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.

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A317316 Multiples of 16 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 16, 3, 32, 5, 48, 7, 64, 9, 80, 11, 96, 13, 112, 15, 128, 17, 144, 19, 160, 21, 176, 23, 192, 25, 208, 27, 224, 29, 240, 31, 256, 33, 272, 35, 288, 37, 304, 39, 320, 41, 336, 43, 352, 45, 368, 47, 384, 49, 400, 51, 416, 53, 432, 55, 448, 57, 464, 59, 480, 61, 496, 63, 512, 65, 528, 67, 544, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 20-gonal numbers (A218864).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 20-gonal numbers.

Links

Crossrefs

Cf. A008598 and A005408 interleaved.
Column 16 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A218864.

Programs

  • Mathematica
    a[n_] := If[OddQ[n], n, 8*n]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)
  • PARI
    concat(0, Vec(x*(1 + 16*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 16*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 16*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 2^(e+3), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 7*2^(1-s)). - Amiram Eldar, Oct 25 2023

A317317 Multiples of 17 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 17, 3, 34, 5, 51, 7, 68, 9, 85, 11, 102, 13, 119, 15, 136, 17, 153, 19, 170, 21, 187, 23, 204, 25, 221, 27, 238, 29, 255, 31, 272, 33, 289, 35, 306, 37, 323, 39, 340, 41, 357, 43, 374, 45, 391, 47, 408, 49, 425, 51, 442, 53, 459, 55, 476, 57, 493, 59, 510, 61, 527, 63, 544, 65, 561, 67, 578, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 21-gonal numbers (A303298).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 21-gonal numbers.

Links

Crossrefs

Cf. A008599 and A005408 interleaved.
Column 17 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303298.

Programs

  • Mathematica
    With[{nn=40},Riffle[17*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,1,17,3},80] (* Harvey P. Dale, Jun 06 2020 *)
  • PARI
    concat(0, Vec(x*(1 + 17*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 17*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 17*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 17*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 15/2^s). - Amiram Eldar, Oct 25 2023

A317318 Multiples of 18 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 18, 3, 36, 5, 54, 7, 72, 9, 90, 11, 108, 13, 126, 15, 144, 17, 162, 19, 180, 21, 198, 23, 216, 25, 234, 27, 252, 29, 270, 31, 288, 33, 306, 35, 324, 37, 342, 39, 360, 41, 378, 43, 396, 45, 414, 47, 432, 49, 450, 51, 468, 53, 486, 55, 504, 57, 522, 59, 540, 61, 558, 63, 576, 65, 594, 67, 612, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 22-gonal numbers (A303299).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 22-gonal numbers.

Links

Crossrefs

Cf. A008600 and A005408 interleaved.
Column 18 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303299.

Programs

  • Mathematica
    a[n_] := If[OddQ[n], n, 9*n]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)
  • PARI
    concat(0, Vec(x*(1 + 18*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 18*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 18*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 9*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 2^(4-s)). - Amiram Eldar, Oct 25 2023

A317319 Multiples of 19 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 19, 3, 38, 5, 57, 7, 76, 9, 95, 11, 114, 13, 133, 15, 152, 17, 171, 19, 190, 21, 209, 23, 228, 25, 247, 27, 266, 29, 285, 31, 304, 33, 323, 35, 342, 37, 361, 39, 380, 41, 399, 43, 418, 45, 437, 47, 456, 49, 475, 51, 494, 53, 513, 55, 532, 57, 551, 59, 570, 61, 589, 63, 608, 65, 627, 67, 646, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 23-gonal numbers (A303303).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 23-gonal numbers.

Links

Crossrefs

Cf. A008601 and A005408 interleaved.
Column 19 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303303.

Programs

  • Mathematica
    a[n_] := If[OddQ[n], n, 19*n/2]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)
  • PARI
    concat(0, Vec(x*(1 + 19*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 19*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 19*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 19*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 17/2^s). - Amiram Eldar, Oct 26 2023

A317320 Multiples of 20 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 20, 3, 40, 5, 60, 7, 80, 9, 100, 11, 120, 13, 140, 15, 160, 17, 180, 19, 200, 21, 220, 23, 240, 25, 260, 27, 280, 29, 300, 31, 320, 33, 340, 35, 360, 37, 380, 39, 400, 41, 420, 43, 440, 45, 460, 47, 480, 49, 500, 51, 520, 53, 540, 55, 560, 57, 580, 59, 600, 61, 620, 63, 640, 65, 660, 67, 680, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 24-gonal numbers (A303814).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 24-gonal numbers.

Links

Crossrefs

Cf. A008602 and A005408 interleaved.
Column 20 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303814.

Programs

  • Mathematica
    With[{nn=40},Riffle[20*Range[0,nn],Range[1,2*nn+1,2]]] (* Harvey P. Dale, Feb 16 2020 *)
  • PARI
    concat(0, Vec(x*(1 + 20*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(n) = n, if n is odd.
a(n) = 10*n, if n is even.
a(2n) = 20*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 20*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 5*2^(e+1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 9*2^(1-s)). - Amiram Eldar, Oct 26 2023

A317321 Multiples of 21 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 21, 3, 42, 5, 63, 7, 84, 9, 105, 11, 126, 13, 147, 15, 168, 17, 189, 19, 210, 21, 231, 23, 252, 25, 273, 27, 294, 29, 315, 31, 336, 33, 357, 35, 378, 37, 399, 39, 420, 41, 441, 43, 462, 45, 483, 47, 504, 49, 525, 51, 546, 53, 567, 55, 588, 57, 609, 59, 630, 61, 651, 63, 672, 65, 693, 67, 714, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 25-gonal numbers (A303304).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 25-gonal numbers.

Links

Crossrefs

Cf. A008603 and A005408 interleaved.
Column 21 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303304.

Programs

  • Mathematica
    a[n_] := If[OddQ[n], n, 21*n/2]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)
  • PARI
    concat(0, Vec(x*(1 + 21*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 21*n, a(2n+1) = 2*n + 1.
Multiplicative with a(2^e) = 21*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 19/2^s). - Amiram Eldar, Oct 26 2023

A317322 Multiples of 22 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 22, 3, 44, 5, 66, 7, 88, 9, 110, 11, 132, 13, 154, 15, 176, 17, 198, 19, 220, 21, 242, 23, 264, 25, 286, 27, 308, 29, 330, 31, 352, 33, 374, 35, 396, 37, 418, 39, 440, 41, 462, 43, 484, 45, 506, 47, 528, 49, 550, 51, 572, 53, 594, 55, 616, 57, 638, 59, 660, 61, 682, 63, 704, 65, 726, 67, 748, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 26-gonal numbers (A316724).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 26-gonal numbers.

Links

Crossrefs

Cf. A008604 and A005408 interleaved.
Column 22 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A316724.

Programs

  • Mathematica
    Module[{nn=40},Riffle[22Range[0,nn],Range[1,2nn,2]]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,1,22,3},80] (* Harvey P. Dale, Dec 12 2021 *)
  • PARI
    concat(0, Vec(x*(1 + 22*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 22*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 22*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 11*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 5*2^(2-s)). - Amiram Eldar, Oct 26 2023

A317323 Multiples of 23 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 23, 3, 46, 5, 69, 7, 92, 9, 115, 11, 138, 13, 161, 15, 184, 17, 207, 19, 230, 21, 253, 23, 276, 25, 299, 27, 322, 29, 345, 31, 368, 33, 391, 35, 414, 37, 437, 39, 460, 41, 483, 43, 506, 45, 529, 47, 552, 49, 575, 51, 598, 53, 621, 55, 644, 57, 667, 59, 690, 61, 713, 63, 736, 65, 759, 67, 782, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 27-gonal numbers (A316725).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 27-gonal numbers.

Links

Crossrefs

Cf. A008605 and A005408 interleaved.
Column 23 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A316725.

Programs

  • Mathematica
    With[{nn=40},Riffle[23*Range[0,nn],Range[1,2*nn,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,23,3},80] (* Harvey P. Dale, May 05 2019 *)
  • PARI
    concat(0, Vec(x*(1 + 23*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 23*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 23*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 23*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 21/2^s). - Amiram Eldar, Oct 26 2023

A317324 Multiples of 24 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 24, 3, 48, 5, 72, 7, 96, 9, 120, 11, 144, 13, 168, 15, 192, 17, 216, 19, 240, 21, 264, 23, 288, 25, 312, 27, 336, 29, 360, 31, 384, 33, 408, 35, 432, 37, 456, 39, 480, 41, 504, 43, 528, 45, 552, 47, 576, 49, 600, 51, 624, 53, 648, 55, 672, 57, 696, 59, 720, 61, 744, 63, 768, 65, 792, 67, 816, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 28-gonal numbers (A303812).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 28-gonal numbers.

Links

Crossrefs

Cf. A008606 and A005408 interleaved.
Column 24 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303812.

Programs

  • Magma
    &cat[[24*n, 2*n + 1]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2018
  • Mathematica
    Table[If[EvenQ[n], 24 (n/2), n], {n, 0, 70}] (* Vincenzo Librandi, Jul 28 2018 *)
With[{nn=40},Riffle[24*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,1,24,3},80] (* Harvey P. Dale, Jul 06 2019 *)
  • PARI
    concat(0, Vec(x*(1 + 24*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 24*n, a(2n+1) = 2*n + 1.
G.f.: x*(1 + 24*x + x^2)/((1-x)^2*(1+x)^2). - Vincenzo Librandi, Jul 28 2018
a(n) = 2*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 29 2018
Multiplicative with a(2^e) = 3*2^(e+2), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 11*2^(1-s)). - Amiram Eldar, Oct 26 2023

A317325 Multiples of 25 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 25, 3, 50, 5, 75, 7, 100, 9, 125, 11, 150, 13, 175, 15, 200, 17, 225, 19, 250, 21, 275, 23, 300, 25, 325, 27, 350, 29, 375, 31, 400, 33, 425, 35, 450, 37, 475, 39, 500, 41, 525, 43, 550, 45, 575, 47, 600, 49, 625, 51, 650, 53, 675, 55, 700, 57, 725, 59, 750, 61, 775, 63, 800, 65, 825, 67, 850, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

Partial sums give the generalized 29-gonal numbers (A303815).
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 29-gonal numbers.

Links

Crossrefs

Cf. A008607 and A005408 interleaved.
Column 25 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303815.

Programs

  • GAP
    Flat(List([0..40],n->[25*n,2*n+1])); # Muniru A Asiru, Jul 28 2018
  • Magma
    &cat[[25*n, 2*n + 1]: n in [0..30]]; // Vincenzo Librandi, Jul 28 2018
  • Maple
    seq(op([25*n,2*n+1]),n=0..40); # Muniru A Asiru, Jul 28 2018
  • Mathematica
    With[{nn=30}, Riffle[25 Range[0, nn], 2 Range[0, nn] + 1]] (* Vincenzo Librandi, Jul 28 2018 *)
  • PARI
    concat(0, Vec(x*(1 + 25*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

Formula

a(2n) = 25*n, a(2n+1) = 2*n + 1.
G.f.: x*(1 + 25*x + x^2)/((1 - x)^2*(1 + x)^2). - Vincenzo Librandi, Jul 28 2018
a(n) = 2*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 29 2018
Multiplicative with a(2^e) = 25*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 23/2^s). - Amiram Eldar, Oct 26 2023
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