The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A002024 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A002024

k appears k times; a(n) = floor(sqrt(2n) + 1/2).
(history; published version)

#292 by Stefano Spezia at Sun Oct 27 15:52:53 EDT 2024

CROSSREFS

Cf. A001462, A002262, A003881, A004736, A127899, A107985, A001563, A014132, A000194, A005145, A131507, A093995, A060432 (partial sums), A375797.

KEYWORD

nonn,easy,nice,tabl,changed

STATUS

proposed

approved

#291 by Boris Putievskiy at Thu Oct 17 14:19:06 EDT 2024

STATUS

editing

proposed

#290 by Boris Putievskiy at Thu Oct 17 14:18:30 EDT 2024

LINKS

Boris Putievskiy, <a href="https://arxiv.org/abs/2310.18466">Integer Sequences: Irregular Arrays and Intra-Block Permutations</a>, arXiv:2310.18466 [math.CO], 2023.

FORMULA

From Boris Putievskiy, Oct 17 2024: (Start)

The general formula for sequence n appears p1*n + p0 times, where p1 and p0 are integer numbers, p1 > 0: a(n) = ceiling((-2*p0 - p1 + sqrt(8*n*p1 + (2*p0 + p1)^2)) / (2*p1)).

The general formula for p0 = 0 see A375797 a(n) = ceiling((- p1 + sqrt(8*n*p1 + p1^2)) / (2*p1)).

Here are some special cases:

p1 = 1, p0 = 0 this sequence; p1 = 1, p0 = 1 A003056; p1 = 2, p0 = 0 A000194; p1 = 2, p0 = 2 A259361; p1 = 3, p0 = 0 A111651; p1 = 4, p0 = -1 A204164. (End)

CROSSREFS

Cf. A001462, A002262, A003881, A004736, A127899, A107985, A001563, A014132, A000194, A005145, A131507, A093995, A060432 (partial sums), A111651, A204164, A259361, A375797.

STATUS

proposed

editing

#289 by Boris Putievskiy at Thu Oct 17 13:53:29 EDT 2024

STATUS

editing

proposed

Discussion

Thu Oct 17

14:07

Andrew Howroyd: p0, p1 suggest primes, so I think these are not the best letters to use. (also why not just m and k which is one letter).
However, I'm inclined to think these kind of generalized formulas don't fit well into linear sequences.
The cross-referencing to other similar sequences seems undesirable - Should each of these sequences xref each other? and who will update all the lists when a similar sequence is added in the future? It's not an indexing strategy that can work in practical terms.
Perhaps just leave the link and nothing else?

14:10

Andrew Howroyd: Does you arxiv even mention any new formulas that are not mentioned in the other references? It seems to me this formula you are adding is very well known.

#288 by Boris Putievskiy at Thu Oct 17 13:51:42 EDT 2024

LINKS

Boris Putievskiy, <a href="https://arxiv.org/abs/2310.18466">Integer Sequences: Irregular Arrays and Intra-Block Permutations</a>, arXiv:2310.18466 [math.CO], 2023.

FORMULA

From Boris Putievskiy, Oct 17 2024: (Start)

The general formula for sequence n appears p1*n + p0 times, where p1 and p0 are integer numbers, p1 > 0: a(n) = ceiling((-2*p0 - p1 + sqrt(8*n*p1 + (2*p0 + p1)^2)) / (2*p1)).

The general formula for p0 = 0 see A375797 a(n) = ceiling((- p1 + sqrt(8*n*p1 + p1^2)) / (2*p1)).

Here are some special cases:

p1 = 1, p0 = 0 this sequence; p1 = 1, p0 = 1 A003056; p1 = 2, p0 = 0 A000194; p1 = 2, p0 = 2 A259361; p1 = 3, p0 = 0 A111651; p1 = 4, p0 = -1 A204164. (End)

CROSSREFS

Cf. A001462, A002262, A003881, A004736, A127899, A107985, A001563, A014132, A000194, A005145, A131507, A093995, A060432 (partial sums), A111651, A204164, A259361, A375797.

STATUS

approved

editing

#287 by N. J. A. Sloane at Wed Apr 24 12:56:54 EDT 2024

STATUS

proposed

approved

#286 by Stefano Spezia at Mon Apr 22 15:57:18 EDT 2024

STATUS

editing

proposed

#285 by Stefano Spezia at Mon Apr 22 15:35:39 EDT 2024

FORMULA

G.f. as array: (x^2*(1 - y)^2 + y^2 + x*y*(1 - 2*y))/((1 - x)^2*(1 - y)^2). - Stefano Spezia, Apr 22 2024

STATUS

approved

editing

#284 by Michael De Vlieger at Tue Feb 27 10:58:03 EST 2024

STATUS

proposed

approved

#283 by Michel Marcus at Tue Feb 27 10:34:27 EST 2024

STATUS

editing

proposed