perplexus.info :: Sequences : Square Crossed Sequence

Square Crossed Sequence (Posted on 2013-12-09)

A sequence {B_p} of positive integers is such that:

B₁ = 20, B₂ = 30, and:

B_p+2 = 3*B_p+1 – B_p, whenever p ≥ 1.

Determine all possible positive integer values of p such that:
1 + 5*B_p+1*B_p is a perfect square.

**** For an extra challenge, derive a non computer program assisted solution.

No Solution Yet Submitted by K Sengupta

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many solutions

| Comment 2 of 3 |

B(p+2) = 3*30 - 20 = 70

B(p+2) = 70, B(p+3) = x, a square y^2

1 + 5*70*x = y^2

Integer solutions:

x = 2(175*n^2 - 349 n + 174), y = 349 - 350*n

x = 2(175*n^2 - 251 n + 90), y = 251 - 350*n

x = 2(175*n^2 - 99 n + 14), y = 99 - 350*n

x = 2(175*n^2 - n), y = 1 - 350*n

n is an element of Z

n = 0

in y = 1 - 350*n ---> y = 1 => x = 0

in y = 99 - 350*n --> y = 99 => x = 28

in y = 251 - 350*n ---> y = 251 => x = 180

in y = 349 - 350*n ---> y = 349 => x = 348

n = 1

we get negative numbers

n < 0

n = -1

y = 351, x = 352

y = 449, x = 201601

y = 601, x = 1032

y = 699, x = 1396

etc.

Posted by Benny on 2013-12-09 19:21:26

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