cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A373021 Decimal expansion of Sum_{k>=0} sin(k*Pi/5)/2^k.

Original entry on oeis.org

6, 6, 6, 4, 4, 8, 8, 7, 0, 8, 1, 2, 3, 1, 3, 9, 1, 4, 8, 6, 1, 6, 3, 5, 7, 3, 2, 8, 5, 0, 1, 7, 8, 6, 5, 3, 2, 0, 0, 7, 9, 1, 7, 4, 2, 0, 3, 2, 8, 9, 7, 8, 9, 4, 2, 0, 2, 0, 7, 7, 9, 5, 1, 1, 1, 4, 9, 3, 4, 8, 6, 5, 9, 3, 7, 7, 1, 6, 8, 8, 6, 5, 3, 8, 7, 4
Offset: 0

Views

Author

Clark Kimberling, Jun 09 2024

Keywords

Comments

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

Examples

			0.666448870812313914861635732850178653200791742032...

Crossrefs

Cf. A373021, A373022, A373023, A373024, A373025, A373026, A373027, A373028.

Programs

  • Mathematica
    {b, m, h} = {2, 5, 1}; s = Sum[Sin[ h  k  Pi/m]/b^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d], 100]

Formula

Equals sqrt(10 - 2*sqrt(5)) / (8 - 2*sqrt(5)).
Equals (-1)*Sum_{k>=0} sin(9*k*Pi/5)/2^k.
Peter J. C. Moses (May 22 2024) found the following generalized summation identities for the eight sequences in Comments and many other sequences:
Sum_{k>=0} sin(h*k + Pi/m)/b^(k+r) = b^(1-r)*(b*sin(Pi/m) + sin(h - Pi/m)/(1 + b^2 - 2*b*cos*(Pi/m)).
Sum_{k>=0} cos(h*k + Pi/m)/b^(k+r) = b^(1-r)*(b*cos(Pi/m) + cos(h - Pi/m)/(1 + b^2 - 2*b*cos*(Pi/m)).

A373026 a(n) is the least positive integer k such that 3*n^2 + 2*n - k is a square.

Original entry on oeis.org

1, 7, 8, 7, 4, 20, 17, 12, 5, 31, 24, 15, 4, 40, 29, 16, 1, 47, 32, 15, 69, 52, 33, 12, 76, 55, 32, 7, 81, 56, 29, 111, 84, 55, 24, 116, 85, 52, 17, 119, 84, 47, 8, 120, 81, 40, 160, 119, 76, 31, 161, 116, 69, 20, 160, 111, 60, 7, 157, 104, 49, 207, 152, 95, 36, 204, 145, 84, 21, 199, 136, 71
Offset: 1

Views

Author

Claude H. R. Dequatre, May 20 2024

Keywords

Comments

The scatterplot shows an interesting structure where terms are on descending hatches.
Terms on each hatch are quite well fitted by a polynomial of degree 2.
The parity of the term indices alternates from one hatch to the next and that of two consecutive terms alternates on the same hatch.
For terms on a given hatch, the differences of order 2 quickly become constant and equal to 2.
The fixed points begin 1, 16, 225, 3136, etc. They appear to be all squares and to come from A098301.

Examples

			a(1) = 1 because 3*1^2 + 2*1 = 5 and 5-1 is a square. So, 1 is a term.
a(2) = 7 because 3*2^2 + 2*2 = 16 and 16-1, 16-2, 16-3, 16-4, 16-5, 16-6 are not squares, but 16-7 is. So, 7 is a term.

Crossrefs

Cf. A000005, A000290, A005408.
Cf. A373016, A373028.

Programs

  • PARI
    a(n) = my(m=3*n^2+2*n-1); m+1-sqrtint(m)^2; \\ Michel Marcus, May 20 2024

A373016 a(n) is the least positive integer k such that 3*n^2 + 2*n + k is a square.

Original entry on oeis.org

4, 9, 3, 8, 15, 1, 8, 17, 28, 4, 15, 28, 43, 9, 24, 41, 60, 16, 35, 56, 4, 25, 48, 73, 11, 36, 63, 92, 20, 49, 80, 113, 31, 64, 99, 9, 44, 81, 120, 20, 59, 100, 143, 33, 76, 121, 3, 48, 95, 144, 16, 65, 116, 169, 31, 84, 139, 196, 48, 105, 164, 8, 67, 128, 191, 25, 88, 153, 220, 44, 111, 180
Offset: 1

Views

Author

Claude H. R. Dequatre, May 20 2024

Keywords

Comments

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.

Examples

			a(1) = 4 because 3*1^2 + 2*1 = 5 and 5 + 1, 5 + 2, 5 + 3 are not squares, but 5 + 4 is. So, 4 is a term.
a(2) = 9 because 3*2^2 + 2*2 = 16 and 16 + 1, 16 + 2, 16 + 3, 16 + 4, 16 + 5, 16 + 6, 16 + 7, 16 + 8 are not squares, but 16 + 9 is. So, 9 is a term.

Crossrefs

Cf. A000005, A000290, A005408.
Cf. A045944, A080883.
Cf. A373026, A373028.
Sequences with similar scatterplot and pin plot graphs: A141130, A141131, A141134, A141135.

Programs

Formula

a(n) is the smallest square greater than 3*n^2 + 2*n, minus 3*n^2 + 2*n. - Charles R Greathouse IV, May 21 2024
1 <= a(n) <= floor(sqrt(12)*n) + 3. I believe both bounds are tight infinitely often. - Charles R Greathouse IV, May 21 2024
a(n) = A080883(A045944(n)). - Michel Marcus, May 22 2024
Showing 1-3 of 3 results.

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