A217894 50k^2-20k-23 interleaved with 50k^2+30k+17 for k=>0.

-23, 17, 7, 97, 137, 277, 367, 557, 697, 937, 1127, 1417, 1657, 1997, 2287, 2677, 3017, 3457, 3847, 4337, 4777, 5317, 5807, 6397, 6937, 7577, 8167, 8857, 9497, 10237, 10927, 11717, 12457, 13297, 14087, 14977, 15817, 16757, 17647, 18637, 19577, 20617, 21607

Offset: 0

Author

Eddie Gutierrez, Oct 14 2012

Keywords

• sign
• easy

Comments

The sequence (the fifth and last 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 fifth sequence has the formulas 50*n^2-20*n-23 and 50*n^2+30*n+17 and is part of a group where each of the sequences are new, except for the parent (in the factored form).

Examples

			a(9) = 2*a(8) - 2*a(6) + a(5) = 1394 - 734 + 277 = 937.
		

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^2-20*k-23, 50*k^2+30*k+17]: k in [0..23]]; // Bruno Berselli, Oct 23 2012

Mathematica

Flatten[Table[{50*n^2 - 20*n - 23, 50*n^2 + 30*n + 17}, {n, 0, 23}]] (* Bruno Berselli, Oct 23 2012 *)
CoefficientList[Series[(-23 + 63*x - 27*x ^ 2 + 37*x^3)/((1+x)*(1-x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 23 2012 *)

PARI

vector(48, n, k=(n-1)\2; if(n%2, 50*k^2-20*k-23, 50*k^2+30*k+17)) \\ Bruno Berselli, Oct 23 2012

Formula

G.f.: (-23+63*x-27*x^2+37*x^3)/((1+x)*(1-x)^3).

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

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

Extensions

Definition rewritten by Bruno Berselli, Nov 22 2012