Pairs of positive integers for which

*3 + 4 + 5 + 6 = 18* =

*15 + 16 + 17 + … + 34 + 35 = 525* =

*85 + 86 + 87 + … + 203 + 204 = 17340* =

*493 + 494 + 495 + … + 1189 = 586177* =

*2871 + 2872 + 2873 + … + 6930 = 19896030* =

*16731 + 16732 + 16733 + … + 40391 = 675781821* =

*97513 + 97514 + 97515 + … + 235416 = 22956120408* =

*568345 + 568346 + 568347 + … + 1372105 = 779829016225* =

*3312555 + 3312556 + 3312557 + … + 7997214*

= 26491211221770

=

*19306983 + 19306984 + 19306985 + … + 46611179*

= 899921240562957

=

*112529341 + 112529342 + 112529343 + … + 271669860*

= 30570830315362260

=

*655869061 + 655869062 + 655869063 + … + 1583407981*

= 1038508305678375841

=

*3822685023 + 3822685024 + 3822685025 + … + 9228778026*

= 35278711540581704598

=

*22280241075 + 22280241076 + 22280241077 + … + 53789260175*

= 1198437683944896688125

=

Find a recursion relation.

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## About benvitalis

math grad - Interest: Number theory

The first number’s in the sequences follow a recurrence of {7,-7,1}. The next number given the first 3 is as follows. given {3,15,85} we have (7 x 85) + (-7 x 15) + (1 x 3) = 493

The last number in the sequences follow a recrrence of {6,-1}, so given the first 2 numbers we get, (6 x 35) + (-1 x 6) = 204

There is also a recurrence for the resultant or (a x b) value, this is

{41,-246, 246, -41, 1}, so given {18,525,17340,586177, 19896030} we get

(1 x 18) + (-41 x 525) + (246 x 17340) + (-246 x 586177) + (41 x 19896030) = 255915594

Using those to find the next sequence gives

129858761425 + ….. + 313506783024 = 40711602541832856049200

here are the recurrence again

{7, -7, 1}..{6, -1}..{41,-246, 246, -41, 1}

Paul

That’s right!