A147658 -id:A147658 - cp's OEIS frontend

A147657 a(1)=1, a(2)=2, thereafter (1, -2, 3, -4, 5, -6, ...) interleaved with (-2, 2, -2, 2, ...).

1, 2, 1, -2, -2, 2, 3, -2, -4, 2, 5, -2, -6, 2, 7, -2, -8, 2, 9, -2, -10, 2, 11, -2, -12, 2, 13, -2, -14, 2, 15, -2, -16, 2, 17, -2, -18, 2, 19, -2, -20, 2, 21, -2, -22, 2, 23, -2, -24, 2, 25, -2, -26, 2, 27, -2, -28, 2, 29, -2, -30, 2, 31, -2, -32, 2, 33

Offset: 1

Gary W. Adamson, Nov 09 2008

Equals POLYMOTZKINTINV [1,2,3,...], such that POLYMOTZKINT A147657 = [1,2,3,...]. A comment accompanying the POLYMOTZKINT operation may be found in A005717.

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A005717, A147658.

Join[{1,2}, LinearRecurrence[{0, -2, 0, -1}, {1, -2, -2, 2}, 65]] (* Georg Fischer, Nov 02 2021 *)

(1, -2, 3, -4, 5,...) interleaved with (1, 2, -2, 2,...) such that the first subsequence starts after the first "2" in the second subsequence.

G.f.: x + x^2*(2+x+2*x^2)/(1+x^2)^2. - R. J. Mathar, Dec 13 2022

Definition and a(28) ff. corrected by Georg Fischer, Nov 02 2021

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

Original entry on oeis.org

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

Paul Curtz, Jan 20 2014

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).

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).

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

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

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

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

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 *)

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

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

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

More terms from Colin Barker, Jan 26 2014

cp's OEIS Frontend

A147657 a(1)=1, a(2)=2, thereafter (1, -2, 3, -4, 5, -6, ...) interleaved with (-2, 2, -2, 2, ...).

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