A335105 -id:A335105 - cp's OEIS frontend

A354445 Number of polynomials per row where the minimum number of rows and polynomials per row necessary to transform A335105 into a triangular array are present.

1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 4, 5, 4, 5, 6, 7, 8, 9, 8, 9, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 17, 18, 19, 20, 21, 20, 21, 20, 21, 22, 23, 24, 25, 26, 29, 28, 29, 28, 29, 30, 31, 32, 33, 32, 33, 32, 31, 34, 35, 36, 37, 38, 37, 40, 41, 42, 43, 44, 45, 44, 45, 46, 47

Offset: 1

David Williams, May 29 2022

nonn

This array treats A335105, an irregular triangle, as a subset of a symmetrical one. It is only necessary to add one row in order to transform A335105 into a triangular array. Rows two, four and six, which correspond to Hydrogen, Lithium and Boron in A335105, are the only rows composed entirely of numerical terms; for these rows the terminal number divided by two and then squared equals the sum of terms left of the right edge. Polynomials within a row may change places with numerical terms within the same row without changing the number of polynomials per row. Given that the summands of A335105 (shell and number of shell's electrons) are necessarily added in multiples of two, the parity of this sequence is alternating.

All the above statements apply to A350597.

			                  X                1
  1 2             1 2              0      Thus, 1, 0, 1, 0, 1, 0, 1, 2, ...
  1 3             1 3 X            1
  1 3 5 6         1 3 5 6          0
  1 3 5 7         1 3 5 7 X        1
  1 3 5 7 9 10    1 3 5 7 9 10     0

David Williams, Table of n, a(n) for n = 1..103

Cf. A335105, A350597.

A350597 Irregular triangle read by rows in which row n lists the partial sums of shell numbers and respective number of electrons for all occupied shells of the n-th element of the periodic table of the elements where electron configurations are ordered according to Madelung's rule.

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 5, 6, 1, 3, 5, 7, 1, 3, 5, 7, 9, 10, 1, 3, 5, 7, 9, 11, 1, 3, 5, 7, 9, 12, 1, 3, 5, 7, 9, 13, 1, 3, 5, 7, 9, 14, 1, 3, 5, 7, 9, 15, 1, 3, 5, 7, 9, 15, 18, 19, 1, 3, 5, 7, 9, 15, 18, 20, 1, 3, 5, 7, 9, 15, 18, 20, 23, 24, 1, 3, 5, 7, 9, 15, 18, 20, 23, 25

Offset: 1

David Williams, Jan 08 2022

This sequence is a variant of A335105.
This sequence agrees with A335105 until the row for scandium; specifically, divergence occurs at t(20,10).
Electron configurations are either written in spectroscopic order as in A335105 where configurations are listed in order of increasing values of the main quantum number, or by order of Madelung's rule, where they are written in order of increasing values of the sums of the main quantum number and respective angular momentum numbers (s,p,d or f) with s=0, p=1, d=2, f=3. Thus in Madelung order 4s (sum 4+0) is written ahead of 3d (sum 3+2).
The final term of each row of this sequence agrees with final term of each row of A335105.

			The configuration for lithium is 1s2 2s1 whether written in spectroscopic order or by order of the Madelung rule. Thus the row for lithium in A335105 and the row for lithium in this sequence agree being derived as follows:
  1s2 2s1
  1+2+2+1
  1, 3, 5, 6
  However: in spectroscopic order scandium is written 1s2, 2s2, 2p6, 3s2, 3p6, 3d1, 4s2 yielding a row in A335105 of 1, 3, 5, 7, 9, 15, 18, 20, 23, 29, 32, 33, 37, 39, whereas in Madelung order the same algorithm applied to the configuration for scandium yields a row of 1, 3, 5, 7, 9, 15, 18, 20, 23, 29, 33, 35, 38, 39.

Gary L. Miessler and Donald A. Tarr, Inorganic Chemistry, 4th Edition. Prentice Hall. Upper Saddle River, New Jersey, 2011.
John R. Rumble (ed.), CRC Handbook of Chemistry and Physics, 100th edition. CRC Press. Boca Raton, Florida, 2019; Section 1, Electron Configurations and Ionization Energy of Neutral Atoms in the Ground State.
Eric Scerri, The Periodic Table, 2nd edition. Oxford University Press. New York, New York, 2019.

Cf. A335105.

cp's OEIS Frontend

A354445 Number of polynomials per row where the minimum number of rows and polynomials per row necessary to transform A335105 into a triangular array are present.

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