A236203 Interleave A005563(n), A028347(n).

0, 0, 3, 5, 8, 12, 15, 21, 24, 32, 35, 45, 48, 60, 63, 77, 80, 96, 99, 117, 120, 140, 143, 165, 168, 192, 195, 221, 224, 252, 255, 285, 288, 320, 323, 357, 360, 396, 399, 437, 440, 480, 483, 525, 528, 572, 575, 621, 624, 672, 675, 725, 728, 780, 783, 837, 840, 896

Offset: 2

Author

Paul Curtz, Jan 20 2014

Keywords

• nonn
• easy

Comments

A175628 gives the numerators of interleaved Lyman and Balmer series, i.e., A005563(n)/A000290(n+1) and A061037(n+2)/A061038(n+2).

Difference table of a(n):

-1, -3, 0, 0, 3, 5, 8, 12, 15, 21, 24, ...

-2, 3, 0, 3, 2, 3, 4, 3, 6, 3, 8, ...

5, -3, 3, -1, 1, 1, -1, 3, -3, 5, -5, ...

-8, 6, -4, 2, 0, -2, 4, -6, 8, -10, 12, ...

14, -10, 6, -2, -2, 6, -10, 14, -18, 22, -26, ...

-24, 16, -8, 0, 8, -16, 24, -32, 40, -48, 56, ... .

a(n+2) gives the numerators of 0/1, 0/16, 3/4, 5/36, 8/9, 12/64, 15/16, 21/100, 24/25, 32/144, ... . The denominators are A097362(n+1)^2. (Compare A097362 to A029578.)

Note the particular distribution of a(-n). Example:

a(n-9) = 12,15, 5,8, 0,3, -3,0, -4,-1, -3,0, 0,3, 5,8, 12,15, ... .

a(2n) + a(2n+1) = a(-2n-1) + a(-2n-2) = -4,0,8,20,36,56,80,... = 4*A000096(n-1).

a(2n) + a(2n-1) = a(-2n) + a(-2n-1) = -5,-3,3,13,... = A001105(n) - A010716(n).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A005843, A008590, A016825, A109613, A147658.

Programs

GAP

List([2..60], n-> (2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8 ); # G. C. Greubel, Dec 04 2019

Magma

[(2*n^2 + 2*n - 19 - (2*n - 11)*(-1)^n)/8: n in [2..60]]; // Vincenzo Librandi, Jul 27 2014

Maple

seq( (2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8, n=2..60); # G. C. Greubel, Dec 04 2019

Mathematica

CoefficientList[Series[x^2(3x^2-2x-3)/((x-1)^3(x+1)^2), {x, 0, 60}], x] (* Vincenzo Librandi, Jul 27 2014 *)
LinearRecurrence[{1,2,-2,-1,1},{0,0,3,5,8},60] (* Harvey P. Dale, Aug 30 2018 *)

PARI

concat([0,0], Vec(x^4*(3*x^2-2*x-3)/((x-1)^3*(x+1)^2) + O(x^60))) \\ Colin Barker, Jan 26 2014

Sage

[(2*n^2 +2*n -19 -(2*n-11)*(-1)^n)/8 for n in (2..60)] # G. C. Greubel, Dec 04 2019

Formula

a(n+2) = (period 8: repeat 1, 16, 1, 1, 1, 4, 1, 1)*A175628(n+1).

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).

a(n+4) - a(n-4) = 0, 8, 8, ... = A168397.

From Colin Barker, Jan 26 2014: (Start)

a(n) = (n^2 -4)/4 for n even, a(n) = (n^2 +2*n -15)/4 for n odd.

G.f.: x^4*(3 + 2*x - 3*x^2)/ ((1-x)^3*(1+x)^2). (End)

a(n) = (2*n^2 + 2*n - 19 - (2*n - 11)*(-1)^n)/8. - Luce ETIENNE, Jul 26 2014

Sum_{n>=4} (-1)^n/a(n) = 11/48. - Amiram Eldar, Aug 21 2022

Extensions

More terms from Colin Barker, Jan 26 2014