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.

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A063929 Radius of A-excircle of Pythagorean triangle with a = (n+1)^2 - m^2, b = 2*(n+1)*m and c = (n+1)^2 + m^2.

Original entry on oeis.org

2, 6, 3, 12, 8, 4, 20, 15, 10, 5, 30, 24, 18, 12, 6, 42, 35, 28, 21, 14, 7, 56, 48, 40, 32, 24, 16, 8, 72, 63, 54, 45, 36, 27, 18, 9, 90, 80, 70, 60, 50, 40, 30, 20, 10, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 132, 120, 108, 96, 84, 72, 60, 48, 36, 24, 12, 156, 143, 130, 117
Offset: 1

Views

Author

Floor van Lamoen, Aug 21 2001

Keywords

Comments

From Wolfdieter Lang, Dec 03 2014: (Start)
For excircles and their radii see the Eric W. Weisstein links. Here the circle radius with center J_A is considered.
Note that not all Pythagorean triangles are covered, e.g., the nonprimitive one (9, 12, 15) does not appear. However, the nonprimitive one (8, 6, 10) does appear as (n, m) = (2, 1). (End)
This triangle T appears also in the problem of finding all positive integer solutions for a and b of the general Fibonacci sequence F(a,b;k+1) = a*F(a,b;k) + b*F(a,b;k-1) (with some inputs F(a,b;0) and F(a,b;1)) such that the limit r = r(a,b) = F(a,b;k+1)/F(a,b;k) for k -> infinity becomes a positive integer r = (a + sqrt(a^2 + 4*b))/2. Namely, for any a = m >= 1 there are infinitely many b solutions b = T(n,m) = (n+1)*(n+1-m) for n >= m. The limit is r(a,b) = n+1 for a = m = 1..n, which is A003057 read as a triangle with offset 1. This entry was motivated by A249973 and A249974 by Kerry Mitchell concerned with real values of r. - Wolfdieter Lang, Jan 11 2015

Examples

			The triangle T(n, m) begins:
n\m   1   2   3   4   5   6   7   8   9 10 11 12 13 14 15 ...
1:    2
2:    6   3
3:   12   8   4
4:   20  15  10  5
5:   30  24  18  12   6
6:   42  35  28  21  14   7
7:   56  48  40  32  24  16   8
8:   72  63  54  45  36  27  18   9
9:   90  80  70  60  50  40  30  20  10
10: 110  99  88  77  66  55  44  33  22 11
11: 132 120 108  96  84  72  60  48  36 24 12
12: 156 143 130 117 104  91  78  65  52 39 26 13
13: 182 168 154 140 126 112  98  84  70 56 42 28 14
14: 210 195 180 165 150 135 120 105  90 75 60 45 30 15
15: 240 224 208 192 176 160 144 128 112 96 80 64 48 32  1
... Formatted and extended by _Wolfdieter Lang_, Dec 02 2014
--------------------------------------------------------------
Example of general (a,b)-Fibonacci sequence positive integer limits r(a,b) (see the Jan 11 2015 comment above):
T(3, 2) = 8, that is a = m = 2 has a solution b = T(3, 2) = 8 with r = r(2,8) = n+1 = 4 = (2 + sqrt(4 + 4*8))/2. The other two solutions with r = 4 appear for b = T(3, m) with m = a = 1 and 3. In general, row n has n times the value n+1 for r, namely r(a=m,b=T(n,m)) = n+1, for m = 1..n. - _Wolfdieter Lang_, Jan 11 2015

Links

  • Eric Weisstein's World of Mathematics, Excircles
  • Eric Weisstein's World of Mathematics, Exradius

Crossrefs

Cf. A003991 (incircle radius), A063930 (B-excircle radius), A001283 (C-excircle radius), A055096 (circumcircle diameter).

Formula

T(n, m) = (n+1)(n-m+1), n >= m >= 1.
T(n, m) = rho_A = sqrt(s*(s-b)*(s-c)/(s-a)) with the semiperimeter s = (a + b + c)/2 and the substituted a, b, c values as given in the name. - Wolfdieter Lang, Dec 02 2014

Extensions

Edited: Crossreferences commented and A055096 added by Wolfdieter Lang, Dec 02 2014

A262373 a(1)=2, a(2)=5, a(3)=3; for n>3, a(n) is the smallest prime that has not already appeared and ends with the first digit in a(n-1) that equals 1, 3, 7 or 9.

Original entry on oeis.org

2, 5, 3, 13, 11, 31, 23, 43, 53, 73, 7, 17, 41, 61, 71, 37, 83, 103, 101, 131, 151, 181, 191, 211, 241, 251, 271, 47, 67, 97, 19, 281, 311, 113, 331, 163, 401, 421, 431, 173, 461, 491, 29, 59, 79, 107, 521, 541, 571, 127, 601, 631, 193, 641, 661, 691, 89, 109
Offset: 1

Views

Author

Vladimir Shevelev, Sep 20 2015

Keywords

Comments

Using Sierpiński's theorems [Sierpiński] (see also [Trost]), it is easy to see that the sequence is a permutation of the sequence of primes (A000040).

References

  • W. Sierpiński, Sur l'existence de nombres premiers avec une suite arbitraire de chiffres initiaux, Le Matematiche Catania, 1951.
  • E. Trost, Primzahlen, Verlag Birkhäuser, 1953, Theorems 20 - 21.

Links

Crossrefs

Cf. A000040, A249974.

Extensions

a(46) corrected by Peter J. C. Moses, Sep 24 2015

A249973 Positive integers A when the positive roots of r^2 = Ar + B are listed in increasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 1, 2, 3, 4, 1, 2, 3, 1, 4, 2, 1, 3, 2, 4, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 5, 3, 1, 4, 2, 1, 3, 5, 2, 4, 1, 3, 2, 5, 1, 4, 3
Offset: 1

Views

Author

Kerry Mitchell, Nov 09 2014

Keywords

Comments

Generalize the Fibonacci sequence recurrence equation as: F_(n+1) = A*F_n + B*F_(n-1), where A and B are positive integers. As n goes to infinity, the ratio F_n / F_(n-1) approaches the positive real number r = (A + sqrt(A*A + 4B))/2. This sequence gives the A values in increasing order of r.
In case of a tie in r values, then sort in increasing order of sqrt(A*A + B*B).
This A sequence appears to be the ordinal transform of the B sequence (A249974) and vice versa. The associative arrays of A and B are transposes. The first row of A's associative array seems to be A006000.
For the A and B values leading to a positive integer limit r see a comment in A063929. - Wolfdieter Lang, Jan 12 2015

Examples

			a(6) = 2 because the 6th smallest value of r (approximately 2.732050808) is that for A=2, B=2.

Crossrefs

Cf. A249974, A006000.

Programs

Extensions

Edited. - Wolfdieter Lang, Jan 11 2015
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