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Basic Summations

These are the basic summations because I was able to derive them with trial and error and intuition so there is no great conceptual write up for them hence the name basic summations.

evens
oeis: A002378 [oblong numbers]
$t=2n$
$s=n(n+1)$
$s=n^2+n$

odds
oeis: A000290 [squares]
$t=2n-1$
$s=n^2$

every other even
oeis: A001105 [—]
$t=4n-2$
$s=2n^2$

every other odd
oeis: A000384 [hexagonal numbers]
$t=4n-3$
$s=n(2n-1)$
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every number
oeis: A000217 [triangular numbers]
$t=\frac{n}{1}$
$s=(\frac{n}{1})(\frac{n+1}{2})$
$s=\frac{n^2+n}{2}$

every other -> other other -> other other other ->…
oeis: A000292 [tetrahedral or triangular pyramidal numbers]
$t=(\frac{n}{1})(\frac{n+1}{2})$ or $t=A000217(n)$
$s=(\frac{n}{1})(\frac{n+1}{2})(\frac{n+2}{3})$
$s=\frac{n^3+3n^2+2n}{6}$

[series name]
oeis: A000332 [pentatope number]
$t=(\frac{n}{1})(\frac{n+1}{2})(\frac{n+2}{3})$ or $t=A000292(n)$
$s=(\frac{n}{1})(\frac{n+1}{2})(\frac{n+2}{3})(\frac{n+3}{4})$
$s=\frac{n^4+6n^3+11n^2+6n}{24}$

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