A217893 50k^2-40k-17 interleaved with 50k^2+10k+13 for k=>0.

-17, 13, -7, 73, 103, 233, 313, 493, 623, 853, 1033, 1313, 1543, 1873, 2153, 2533, 2863, 3293, 3673, 4153, 4583, 5113, 5593, 6173, 6703, 7333, 7913, 8593, 9223, 9953, 10633, 11413, 12143, 12973, 13753, 14633, 15463, 16393, 17273, 18253, 19183, 20213, 21193

Offset: 0

Author

Eddie Gutierrez, Oct 14 2012

Keywords

• sign
• easy

Comments

The sequence (the fourth in the family) is present as a family of interleaved sequences (five in total) which are separated or factored out to give individual sequences. The first sequence is the parent having the formulas: 50*n^2-100*n+25 and 50*n^2-50*n+25 whose entries are all divisible by 25 and is identical to A178218. The fourth sequence has the formulas 50*n^2-40*n-17 and 50*n^2+10*n+13 and is part of a group where each of the sequences are new, except for the parent (in the factored form).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eddie Gutierrez New Interleaved Sequences Part H or Oddwheel.com, Section B1 Line 28 (square_sequencesVIII.html), Part H.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A178218, A214345, A214393, A214405, A216876.

Programs

Magma

&cat[[50*k^2-40*k-17,50*k^2+10*k+13]: k in [0..23]]; // Bruno Berselli, Oct 23 2012

Mathematica

Flatten[Table[{50 n^2 - 40 n - 17, 50 n^2 + 10 n + 13}, {n, 0, 23}]] (* Bruno Berselli, Oct 23 2012 *)
CoefficientList[Series[(-17 + 47*x - 33*x^2 + 53*x^3)/((1+x)*(1-x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 15 2012 *)

PARI

vector(48, n, k=(n-1)\2; if(n%2, 50*k^2-40*k-17, 50*k^2+10*k+13)) \\ Bruno Berselli, Oct 23 2012

Formula

G.f.: (-17+47*x-33*x^2+53*x^3)/((1+x)*(1-x)^3).

a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).

a(n) = 1+(10*n*(5*n-8)-75*(-1)^n+3)/4. [Bruno Berselli, Oct 15 2012]

Extensions

Definition rewritten by Bruno Berselli, Nov 09 2012