Sums of consecutive triangular numbers — Part 2

 

Let   N_3   be the sums of 3 consecutive triangular numbers, and   N_6   the sums of 6 consecutive triangular numbers.

Let’s find   N_3   and   N_6   such that    N_3 - N_6 = 2

N_3 = T_{x-1} + T_{x} + T_{x+1}

N_6 = T_{y-2} + T_{y-1} + T_{y} + T_{y+1} + T_{y+2} + T_{y+3}

 
Here are the first few solutions:
 

85 = T_6 + T_7 + T_8
83 = T_2 + T_3 + T_4 + T_5 + T_6 + T_7

2710 = T_{41} + T_{42} + T_{43}
2708 = T_{27} + T_{28} + T_{29} + T_{30} + T_{31} + T_{32}

91885 = T_{246} + T_{247} + T_{248}
91883 = T_{172} + T_{173} + T_{174} + T_{175} + T_{176} + T_{177}

3121210 = T_{1441} + T_{1442} + T_{1443}
3121208 = T_{1017} + T_{1018} + T_{1019} + T_{1020} + T_{1021} + T_{1022}

106029085 = T_{8406} + T_{8407} + T_{8408}
106029083 = T_{5942} + T_{5943} + T_{5944} + T_{5945} + T_{5946} + T_{5947}

3601867510 = T_{49001} + T_{49002} + T_{49003}
3601867508 = T_{34647} + T_{34648} + T_{34649} + T_{34650} + T_{34651} + T_{34652}

122357466085 = T_{285606} + T_{285607} + T_{285608}
122357466083 = T_{201952} + T_{201953} + ... + T_{201957}

4156551979210 = T_{1664641} + T_{1664642} + T_{1664643}
4156551979208 = T_{1177077} + T_{1177078} + ... + T_{1177082}

141200409826885 = T_{9702246} + T_{9702247} + T_{9702248}
141200409826883 = T_{6860522} + T_{6860523} + ... + T_{6860527}

4796657382134710 = T_{56548841} + T_{56548842} + T_{56548843}
4796657382134708 = T_{39986067} + T_{39986068} + ... + T_{39986072}

162945150582753085 = T_{329590806} + T_{329590807} + T_{329590808}
162945150582753083 = T_{233055892} + T_{233055893} + ... + T_{233055897}

5535338462431470010 = T_{1920996001} + T_{1920996002} + T_{1920996003}
5535338462431470008 = T_{1358349297} + T_{1358349298} + ... + T_{1358349302}

188038562572087227085 = T_{11196385206} + T_{11196385207} + T_{11196385208}
188038562572087227083 = T_{7917039902} + T_{7917039903} + ... + T_{7917039907}

6387775788988534250710 = T_{65257315241} + T_{65257315242} + T_{65257315243}
6387775788988534250708 = T_{46143890127} + T_{46143890128} + ... + T_{46143890132}

216996338263038077296885 = T_{380347506246} + T_{380347506247} + T_{380347506248}
216996338263038077296883 = T_{268946300872} + T_{268946300873} + ... + T_{268946300877}

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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