jan 02 2004

Here "original configuration" seems to mean each coin in original orientation and either original or reverse order; for the original order and either original or reverse orientation n*A002326(n) flips required, while for both original order and original orientation n*A003558(n) required. - Henry Bottomley, Jan 19 2007

Birtwistle (1973) attributes the problem to Iain Bride and John Gilder of UMIST (University of Manchester Institute of Science and Technology).

			For 3 coins (starting with HHH) the flips move the stack through the sequence: HHH -1-> THH -2-> THH -3-> TTH -1-> HTH -2-> HTH -3-> THT -1-> HHT -2-> TTT -3-> HHH. (-n-> indicates n coins are flipped)

G. M. Birtwistle, Simula Begin, Auerbach Publishers, Philadelphia, 1973 [Uses this problem to illustrate the power of the Simula language]
Popular Computing (Calabasas, CA), Penny Flipping, Vol. 3 (No. 23, Feb 1973), pages PC23-10 to PC23-13) and Vol. 3 (No. 29, Aug 1975), pages PC29-6 to PC29-8. Gives first 32 terms.

B. B. Newman, The Flippin' Coins Problem, Mathematics Magazine, Vol. 54 (1981), pp. 51-59.

Cf. A002326, A003558, A056951.

b[n_] := MultiplicativeOrder[2, 2n+1];
c[n_] := MultiplicativeOrder[2, 2n+1, {-1, 1}];
a[1] = 2; a[n_] := If[b[n] == c[n], n*b[n], n*b[n]/2 - 1];
Array[a, 50] (* Jean-François Alcover, Oct 29 2017, after Henry Bottomley *)

For n>1, if A002326(n)=A003558(n) then a(n)=n*A002326(n), otherwise a(n)=n*A002326(n)-1. - Henry Bottomley, Jan 19 2007

cp's OEIS Frontend

A130517 Triangle read by rows: row n counts down from n in steps of 2, then counts up the remaining elements in the set {1,2,...,n}, again in steps of 2.

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A130598 A shell geometric model of the nucleus. The location of the magic numbers. A triangle.

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A130556 A model of the atomic nucleus (Shell model of nucleus). A triangle.

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A130602 A shell geometric model of the atomic nucleus.

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