Sums of consecutive triangular numbers — Part 1

 

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_6 - N_3 = 1

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:
 

19 = T_2 + T_3 + T_4
20 = T_1 + T_2 + T_3 + T_4 + T_5 + T_6

199 = T_{10} + T_{11} + T_{12}
200 = T_5 + T_6 + T_7 + T_8 + T_9 + T_{10}

 

514 = T_{17} + T_{18} + T_{19}
515 = T_{10} + T_{11} + T_{12} + T_{13} + T_{14} + T_{15}

6634 = T_{65} + T_{66} + T_{67}
6635 = T_{44} + T_{45} + T_{46} + T_{47} + T_{48} + T_{49}

 

17335 = T_{106} + T_{107} + T_{108}
17336 = T_{73} + T_{74} + T_{75} + T_{76} + T_{77} + T_{78}

225235 = T_{386} + T_{387} + T_{388}
225236 = T_{271} + T_{272} + T_{273} + T_{274} + T_{275} + T_{276}

 

588754 = T_{625} + T_{626} + T_{627}
588755 = T_{440} + T_{441} + T_{442} + T_{443} + T_{445} + T_{446}

7651234 = T_{2257} + T_{2258} + T_{2259}
7651235 = T_{1594} + T_{1595} + T_{1596} + T_{1597} + T_{1598} + T_{1599}

 

20000179 = T_{3650} + T_{3651} + T_{3652}
20000180 = T_{2579} + T_{2580} + T_{2581} + T_{2582} + T_{2583} + T_{2584}

259916599 = T_{13162} + T_{13163} + T_{13164}
259916600 = T_{9305} + T_{9306} + T_{9307} + T_{9308} + T_{9309} + T_{9310}

 

679417210 = T_{21281} + T_{21282} + T_{21283}
679417211 = T_{15046} + T_{15047} + T_{15048} + T_{15049} + T_{15050} + T_{15051}

8829513010 = T_{76721} + T_{76722} + T_{76723}
8829513011 = T_{54248} + T_{54249} + T_{54250} + T_{54251} + T_{54252} + T_{54253}

 

23080184839 = T_{124042} + T_{124043} + T_{124044}
23080184840 = T_{87709} + T_{87710} + T_{87711} + T_{87712} + T_{87713} + T_{87714}

299943525619 = T_{447170} + T_{447171} + T_{447172}
299943525620 = T_{316195} + T_{316196} + T_{316197} + T_{316198} + T_{316199} + T_{316200}

 

784046867194 = T_{722977} + T_{722978} + T_{722979}
784046867195 = T_{511220} + T_{511221} + T_{511222} + T_{511223} + T_{511224} + T_{511225}

10189250357914 = T_{2606305} + T_{2606306} + T_{2606307}
10189250357915 = T_{1842934} + T_{1842935} + T_{1842936} + T_{1842937} + T_{1842938} + T_{1842939}

 

26634513299635 = T_{4213826} + T_{4213827} + T_{4213828}
26634513299636 = T_{2979623} + T_{2979624} + T_{2979625} + T_{2979626} + T_{2979627} + T_{2979628}

346134568643335 = T_{15190666} + T_{15190667} + T_{15190668}
346134568643336 = T_{10741421} + T_{10741422} + ... + T_{10741426}

 

904789405320274 = T_{24559985} + T_{24559986} + T_{24559987}
904789405320275 = T_{17366530} + T_{17366531} + ... + T_{17366535}

11758386083515354 = T_{88537697} + T_{88537698} + T_{88537699}
11758386083515355 = T_{62605604} + T_{62605605} + ... + T_{62605609}

……………………….
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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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