## 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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