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 1-5 of 5 results.

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

A363282 Squares (A000290) and centered squares (A001844), in increasing order (i.e., sorted and without duplicates).

Original entry on oeis.org

1, 4, 5, 9, 13, 16, 25, 36, 41, 49, 61, 64, 81, 85, 100, 113, 121, 144, 145, 169, 181, 196, 221, 225, 256, 265, 289, 313, 324, 361, 365, 400, 421, 441, 481, 484, 529, 545, 576, 613, 625, 676, 685, 729, 761, 784, 841, 900, 925, 961, 1013, 1024, 1089, 1105
Offset: 1

Views

Author

Clark Kimberling, May 25 2023

Keywords

Comments

This sequence consists of the numbers in A363267 arranged in increasing order. Unlike A363267, this is not a linear recurrence sequence; see A363319.

Crossrefs

Cf. A000290, A001844, A363267, A363319.

Programs

  • Mathematica
    c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, 2 c[n - 1] - n + 1]
u = Table[c[n], {n, 1, 120}]  (* A363267 *)
Union[u] (* A363282 *)

Extensions

Definition corrected by N. J. A. Sloane, Jun 12 2023

A363268 Squares (A000290) alternating with 1+squares (A002522).

Original entry on oeis.org

1, 1, 4, 2, 9, 5, 16, 10, 25, 17, 36, 26, 49, 37, 64, 50, 81, 65, 100, 82, 121, 101, 144, 122, 169, 145, 196, 170, 225, 197, 256, 226, 289, 257, 324, 290, 361, 325, 400, 362, 441, 401, 484, 442, 529, 485, 576, 530, 625, 577, 676, 626, 729, 677, 784, 730, 841
Offset: 1

Views

Author

Clark Kimberling, May 24 2023

Keywords

Links

Crossrefs

Cf. A000290, A002522, A123596, A363267, A363269, A363283.

Programs

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

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: (-1 - x^2 + 2 x^3 - 2 x^4)/((-1 + x)^3 (1 + x)^2).

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

A363319 Squares (A000290) and centered squares (A001844) sorted, including duplicates.

Original entry on oeis.org

1, 1, 4, 5, 9, 13, 16, 25, 25, 36, 41, 49, 61, 64, 81, 85, 100, 113, 121, 144, 145, 169, 181, 196, 221, 225, 256, 265, 289, 313, 324, 361, 365, 400, 421, 441, 481, 484, 529, 545, 576, 613, 625, 676, 685, 729, 761, 784, 841, 841, 900, 925, 961, 1013, 1024
Offset: 1

Views

Author

Clark Kimberling, May 27 2023

Keywords

Comments

This sequence consists of the numbers in A363267 arranged in nondecreasing order, including duplicates, which are given by A008844 = (1, 25, 841, 28561, ...).

Crossrefs

Cf. A000290, A001844, A363267, A363282.

Programs

  • Mathematica
    c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, 2 c[n - 1] - n + 1]
u = Table[c[n], {n, 1, 120}]  (* A363267 *)
s = Sort[u] (* this sequence *)

Extensions

Incorrect recurrence removed by Georg Fischer, Aug 14 2023
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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