cp's OEIS Frontend

The golden ratio phi is the real part of 2*exp(i*Pi/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron and dodecahedron).

Note that c^2+A001622^2 = 4; c*A001622 = A188593 = 2*A019881; c = 2*A019845.

Edge length of a regular pentagon with unit circumradius. - Stanislav Sykora, May 07 2014

This is a constructible number (see A003401 for more details). Moreover, since phi is also constructible, (2^k)*exp(i*Pi/5), for any integer k, is a constructible complex number. - Stanislav Sykora, May 02 2016

rms(c, phi) := sqrt((c^2+phi^2)/2) = sqrt(2) = A002193.

1 9 11 19 21 29 31 39... (A090771)

2 8 12 18 22 28 32 38... (A090772)

3 7 13 17 23 27 33 37... (A063226)

4 6 14 16 24 26 34 36... (A090773)

5 10 15 20 25 30 35 40... (A008587, excluding initial term)

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For such arrays A_k, here A_5, see a W. Lang comment on A113807, the A_7 case. However, in order to obtain A_5 one should take the last row as the first one after adding a 0 in front (thus getting a permutation of the nonnegative integers). - Wolfdieter Lang, Feb 02 2012

Also the decimal expansion of cotangent of 54 degrees. - Mohammad K. Azarian, Jun 30 2013

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

A quasipolynomial of order two and degree two: a(n) = 25n^2 - 30n + 9 if n is even and 25n^2 - 20n + 4 if n is odd. - Charles R Greathouse IV, Nov 03 2021

Equivalently, numbers k such that k and (7^h)^k end with the same digit, where h == 1 (mod 4).

Also, numbers k such that k and (3^h)^k end with the same digit, where h == 3 (mod 4).

Numbers congruent to {3, 17} mod 20. - Amiram Eldar, Feb 27 2023

For any m >= 0, if F(m) = 2^(2^m) + 1 has a factor of the form b = a(n)*2^k + 1 with even k >= m + 2 and n >= 1, then the cofactor of F(m) is equal to F(m)/b = j*2^k + 1, where j is congruent to 7 mod 10 if n is odd, or j is congruent to 3 mod 10 if n is even. That is, the integer a(n) + j must be divisible by 10.

In A302717, if we count the terms added from each 4-tuple during each iteration we find that either two or three terms are added: 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, ... where the set of three twos (2, 2, 2) appears with decreasing frequency. a(n) is the index into A302717 where each such set begins.

40*a(n) + 169 is a square, whose square root belongs to A063226 and A063206. - Bruno Berselli, Apr 17 2018

When a square ends with 9, it ends with only one 9.

From Marius A. Burtea, Nov 02 2021 : (Start)

The sequence is infinite because the numbers 303, 3003, 30003, ..., 3*(10^k + 1), k >= 2, are terms with squares 91809, 9018009, 900180009, 90001800009, ... 9*(10^(2*k) + 2*10^k + 1), k >= 2.

Numbers 97, 967, 9667, 96667, 966667, ..., (29*10^n + 1) / 3, k >= 1, are terms and have no digits 0, because their squares are 9409, 935089, 93450889, 9344508889, 934445088889, ...

Also 963, 9663, 96663, 966663, 9666663, 96666663, ... (29*10^k - 11) / 3, k >= 2, are terms and have no digits 0, because their squares are 927369, 93373569, 9343735569, 934437355569, 93444373555569, 9344443735555569, ... (End)

At i = 0 and at j = 0 solutions include all odd numbers, so these have not been listed above, but the full table (for all i,j) is considered for patterns later below.

If we exclude multiples of the same k in the listed sequence it equals A180263, which uses a different set of criteria related to nonprimes.

The results may be viewed as a "distance" from a point on the real x-y plane to the imaginary point sqrt(-1) on a complex z-axis (or plane).

These results occur in a number of infinite and branching "subsequences" with distinct but usually simple patterns that are discernible when viewing the triangle. The main subsequence (#1) begins at value 3. Other subsequences branch off from it. Patterns are evident in both the values of results (by examining differences) and in their i and j positions. Example subsequences are numbered in the order of lowest value for their starting terms:

Subseq #1: 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, etc.

Subseq #2: 7, 13, 17, 23, 27, 33, 37, 43, 47, 53, 57, 63, 67, 73, etc.

Subseq #3: 7, 41, 75, 99, 133, 157, 191, 215, 249, 273, 307, etc.

Subseq #4: 13, 21, 47, 55, 81, 89, 115, 123, 149, 157, 183, 191, etc.

Subseq #5: 17, 23, 47, 57, 93, 107, 155, 173, 233, 255, 327, 353, etc.

Subseq #6: 21, 31, 47, 57, 73, 83, 99, 109, 125, 135, 151, 161, etc.

Subsequence #1 conforms to A002061 (when excluding its two initial 1's) with the 3 at i=1, j=0 (on other side of diagonal). Subsequence #2 conforms to A063226, when excluding its initial term 3. The other examples given are not in the OEIS currently.

Branching can be seen as follows: subsequence #3 starts on the diagonal at i,j = 2 and it branches off #1 (or #2). Subsequence #4 branches off #1. Subsequence #5 branches off of #2. Subsequence #6 branches off #1. Subsequence #5 and #6 cross over each other, at (47,57) which occurs at the same i,j, values.

The full set of odd numbers along the axes for i,j = 0, in all four quadrants are also simple patterns.

Additional subsequences are spawned at higher values of i and j, apparently without end, evoking the appearance of linear and curved fractals. It is NOT clear whether all instances of k occur in such infinite subsequences.

Results are prolific because of the -1 in the equation. Without it (i.e., using only Pythagorean distance) there are no integer solutions for k with two odd legs to the triangle. Though certain other constants (added or subtracted) produce multiple obvious subsequences as well.