A061038 -id:A061038 - cp's OEIS frontend

A174381 Triangle for denominators of half extended Rydberg-Ritz spectrum of the hydrogenic spectra. a(n) is an antidiagonal writing of array in A171522 without first column.

0, 0, 4, 0, 9, 36, 0, 16, 16, 144, 0, 25, 100, 225, 400, 0, 36, 9, 12, 144, 900, 0, 49, 196, 441, 784, 1225, 1764, 0, 64, 64, 576, 64, 1600, 576, 3136, 0, 81, 324, 81, 1296, 2025, 324, 3969, 5184, 0, 100, 25, 900, 400, 100, 225, 4900, 1600, 8100, 0, 121, 484, 1089

Offset: 0

Paul Curtz, Mar 17 2010

Companion to A172157 (numerators). Hence -1/0; -1/0,-3/4; -1/0,-8/9,-5/36; -1/0,-15/16,-3/16,-7/144; -1/0,-24/25,-21/100,-16/225,-9/400; for 1) (-1/0, A005563/A000290(n+1))=A067998(n+1)/A000290 Lyman; 2) -1/0, -3/4, A061037/A061038 Balmer ; 3) -1/0, -8/9, -5/36, A061039/A061040)=A171709(n+3)/ Paschen; 4) (-1/0, -15/16, -3/16, -7/144, A061041/A061042 Brackett; .. .

a(n)= 0, (mix 0 or A000004 , n-th row of A120073)

A177842 Period 27: repeat 1, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 1, 81, 81, 9, 81, 81.

Original entry on oeis.org

1, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 1, 81, 81, 9, 81, 81, 1, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 1, 81, 81, 9, 81, 81, 1, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 1, 81, 81

Offset: 0

Paul Curtz, May 14 2010

The generating formula is a(n) = A061040(n+3) - 9*A061039(n+3). This is a member of the family of sequences with A000012(n) = A000290(n+1) -A005563(n+1), with period length 1, and A177499(n) = A061038(n+2) -4*A061037(n+2), with period length 4.

a(n) here has period length 3^3 and the general series of this family has period length k^k.

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

a(n)=3^[0, 4, 4, 1, 4, 4, 2, 4, 4, 1, 4, 4, 1, 4, 4, 2, 4, 4, 1, 4, 4, 0, 4, 4, 2, 4, 4][n%27+1] \\ Charles R Greathouse IV, Jul 17 2016

G.f.: ( -1 -81*x -3*x^9 -3*x^3 -81*x^4 -81*x^5 -9*x^6 -81*x^7 -81*x^8 -81*x^10 -3*x^12 -81*x^13 -81*x^14 -9*x^15 -81*x^16 -81*x^17 -3*x^18 -81*x^19 -81*x^20 -x^21 -81*x^22 -81*x^23 -9*x^24 -81*x^25 -81*x^26 -81*x^11 -81*x^2 ) / ( (x-1) *(1+x+x^2) *(1+x^3+x^6) *(1+x^9+x^18) ). - R. J. Mathar, Dec 09 2010

a(n) = a(n+27).

cp's OEIS Frontend

A174381 Triangle for denominators of half extended Rydberg-Ritz spectrum of the hydrogenic spectra. a(n) is an antidiagonal writing of array in A171522 without first column.

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A177842 Period 27: repeat 1, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 3, 81, 81, 9, 81, 81, 3, 81, 81, 1, 81, 81, 9, 81, 81.

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