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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A303302 a(n) = 34*n^2.

Original entry on oeis.org

0, 34, 136, 306, 544, 850, 1224, 1666, 2176, 2754, 3400, 4114, 4896, 5746, 6664, 7650, 8704, 9826, 11016, 12274, 13600, 14994, 16456, 17986, 19584, 21250, 22984, 24786, 26656, 28594, 30600, 32674, 34816, 37026, 39304, 41650, 44064, 46546, 49096, 51714, 54400, 57154, 59976, 62866, 65824, 68850, 71944
Offset: 0

Views

Author

Omar E. Pol, May 13 2018

Keywords

Comments

Sequence found by reading the line from 0, in the direction 0, 34, ..., in the square spiral whose vertices are the generalized 19-gonal numbers A303813.

Links

Crossrefs

Cf. A005843, A008599, A303813.
Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30), A244082 (k=32), this sequence (k=34), A016910 (k=36), A016982 (k=49), A017066 (k=64), A017162 (k=81), A017270 (k=100), A017390 (k=121), A017522 (k=144).

Programs

  • Magma
    [34*n^2: n in [0..50]]; // Vincenzo Librandi Jun 07 2018
  • Mathematica
    Table[34 n^2, {n, 0, 40}]
LinearRecurrence[{3,-3,1},{0,34,136},50] (* Harvey P. Dale, Jul 23 2018 *)
  • PARI
    a(n) = 34*n^2;
  • PARI
    concat(0, Vec(34*x*(1 + x) / (1 - x)^3 + O(x^40))) \\ Colin Barker, Jun 12 2018

Formula

a(n) = 34*A000290(n) = 17*A001105(n) = 2*A244630(n).
G.f.: 34*x*(1 + x)/(1 - x)^3. - Vincenzo Librandi, Jun 07 2018
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 34*x*(1 + x)*exp(x).
a(n) = A005843(n)*A008599(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A330133 a(n) = (1/16)*(5 + (-1)^(1+n) - 4*cos(n*Pi/2) + 10*n^2).

Original entry on oeis.org

0, 1, 3, 6, 10, 16, 23, 31, 40, 51, 63, 76, 90, 106, 123, 141, 160, 181, 203, 226, 250, 276, 303, 331, 360, 391, 423, 456, 490, 526, 563, 601, 640, 681, 723, 766, 810, 856, 903, 951, 1000, 1051, 1103, 1156, 1210, 1266, 1323, 1381, 1440, 1501, 1563, 1626, 1690, 1756
Offset: 0

Views

Author

Stefano Spezia, Dec 02 2019

Keywords

Comments

For n > 0, partial sums of A047201.

Links

Crossrefs

Cf. A005891, A033583 (10*n^2), A047201.

Programs

  • Magma
    I:=[0, 1, 3, 6, 10, 16]; [n le 6 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-4)-2*Self(n-5)+Self(n-6): n in [1..54]];
  • Maple
    gf:=(1/16)*(-exp(-x) + 5*exp(x)*(1 + 2*x + 2*x^2) - 4*cos(x)); ser := series(gf, x, 54):
seq(factorial(n)*coeff(ser, x, n), n = 0 .. 53)
  • Mathematica
    Table[(1/16)*(5+(-1)^(1+n)-4*Cos[n*Pi/2]+10*n^2),{n,0,53}]
LinearRecurrence[{2,-1,0,1,-2,1},{0,1,3,6,10,16},60] (* Harvey P. Dale, Jul 21 2021 *)
  • PARI
    concat([0], Vec(-x*(1 + x + x^2 + x^3 + x^4)/((-1 + x)^3*(1 + x)*(1 + x^2))+O(x^54)))

Formula

O.g.f.: -x*(1 + x + x^2 + x^3 + x^4)/((-1 + x)^3*(1 + x)*(1 + x^2)).
E.g.f.: (1/16)*(-exp(-x) + 5*exp(x)*(1 + 2*x + x^2) - 4*cos(x)).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) -2*a(n-5) + a(n-6) for n > 5.
a(2*n-1) = A005891(n-1) for n > 0.
a(4*n) = 10*n^2. - Bernard Schott, Dec 06 2019

A363436 Array read by ascending antidiagonals: A(n, k) = k*n^2, with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 2, 0, 0, 9, 8, 3, 0, 0, 16, 18, 12, 4, 0, 0, 25, 32, 27, 16, 5, 0, 0, 36, 50, 48, 36, 20, 6, 0, 0, 49, 72, 75, 64, 45, 24, 7, 0, 0, 64, 98, 108, 100, 80, 54, 28, 8, 0, 0, 81, 128, 147, 144, 125, 96, 63, 32, 9, 0, 0, 100, 162, 192, 196, 180, 150, 112, 72, 36, 10, 0
Offset: 0

Views

Author

Stefano Spezia, Jul 08 2023

Keywords

Examples

			The array begins:
  0,  0,  0,   0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5,   6, ...
  0,  4,  8,  12,  16,  20,  24, ...
  0,  9, 18,  27,  36,  45,  54, ...
  0, 16, 32,  48,  64,  80,  96, ...
  0, 25, 50,  75, 100, 125, 150, ...
  0, 36, 72, 108, 144, 180, 216, ...
  ...

Crossrefs

Cf. A000290 (k = 1), A001105 (k = 2), A033428 (k = 3), A016742 (k = 4), A033429 (k = 5), A033581 (k = 6), A033582 (k = 7), A139098 (k = 8), A016766 (k = 9), A033583 (k = 10), A033584 (k = 11), A135453 (k = 12), A152742 (k = 13), A144555 (k = 14), A064761 (k = 15), A016802 (k = 16), A244630 (k = 17), A195321 (k = 18), A244631 (k = 19), A195322 (k = 20), A064762 (k = 21), A195323 (k = 22), A244632 (k = 23), A195824 (k = 24), A016850 (k = 25), A244633 (k = 26), A244634 (k = 27), A064763 (k = 28), A244635 (k = 29), A244636 (k = 30).
Cf. A001477 (n = 1), A008586 (n = 2), A008591 (n = 3), A008598 (n = 4), A008607 (n = 5), A044102 (n = 6), A152691 (n = 8).
Cf. A000007 (n = 0 or k = 0), A000578 (main diagonal), A002415 (antidiagonal sums), A004247.

Programs

  • Mathematica
    A[n_,k_]:=k n^2; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

Formula

O.g.f.: x*y*(1 + x)/((1 - x)^3*(1 - y)^2).
E.g.f.: x*y*(1 + x)*exp(x + y).
A(n, k) = n*A004247(n, k).
Previous Showing 31-33 of 33 results.

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