cp's OEIS Frontend

Guide to related sequences:

sequence summand approximation minimal polynomial

(a(n)) sin(k*Pi/5)/2^k 0.6664488708 5 - 65*x^2 + 121*x^4

A373022 sin(2*k*Pi/5)/2^k 0.5053526528 5 - 265*x^2 + 961*x^4

A373023 sin(3*k*Pi/5)/2^k 0.3050180080 5 - 65*x^2 + 121*x^4

A373024 sin(4*k*Pi/5)/2^k 0.1427344344 5 - 265*x^2 + 961*x^4

A373025 cos(k*Pi/5)/2^k 1.3503729060 11 - 23*x + 11*x^2

A373026 cos(2*k*Pi/5)/2^k 0.8985194182 19 - 49*x + 31*x^2

A373027 cos(3*k*Pi/5)/2^k 0.7405361848 11 - 23*x + 11*x^2

A373028 cos(4*k*Pi/5)/2^k 0.6821257430 19 - 49*x + 31*x^2

Many runs of consecutive integers appear in the sequence. The first ones are [2,3,4,5,6], [8,9,10], [12,13,14], [20,21], [24,25], [28,29,30], ...

The scatterplot shows an interesting structure where distinct sets of terms can be seen on three straight lines. The greater the slope of the line, the higher the density of terms. The remaining terms are more randomly distributed between the three lines.

More detailed observations:

For the line of lowest slope, the term and term index parity alternates from one term to the next and if the term index is even, the term is odd and reciprocally. Terms of the same set and their indices are in arithmetic progression of respective common difference of 1 and 15.

For the line of medium slope, all term indices are even and the term parity alternates from one term to the next. Terms of the same set and their indices are in arithmetic progression of respective common difference 1 and 4.

For the line of greatest slope, if the term index is even, then the term is odd and reciprocally. All terms are equal to their index + 1.

There are no fixed points a(n) = n since 3*n^2+2*n + n^2 = (2*n)^2 + 2*n falls between (2*n)^2 and (2*n+1)^2, so not square.

The scatterplot shows an interesting crosshatch structure where all terms are at the intersection of ascending and descending hatches.

Terms on each hatch are quite well fitted by a polynomial of degree 2.

For terms on ascending hatches, the parity of the term indices does not change on a given hatch but alternates from one hatch to the next and on the same hatch, the parity of two consecutive terms alternates.

For terms on descending hatches, the parity of the indices of two consecutive terms alternates on the same hatch and that of terms does not change on the same hatch but alternates from one hatch to the next.

All squares exclusively are in ascending order on the same ascending hatch at n = 6, 10, 14, 18, 22, ... but some squares can be also found at the intersection of other hatches.

The first differences of the indices of the terms located on ascending and descending hatches are respectively equal to 4 and 3. For terms that are on the ascending and descending hatches, the differences of order 2 quickly become constant and equal to 2 and 4, respectively.

The fixed points begin 3, 48, 675, 9408, etc. They are all divisible by 3 and their parity seems to alternate. It appears that they are the positive terms of A007654.