The On-Line Encyclopedia of Integer Sequences (OEIS)

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A033996

8 times triangular numbers: a(n) = 4*n*(n+1).
(history; published version)

#214 by Charles R Greathouse IV at Fri Jul 26 21:16:31 EDT 2024

COMMENTS

The number of active (ON, black) cells in the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood. - Robert Price, May 19 2016

Discussion

Fri Jul 26

21:16

OEIS Server: https://oeis.org/edit/global/2997

#213 by Michel Marcus at Tue Feb 21 02:14:49 EST 2023

STATUS

reviewed

approved

#212 by Joerg Arndt at Tue Feb 21 02:05:23 EST 2023

STATUS

proposed

reviewed

#211 by Amiram Eldar at Tue Feb 21 01:05:05 EST 2023

STATUS

editing

proposed

#210 by Amiram Eldar at Tue Feb 21 00:52:19 EST 2023

LINKS

Leo Tavares, <a href="/A033996/a033996.jpg">Illustration: Centroid Diamonds</a>.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KnightGraphHamiltonianPath.html">Knight GraphHamiltonian Path</a>.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPathKnightGraph.html">Hamiltonian PathKnight Graph</a>.

#209 by Amiram Eldar at Tue Feb 21 00:51:19 EST 2023

LINKS

<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

#208 by Amiram Eldar at Tue Feb 21 00:50:36 EST 2023

REFERENCES

S. Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

LINKS

S. Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

<a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>.

<a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>.

#207 by Amiram Eldar at Tue Feb 21 00:49:41 EST 2023

FORMULA

From Amiram Eldar, Feb 21 2023: (Start)

Product_{n>=1} (1 - 1/a(n)) = -(4/Pi)*cos(Pi/sqrt(2)).

Product_{n>=1} (1 + 1/a(n)) = 4/Pi (A088538). (End)

CROSSREFS

Cf. A000217, A016754, A002378, A024966, A027468, A028895, A028896, A045943, A046092, A049598, A088538, A124080, A008590 (first differences), A130809 (partial sums).

STATUS

approved

editing

#206 by Michael De Vlieger at Sat Mar 05 23:01:39 EST 2022

STATUS

proposed

approved

#205 by Jon E. Schoenfield at Sat Mar 05 20:18:18 EST 2022

STATUS

editing

proposed