# What is the closed form for this sequence, powers of $4$?

What is the closed form for this sequence:

1, 4, 12, 40, 148, 576, 2284, 9112, 36420, 145648, 582556, 2330184, 9320692, 37282720, 149130828, 596523256, 2386092964, 9544371792, 38177487100, 152709948328, ?

I know it has something to do with powers of $4$, possibly also powers of $2$. It turns up in an attempt to find an integral related to $\zeta(2) = \frac{\pi^2}{6}$.

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try oeis.org, but it seems does not work... – ougao Aug 11 '14 at 13:19
All but the last 3 terms are odd terms of oeis.org/A068018 but that may just be coincidence. – Jam Aug 11 '14 at 13:25
This is probably stupid, but Wolfram Alpha suggests a closed form being $$a_n = 1/36 (48 n+5 4^n-32) \text{ (for all terms given)}$$ – String Aug 11 '14 at 13:28
Thank you for the comments. – Mats Granvik Aug 11 '14 at 13:33
The ratios between the numbers fluctuates a bit in the start but then converges to $4$, here's the ratios: 4, 3, 3.33333, 3.7, 3.89189, 3.96528, 3.98949, 3.99693, 3.99912, 3.99975, 3.99993, 3.99998, 3.99999, 4, 4, 4, 4, 4, 4 – Kristoffer Ryhl Aug 11 '14 at 13:34

Since the first differences are given by A135583 i.e. $$a(n+1)-a(n)=\frac{4+5\cdot 4^n}3$$ we may conjecture (with some tuning) that $$a(n)=1+\frac 13\left(4n+5\frac{4^{n}-1}{4-1}\right)$$ which appears right.