cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A216865 16k^2-32k+8 interleaved with 16k^2-16k+8 for k>=0.

Original entry on oeis.org

8, 8, -8, 8, 8, 40, 56, 104, 136, 200, 248, 328, 392, 488, 568, 680, 776, 904, 1016, 1160, 1288, 1448, 1592, 1768, 1928, 2120, 2296, 2504, 2696, 2920, 3128, 3368, 3592, 3848, 4088, 4360, 4616, 4904, 5176, 5480, 5768, 6088, 6392, 6728, 7048, 7400, 7736
Offset: 0

Views

Author

Eddie Gutierrez, Sep 18 2012

Keywords

Comments

The sequence (the first in the family) is present as a family of single interleaved sequence of which are separated or factored out of the larger sequence to give individual sequences. The larger sequence produces four smaller interleaved sequences where one of them has the formula above and a second interleaved sequences having the formulas (16n^2-24n+1) and (16n^2-6n+5). This interleaved sequence is A214393. The fourth interleaved sequence in the group has the formulas (16n^2-8n-7) and (16n^2+2n+5) and it is A214405. There are a total of four sequences in this family.

Links

Crossrefs

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

Programs

  • Magma
    &cat[[16*k^2-32*k+8, 16*k^2-16*k+8]: k in [0..23]]; // Bruno Berselli, Oct 01 2012
  • Mathematica
    Flatten[Table[{16 n^2 - 32 n + 8, 16 n^2 - 16 n + 8}, {n, 0, 23}]] (* Bruno Berselli, Sep 30 2012 *)
  • PARI
    vector(47, n, k=(n-1)\2; if(n%2, 16*k^2-32*k+8, 16*k^2-16*k+8)) \\ Bruno Berselli, Oct 01 2012

Formula

G.f.: 8*(1-x-3*x^2+5*x^3)/((1+x)*(1-x)^3). [Bruno Berselli, Sep 30 2012]
a(n) = 2*(2*n*(n-4)-3*(-1)^n+7). [Bruno Berselli, Sep 30 2012]
a(n) = 8*A178218(n-3) with A178218(-3)=1, A178218(-2)=1, A178218(-1)=-1, A178218(0)=1. [Bruno Berselli, Oct 01 2012]

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2012

A216871 16k^2-16k-4 interleaved with 16k^2+4 for k>=0.

Original entry on oeis.org

-4, 4, -4, 20, 28, 68, 92, 148, 188, 260, 316, 404, 476, 580, 668, 788, 892, 1028, 1148, 1300, 1436, 1604, 1756, 1940, 2108, 2308, 2492, 2708, 2908, 3140, 3356, 3604, 3836, 4100, 4348, 4628, 4892, 5188, 5468, 5780, 6076, 6404, 6716, 7060, 7388, 7748, 8092
Offset: 0

Views

Author

Eddie Gutierrez, Sep 18 2012

Keywords

Comments

The sequence (the third in the family) is present as a family of single interleaved sequence of which are separated or factored out of the larger sequence to give individual sequences. The larger sequence produces four smaller interleaved sequences where one of them has the formula above and a second interleaved sequences having the formulas (16n^2-24n+1) and (16n^2-6n+5). This interleaved sequence is A214393. The fourth interleaved sequence in the group has the formulas (16n^2-8n-7) and (16n^2+2n+5) and it is A214405. There are a total of four sequences in this family.

Links

Crossrefs

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

Programs

  • Magma
    &cat[[16*k^2-16*k-4, 16*k^2+4]: k in [0..23]]; // Bruno Berselli, Sep 27 2012
  • Mathematica
    Flatten[Table[{16 n^2 - 16 n - 4, 16 n^2 + 4}, {n, 0, 23}]] (* Bruno Berselli, Sep 26 2012 *)
LinearRecurrence[{2,0,-2,1},{-4,4,-4,20},50] (* Harvey P. Dale, Dec 09 2015 *)
  • PARI
    vector(47, n, k=(n-1)\2; if(n%2, 16*k^2-16*k-4, 16*k^2+4)) \\ Bruno Berselli, Sep 28 2012

Formula

Contribution from Bruno Berselli, Sep 27 2012: (Start)
G.f.: -4*(1-3*x+3*x^2-5*x^3)/((1+x)*(1-x)^3).
a(n) = 2*(2*n*(n-2)-3*(-1)^n+1).
a(n) = 4*A214345(n-3) with A214345(-3)=-1, A214345(-2)=1, A214345(-1)=-1. (End)

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2012

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

Original entry on oeis.org

-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

Views

Author

Eddie Gutierrez, Oct 14 2012

Keywords

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

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

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

Views

Author

Eddie Gutierrez, Oct 14 2012

Keywords

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

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

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

Views

Author

Eddie Gutierrez, Sep 17 2012

Keywords

Comments

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.

Links

Crossrefs

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

Programs

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

Formula

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

Extensions

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

Views

Author

Eddie Gutierrez, Sep 17 2012

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Magma
    &cat[[18*k^2-12*k-7, 18*k^2+6*k+5]: k in [0..22]]; // Bruno Berselli, Oct 05 2012
  • Mathematica
    Flatten[Table[{18 n^2 - 12 n - 7, 18 n^2 + 6 n + 5}, {n, 0, 22}]] (* Bruno Berselli, Oct 05 2012 *)
  • PARI
    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

Formula

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

Extensions

Definition rewritten by Bruno Berselli, Oct 25 2012
Showing 1-6 of 6 results.

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