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.

Previous Showing 21-25 of 25 results.

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

A317326 Multiples of 26 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 26, 3, 52, 5, 78, 7, 104, 9, 130, 11, 156, 13, 182, 15, 208, 17, 234, 19, 260, 21, 286, 23, 312, 25, 338, 27, 364, 29, 390, 31, 416, 33, 442, 35, 468, 37, 494, 39, 520, 41, 546, 43, 572, 45, 598, 47, 624, 49, 650, 51, 676, 53, 702, 55, 728, 57, 754, 59, 780, 61, 806, 63, 832, 65, 858, 67, 884, 69
Offset: 0

Views

Author

Omar E. Pol, Jul 25 2018

Keywords

Comments

a(n) is the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 30-gonal numbers (A316729).
Partial sums give the generalized 30-gonal numbers.
More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
From Bruno Berselli, Jul 27 2018: (Start)
Also, this type of sequence is characterized by:
O.g.f.: x*(1 + m*x + x^2)/(1 - x^2)^2;
E.g.f.: x*(2 - m + (2 + m)*exp(2*x))*exp(-x)/4;
a(n) = -a(-n) = (2 + m - (2 - m)*(-1)^n)*n/4;
a(n) = (m/2)^((1 + (-1)^n)/2)*n;
a(n) = 2*a(n-2) - a(n-4), with signature (0,2,0,-1). (End)

Links

Crossrefs

Cf. A252994 and A005408 interleaved.
Column 26 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), A317311 (k=15), A317312 (k=16), A317313 (k=17), A317314 (k=18), A317315 (k=19), A317316 (k=20), A317317 (k=21), A317318 (k=22), A317319 (k=23), A317320 (k=24), A317321 (k=25), A317322 (k=26), A317323 (k=27), A317324 (k=28), A317325 (k=29), this sequence (k=30).
Cf. A316729.

Programs

  • Julia
    [13^div(1+(-1)^n,2)*n for n in 0:70] |> println # Bruno Berselli, Jul 28 2018
  • Mathematica
    Table[(7 + 6 (-1)^n) n, {n, 0, 70}] (* Bruno Berselli, Jul 27 2018 *)

Formula

a(2*n) = 26*n, a(2*n+1) = 2*n + 1.
From Bruno Berselli, Jul 27 2018: (Start)
O.g.f.: x*(1 + 26*x + x^2)/(1 - x^2)^2.
E.g.f.: x*(-6 + 7*exp(2*x))*exp(-x).
a(n) = -a(-n) = (7 + 6*(-1)^n)*n.
a(n) = 13^((1 + (-1)^n)/2)*n.
a(n) = 2*a(n-2) - a(n-4). (End)
Multiplicative with a(2^e) = 13*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 + 3*2^(3-s)). - Amiram Eldar, Oct 26 2023

A280579 Square array read by antidiagonals downwards giving the first differences A261327(n+p) - A261327(n), with p >= 0.

Original entry on oeis.org

0, 0, 4, 0, -3, 1, 0, 11, 8, 12, 0, -8, 3, 0, 4, 0, 24, 16, 27, 24, 28, 0, -19, 5, -3, 8, 5, 9, 0, 43, 24, 48, 40, 51, 48, 52, 0, -36, 7, -12, 12, 4, 15, 12, 16, 0, 68, 32, 75, 56, 80, 72, 83, 80, 84, 0, -59, 9, -27
Offset: 0

Views

Author

Paul Curtz, Jan 05 2017

Keywords

Comments

Successive rows:
p
0: 0, 0, 0, 0, 0, 0, 0, ...
1: 4, -3, 11, -8, 24, -19, 43, ...
2: 1, 8, 3, 16, 5, 24, 7, ...
3: 12, 0, 27, -3, 48, -12, 75, ...
4: 4, 24, 8, 40, 12, 56, 16, ...
5: 28, 5, 51, 4, 80, -3, 115, ...
6: 9, 48, 15, 72, 21, 96, 27, ...
... .
Main diagonal: alternate 3*n^2, -3.
From p>0, the rows are multiples of 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, ... . Sequences appearing after division: shifted A144433 or A195161, A064680. For p=3, we have (n+2)^2, -n^2.
First column: alternate n^2, 4*(n^2 + n + 1). Its first differences (4, -3, 11, -8, 24, ...) is the sequence of the square array for p=1.
Third column: 0, 3, 8, 15, ... is A005563(n).
Fifth column: 5, 21, 45, 77, ... is a bisection of A061037(n).
Seventh column: 7, 16, 40, 55, 91, 112, ... is a subsequence of A061039(n).
Etc. From the Rydberg spectra of the hydrogen atom (mentioned in A261327).
Starting for instance from p=-3,at the main antidiagonal,yields:
-3: -12, 0, -27, 3, ... see p=3
-2: -1, -8, -3, -16, -5, ... p=2
-1: -4, 3, -11, 8, -24, 19, ... p=1.

Crossrefs

Cf. A000290, A002061, A005563, A061037, A061039, A064680, A112087, A144433, A195161, A261327.

A281098 a(n) is the GCD of the sequence d(n) = A261327(k+n) - A261327(k) for all k.

Original entry on oeis.org

0, 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, 9, 1, 20, 3, 11, 1, 24, 1, 13, 3, 28, 1, 15, 1, 32, 3, 17, 1, 36, 1, 19, 3, 40, 1, 21, 1, 44, 3, 23, 1, 48, 1, 25, 3, 52, 1, 27, 1, 56, 3, 29, 1, 60, 1, 31, 3, 64, 1, 33, 1, 68, 3, 35, 1, 72, 1, 37, 3, 76, 1, 39, 1
Offset: 0

Views

Author

Paul Curtz, Jan 14 2017

Keywords

Comments

Successive sequences:
0: 0, 0, 0, 0, ... = 0 * ( )
1: 4, -3, 11, -8, ... = 1 * ( )
2: 1, 8, 3, 16, ... = 1 * ( ) A195161
3: 12, 0, 27, -3, ... = 3 * (4, 0, 9, -1, ...)
4: 4, 24, 8, 40, ... = 4 * (1, 6, 2, 10, ...) A064680
5; 28, 5, 51, 4, ... = 1 * ( )
6: 9, 48, 15, 72, ... = 3 * (3, 16, 5, 24, ...) A195161
7: 52, 12, 83, 13, ... = 1 * ( )
8: 16, 80, 24, 112, ... = 8 * (2, 10, 3, 14, ...) A064080
9: 84 21, 123, 24, ... = 3 * (28, 7, 41, 8, ...)
10: 25, 120, 35, 160, ... = 5 * (5, 24, 7, 32, ...) A195161

Links

Crossrefs

Cf. A064680, A144433 or A195161.
Cf. A000012, A005408, A008586, A010701, A109007 (bisection), A016825, A165988 (via A022998), A166138, A166304, A280579.

Programs

  • Mathematica
    CoefficientList[Series[(-x (-1 - x - 4 x^2 - 5 x^3 - 3 x^4 - 6 x^5 + 3 x^6 - 5 x^7 + 4 x^8 - x^9 + x^10))/((x^2 - x + 1) (1 + x + x^2) (x - 1)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 79}], x] (* Michael De Vlieger, Feb 02 2017 *)
  • PARI
    f(n) = numerator((4 + n^2)/4);
a(n) = gcd(vector(1000, k, f(k+n) - f(k))); \\ Michel Marcus, Jan 15 2017
  • PARI
    A281098(n) = if(n%2, gcd((n\2)-1,3), n>>(bitand(n,2)/2)); \\ Antti Karttunen, Feb 15 2023

Formula

G.f.: -x*( -1 - x - 4*x^2 - 5*x^3 - 3*x^4 - 6*x^5 + 3*x^6 - 5*x^7 + 4*x^8 - x^9 + x^10 )/( (x^2 - x + 1)*(1 + x + x^2)*(x - 1)^2*(1 + x)^2*(1 + x^2)^2 ). - R. J. Mathar, Jan 31 2017
a(2*k) = A022998(k).
a(2*k+1) = A109007(k-1).
a(3*k) = interleave 3*k*(3 +(-1)^k)/2, 3.
a(3*k+1) = interleave 1, A166304(k).
a(3*k+2) = interleave A166138(k), 1.
a(4*k) = 4*k.
a(4*k+1) = period 3: repeat [1, 1, 3].
a(4*k+2) = 1 + 2*k.
a(4*k+3) = period 3: repeat [3, 1, 1].
a(n+12) - a(n) = 6*A131743(n+3).
a(n) = (18*n + 40 - 16*cos(n*Pi/3) + 9*n*cos(n*Pi/2) + 32*cos(2*n*Pi/3) + (18*n - 40)*cos(n*Pi) + 3*n*cos(3*n*Pi/2) - 16*cos(5*n*Pi/3))/48. - Wesley Ivan Hurt, Oct 04 2018

Extensions

Corrected and extended by Michel Marcus, Jan 15 2017
Previous Showing 21-25 of 25 results.

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