Square triangular number: Difference between revisions

Revision as of 06:04, 28 May 2024

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The [[generating function]] for the square triangular numbers is:<ref>{{cite web |first=Simon |last=Plouffe |author-link=Simon Plouffe |title=1031 Generating Functions |url=http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf |publisher=University of Quebec, Laboratoire de combinatoire et d'informatique mathématique |page=A.129 |date=August 1992 |access-date=2009-05-11 |archive-date=2012-08-20 |archive-url=https://web.archive.org/web/20120820012535/http://www.plouffe.fr/simon/articles/FonctionsGeneratrices.pdf |url-status=dead }}</ref>

:<math>\frac{1+z}{(1-z)\left(z^2 - 34z + 1\right)} = 1 + 36z + 1225 z^2 + \cdots</math>

==Numerical data==

As {{mvar|k}} becomes larger, the ratio {{math|{{sfrac|''t_k''|''s_k''}}}} approaches [[square root of 2|{{sqrt|2}}]] ≈&nbsp;{{val|1.41421356}}, and the ratio of successive square triangular numbers approaches {{nowrap|(1 + {{sqrt|2}})⁴}} {{nowrap|{{=}} 17 + 12{{sqrt|2}}}} ≈&nbsp;{{val|33.970562748}}. The table below shows values of {{mvar|k}} between 0 and 11, which comprehend all square triangular numbers up to {{val|e=16}}.

:{| class="wikitable" border="1" style="text-align:right"

|-

! {{mvar|k}}

! {{math|''N_k''}}

! {{math|''s_k''}}

! {{math|''t_k''}}

!rowspan=2 valign=top| {{math|{{sfrac|''t_k''|''s_k''}}}}

!rowspan=3 valign=top| {{math|{{sfrac|''N_k''|''N''_{''k'' − 1}}}}}

|-

|0

|0

|0

|0

|-

|1

|1

|1

|1

|align=left|1

|-

|2

|36

|6

|8

|align=left|{{val|1.33333333}}

|align=left|36

|-

|3

|{{val|1225|fmt=gaps}}

|35

|49

|align=left|1.4

|align=left|{{val|34.027777778}}

|-

|4

|{{val|41616}}

|204

|288

|align=left|{{val|1.41176471}}

|align=left|{{val|33.972244898}}

|-

|5

|{{val|1413721}}

|{{val|1189|fmt=gaps}}

|{{val|1681|fmt=gaps}}

|align=left|{{val|1.41379310}}

|align=left|{{val|33.970612265}}

|-

|6

|{{val|48024900}}

|{{val|6930|fmt=gaps}}

|{{val|9800|fmt=gaps}}

|align=left|{{val|1.41414141}}

|align=left|{{val|33.970564206}}

|-

|7

|{{val|1631432881}}

|{{val|40391}}

|{{val|57121}}

|align=left|{{val|1.41420118}}

|align=left|{{val|33.970562791}}

|-

|8

|{{val|55420693056}}

|{{val|235416}}

|{{val|332928}}

|align=left|{{val|1.41421144}}

|align=left|{{val|33.970562750}}

|-

|9

|{{val|1882672131025}}

|{{val|1372105}}

|{{val|1940449}}

|align=left|{{val|1.41421320}}

|align=left|{{val|33.970562749}}

|-

|10

|{{val|63955431761796}}

|{{val|7997214}}

|{{val|11309768}}

|align=left|{{val|1.41421350}}

|align=left|{{val|33.970562748}}

|-

|11

|{{val|2172602007770041}}

|{{val|46611179}}

|{{val|65918161}}

|align=left|{{val|1.41421355}}

|align=left|{{val|33.970562748}}

|}<!-- The table was generated in 23-jul-2016 using a Python script available at http://pastebin.com/sWyesrR8 -->

==See also==