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 1-3 of 3 results.

A363267 Squares (A000290) alternating with centered squares (A001844).

Original entry on oeis.org

1, 1, 4, 5, 9, 13, 16, 25, 25, 41, 36, 61, 49, 85, 64, 113, 81, 145, 100, 181, 121, 221, 144, 265, 169, 313, 196, 365, 225, 421, 256, 481, 289, 545, 324, 613, 361, 685, 400, 761, 441, 841, 484, 925, 529, 1013, 576, 1105, 625, 1201, 676, 1301, 729, 1405, 784
Offset: 1

Views

Author

Clark Kimberling, May 24 2023

Keywords

Comments

This is a linear recurrence sequence. If the terms are arranged in nondecreasing order, the result, A363319, is linearly recurrent. If the terms are arranged in increasing order, so that there are no duplicates, the result, A363282, is not linearly recurrent.

Links

Crossrefs

Cf. A000290, A001844, A123596, A363268, A363269, A363282, A363319.

Programs

  • Mathematica
    c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, 2 c[n - 1] - n + 1]
Table[c[n], {n, 1, 120}]

Formula

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
G.f.: x*(-1 - x - x^2 - 2 x^3 - x^5)/(-1 + x^2)^3.
a(n+1) = n/2+3*n^2/8+3/4+(-1)^n*(1/4+n/2-n^2/8). - R. J. Mathar, Jun 15 2023

A363269 Positive squares (A000290) alternating with positive square pyramidal numbers (A000330).

Original entry on oeis.org

1, 1, 4, 5, 9, 14, 16, 30, 25, 55, 36, 91, 49, 140, 64, 204, 81, 285, 100, 385, 121, 506, 144, 650, 169, 819, 196, 1015, 225, 1240, 256, 1496, 289, 1785, 324, 2109, 361, 2470, 400, 2870, 441, 3311, 484, 3795, 529, 4324, 576, 4900, 625, 5525, 676, 6201, 729
Offset: 1

Views

Author

Clark Kimberling, May 24 2023

Keywords

Links

Crossrefs

Cf. A000290, A000330, A123596, A363267, A363268.
Range of terms: A363284\{0}.

Programs

  • Mathematica
    c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, c[n - 2] + c[n - 1]]
Table[c[n], {n, 1, 120}]
  • PARI
    a(n) = if(n%2, (n+1)^2/4, n*(n+1)*(n+2)/24); \\ Kevin Ryde, Jun 10 2023

Formula

a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8).
G.f.: x*(1 + x + x^3 - x^4)/(-1 + x^2)^4.
E.g.f.: ((18*x + 6*x^2)*cosh(x) + (6 + 6*x + 6*x^2 + x^3)*sinh(x))/24. - Stefano Spezia, Jun 10 2023
48*a(n) = (n+1) * (n^2 +(-1)^n*n^2 -4*(-1)^n*n +8*n -6*(-1)^n +6). - R. J. Mathar, Jun 22 2023

A363283 Squares (A000290) and (1+squares) (A002522), in increasing order.

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 16, 17, 25, 26, 36, 37, 49, 50, 64, 65, 81, 82, 100, 101, 121, 122, 144, 145, 169, 170, 196, 197, 225, 226, 256, 257, 289, 290, 324, 325, 361, 362, 400, 401, 441, 442, 484, 485, 529, 530, 576, 577, 625, 626, 676, 677, 729, 730, 784, 785
Offset: 1

Views

Author

Clark Kimberling, May 25 2023

Keywords

Comments

This sequence consists of the numbers in A363268 arranged in increasing order. This sequence and A363268 have the same linear recurrence (in contrast to these pairs: A363267 and A363282; and A363269 and A363283).

Links

Crossrefs

Cf. A000290, A002522, A363267, A363282, A363268.

Programs

  • Mathematica
    c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, c[n - 1] - n + 2]
u = Table[c[n], {n, 1, 120}] (* A363268 *)
Union[u]   (* this sequence *)

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: x*(-1 - x + x^3 - x^4)/((-1 + x)^3 (1 + x)^2).
a(n) = ((2n^2 + 2n + 5) - (2n - 3)*(-1)^n)/8. - Aaron J Grech, Aug 26 2024
E.g.f.: ((4 + 3*x + x^2)*cosh(x) + (1 + x + x^2)*sinh(x) - 4)/4. - Stefano Spezia, Aug 27 2024

Extensions

Definition corrected by N. J. A. Sloane, Jun 12 2023
Showing 1-3 of 3 results.

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