A214345 -id:A214345 - cp's OEIS frontend

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

Eddie Gutierrez, Oct 14 2012

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).

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).

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

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

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 *)

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

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]

Definition rewritten by Bruno Berselli, Nov 09 2012

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

Original entry on oeis.org

-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

Eddie Gutierrez, Oct 14 2012

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).

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

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).

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

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

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 *)

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

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]

Definition rewritten by Bruno Berselli, Nov 22 2012

A216852 18k^2-36k+9 interleaved with 18k^2-18k+9 for k>=0.

Original entry on oeis.org

9, 9, -9, 9, 9, 45, 63, 117, 153, 225, 279, 369, 441, 549, 639, 765, 873, 1017, 1143, 1305, 1449, 1629, 1791, 1989, 2169, 2385, 2583, 2817, 3033, 3285, 3519, 3789, 4041, 4329, 4599, 4905, 5193, 5517, 5823, 6165, 6489, 6849, 7191, 7569, 7929, 8325, 8703

Offset: 0

Eddie Gutierrez, Sep 17 2012

The sequence is present as a family of single interleaved sequence of which there are many which are separated or factored out to give individual sequences. The larger sequence produces two smaller interleaved sequences where one of them has the formulas above and the other interleaved sequence has the formulas (18n^2-24n+1) and (18n^2-6n+5). The latter interleaved sequence is A214493. There are three sequences in this family.

Eddie Gutierrez New Interleaved Sequences Part B on oddwheel.com, Section B1 Line No. 22 (square_sequencesII.html) Part B
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Cf. A178218, A214345, A214393, A214405, A216844, A216865, A216875, A216876.

&cat[[18*k^2-36*k+9, 18*k^2-18*k+9]: k in [0..23]]; // Bruno Berselli, Oct 01 2012

Flatten[Table[{18 n^2 - 36 n + 9, 18 n^2 - 18 n + 9}, {n, 0, 23}]] (* Bruno Berselli, Oct 01 2012 *)
Flatten[Table[18n^2+9-{36n,18n},{n,0,50}]] (* or *) LinearRecurrence[ {2,0,-2,1},{9,9,-9,9},100] (* Harvey P. Dale, Apr 26 2014 *)

vector(47, n, k=(n-1)\2; if(n%2, 18*k^2-36*k+9, 18*k^2-18*k+9)) \\ Bruno Berselli, Oct 01 2012

From Bruno Berselli, Oct 01 2012: (Start)
G.f.: 9*(1-x-3*x^2+5*x^3)/((1+x)*(1-x)^3).
a(n) = (9/4)*(2*n*(n-4)-3*(-1)^n+7).
a(n) = 9*A178218(n-3) with A178218(-3)=1, A178218(-2)=1, A178218(-1)=-1, A178218(0)=1. (End)
a(0)=9, a(1)=9, a(2)=-9, a(3)=9, a(n)=2*a(n-1)-2*a(n-3)+a(n-4). - Harvey P. Dale, Apr 26 2014

Definition rewritten by Bruno Berselli, Oct 25 2012

A216853 18k^2-12k-7 interleaved with 18k^2+6k+5 for k>=0.

Original entry on oeis.org

-7, 5, -1, 29, 41, 89, 119, 185, 233, 317, 383, 485, 569, 689, 791, 929, 1049, 1205, 1343, 1517, 1673, 1865, 2039, 2249, 2441, 2669, 2879, 3125, 3353, 3617, 3863, 4145, 4409, 4709, 4991, 5309, 5609, 5945, 6263, 6617, 6953, 7325, 7679, 8069, 8441, 8849

Offset: 0

Eddie Gutierrez, Sep 17 2012

The sequence (the third in the family) is present as a family of single interleaved sequence of which there are many which are separated or factored out to give individual sequences. The larger sequence produces two smaller interleaved sequences where one of them has the formulas above and the other interleaved sequence has the formulas (18n^2-24n-1) and (18n^2-6n+5). The latter interleaved sequence is A214493. There are three sequences in this family.

Eddie Gutierrez New Interleaved Sequences Part B on oddwheel.com, Section B1 Line No. 22 (square_sequencesII.html) Part B
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

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

&cat[[18*k^2-12*k-7, 18*k^2+6*k+5]: k in [0..22]]; // Bruno Berselli, Oct 05 2012

Flatten[Table[{18 n^2 - 12 n - 7, 18 n^2 + 6 n + 5}, {n, 0, 22}]] (* Bruno Berselli, Oct 05 2012 *)

vector(46, n, k=(n-1)\2; if(n%2, 18*k^2-12*k-7, 18*k^2+6*k+5)) \\ Bruno Berselli, Oct 05 2012

G.f.: -(7-19*x+11*x^2-17*x^3)/((1+x)*(1-x)^3). - Bruno Berselli, Oct 05 2012

a(n) = (6*n*(3*n-4)-27*(-1)^n-1)/4. - Bruno Berselli, Oct 05 2012

Definition rewritten by Bruno Berselli, Oct 25 2012

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A216852 18k^2-36k+9 interleaved with 18k^2-18k+9 for k>=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A216853 18k^2-12k-7 interleaved with 18k^2+6k+5 for k>=0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions