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 1-4 of 4 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

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

A363284 Numbers that are square or square pyramidal.

Original entry on oeis.org

0, 1, 4, 5, 9, 14, 16, 25, 30, 36, 49, 55, 64, 81, 91, 100, 121, 140, 144, 169, 196, 204, 225, 256, 285, 289, 324, 361, 385, 400, 441, 484, 506, 529, 576, 625, 650, 676, 729, 784, 819, 841, 900, 961, 1015, 1024, 1089, 1156, 1225, 1240, 1296, 1369, 1444, 1496
Offset: 1

Views

Author

Clark Kimberling, May 25 2023

Keywords

Comments

This sequence essentially consists of the numbers in A363269 arranged in increasing order. Although A363269 is a linear recurrence sequence, it appears that this sequence is not.
4900 is the only nontrivial case of a square number that is also square pyramidal (proved by Watson). - Peter Munn, Jul 30 2023

References

  • W. Ljunggren, New solution of a problem proposed by E. Lucas, Norsk Mat. Tidsskr. 34 (1952), pp 65-72.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, entry 24, p 101.

Links

Crossrefs

Cf. A000290, A000330, A363269.

Programs

  • Mathematica
    c[1] = 1; c[2] = 1;
c[n_] := If[OddQ[n], c[n - 2] + n, c[n - 2] + c[n - 1]]
u = Table[c[n], {n, 1, 120}]  (* A363269 *)
FindLinearRecurrence[u]
Union[u] (* this sequence *)

Extensions

Name simplified and 0 prefixed to data by Peter Munn, Jul 30 2023
Showing 1-4 of 4 results.

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

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