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 1-5 of 5 results.

A265408 Prime factorization representation of Spironacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(A265409(n)).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 38, 138, 870, 9765, 213675, 4309305, 201226025, 9679967297, 810726926009, 40855897091009, 4259653632223561, 380804291082185737, 44319264099050115071, 4644246052673250585913
Offset: 0

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Author

Antti Karttunen, Dec 13 2015

Keywords

Comments

The polynomials encoded by these numbers could also be called "Fernandez spiral polynomials" after Neil Fernandez, who discovered sequence A078510, which is obtained when they are evaluated at X=1.
The polynomial recurrence uses the same composition rules as the Fibonacci polynomials (A206296), but with the neighborhood rules of A078510, where the other polynomial is taken from the nearest inner neighbor (A265409) when the polynomials are arranged as a spiral into a square grid. See A265409 for the illustration.

Examples

			n    a(n)   prime factorization    Spironacci polynomial
------------------------------------------------------------
0       1   (empty)                S_0(x) = 0
1       2   p_1                    S_1(x) = 1
2       3   p_2                    S_2(x) = x
3       5   p_3                    S_3(x) = x^2
4       7   p_4                    S_4(x) = x^3
5      11   p_5                    S_5(x) = x^4
6      13   p_6                    S_6(x) = x^5
7      17   p_7                    S_7(x) = x^6
8      38   p_8 * p_1              S_8(x) = x^7 + 1
9     138   p_9 * p_2 * p_1        S_9(x) = x^8 + x + 1

Crossrefs

Cf. A003961, A265407, A265409.
Cf. also A078510, A206296 and A265752, A265753.

Formula

a(0) = 1, a(1) = 2, and for n >= 2, a(n) = A003961(a(n-1)) * a(A265409(n)).
Other identities. For all n >= 0:
A078510(n) = A001222(a(n)). [when each polynomial is evaluated at x=1]
A265407(n) = A248663(a(n)). [at x=2 over the field GF(2)]

A265407 Spironacci-style recurrence: a(0)=0, a(1)=1, a(n) = 2*a(n) XOR a(A265409(n)).

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 64, 129, 259, 519, 1036, 2074, 4150, 8296, 16600, 33208, 66424, 132832, 265696, 531424, 1062880, 2125696, 4251521, 8502785, 17005825, 34011905, 68023301, 136047622, 272093206, 544188470, 1088378998, 2176753882, 4353515996, 8707015520, 17414063992, 34828160840, 69656354600, 139312643368
Offset: 0

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Author

Antti Karttunen, Dec 13 2015

Keywords

Comments

Spironacci-polynomials evaluated at X=2 over the field GF(2).
This is otherwise computed like A078510, which starts with a(0)=0 placed in the center of spiral (in square grid), followed by a(1) = 1, after which each term is a sum of two previous terms that are nearest when terms are arranged in a spiral, that is terms a(n-1) and a(A265409(n)), except here we first multiply the term a(n-1) by 2, and use carryless XOR (A003987) instead of normal addition.

Links

Crossrefs

Cf. A003987, A248663, A265408, A265409.
Cf. also A078510, A264977.

Formula

a(0)=0, a(1)=1; after which, a(n) = 2*a(n) XOR a(A265409(n)).
a(n) = A248663(A265408(n)).

A265359 Spiralwise distance to the nearest inner neighbor in Ulam-style square-spirals using zero-based indexing: a(0) = 0, for n >= 1, a(n) = n - A265409(n).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 9, 10, 11, 11, 11, 12, 13, 13, 13, 14, 15, 15, 15, 15, 16, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19, 20, 21, 21, 21, 21, 21, 22, 23, 23, 23, 23, 23, 23, 24, 25, 25, 25, 25, 25, 25, 26, 27, 27, 27, 27, 27, 27, 27, 28, 29, 29, 29, 29, 29, 29, 29, 30, 31, 31, 31, 31, 31, 31, 31, 31, 32
Offset: 0

Views

Author

Antti Karttunen, Dec 16 2015

Keywords

Links

Crossrefs

Cf. A000267, A002620, A240025, A265409.

Formula

a(0) = 0, for n >= 1, a(n) = n - A265409(n).
If n <= 7, then a(n) = n, otherwise a(n) = a(n-1) + A240025(n) + A240025(n-1).

A078510 Spiro-Fibonacci numbers, a(n) = sum of two previous terms that are nearest when terms arranged in a spiral.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 21, 24, 27, 31, 36, 42, 48, 54, 61, 69, 78, 88, 98, 108, 119, 131, 144, 158, 172, 186, 201, 217, 235, 256, 280, 304, 328, 355, 386, 422, 464, 512, 560, 608, 662, 723, 792, 870, 958, 1056
Offset: 0

Views

Author

Neil Fernandez, Jan 05 2003

Keywords

Comments

Or "Spironacci numbers" for short. See also Spironacci polynomials, A265408. This sequence has an interesting growth rate, see A265370 and A265404. - Antti Karttunen, Dec 13 2015

Examples

			Terms are written in square boxes radiating spirally (cf. Ulam prime spiral). a(0)=0 and a(1)=1, so write 0 and then 1 to its right. a(2) goes in the box below a(1). The nearest two filled boxes contain a(0) and a(1), so a(2)=a(0)+a(1)=0+1=1. a(3) goes in the box to the left of a(2). The nearest two filled boxes contain a(0) and a(2), so a(3)=a(0)+a(2)=0+1=1.
From _Antti Karttunen_, Dec 17 2015: (Start)
The above description places cells in clockwise direction. However, for the computation of this sequence the actual orientation of the spiral is irrelevant. Following the convention used at A265409, we draw this spiral counterclockwise:
+--------+--------+--------+--------+
|a(15)   |a(14)   |a(13)   |a(12)   |
| = a(14)| = a(13)| = a(12)| = a(11)|
| + a(4) | + a(3) | + a(2) | + a(2) |
| = 9    | = 8    | = 7    | = 6    |
+--------+--------+--------+--------+
|a(4)    |a(3)    |a(2)    |a(11)   |
| = a(3) | = a(2) | = a(1) | = a(10)|
| + a(0) | + a(0) | + a(0) | + a(2) |
| = 1    | = 1    | = 1    | = 5    |
+--------+--------+--------+--------+
|a(5)    | START  |   ^    |a(10)   |
| = a(4) | a(0)=0 | a(1)=1 | = a(9) |
| + a(0) |   -->  |        | + a(1) |
| = 1    |        |        | = 4    |
+--------+--------+--------+--------+
|a(6)    |a(7)    |a(8)    |a(9)    |
| = a(5) | = a(6) | = a(7) | = a(8) |
| + a(0) | + a(0) | + a(1) | + a(1) |
| = 1    | = 1    | = 2    | = 3    |
+--------+--------+--------+--------+
(End)

Links

Crossrefs

Cf. A000045, A001222, A033951, A063826, A265407, A265408, A265409, A265359, A265370, A265404.

Formula

From Antti Karttunen, Dec 13 2015: (Start)
a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) + a(A265409(n)).
equally, for n > 1, a(n) = a(n-1) + a(n - A265359(n)).
a(n) = A001222(A265408(n)).
(End)

A265410 a(n) = one-based index to the nearest horizontally or vertically adjacent inner neighbor in square-grid spirals, and to the nearest diagonally adjacent inner neighbor when n is one of the corner cases A033638.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 10, 10, 10, 11, 12, 13, 13, 13, 14, 15, 16, 17, 17, 17, 18, 19, 20, 21, 21, 21, 22, 23, 24, 25, 26, 26, 26, 27, 28, 29, 30, 31, 31, 31, 32, 33, 34, 35, 36, 37, 37, 37, 38, 39, 40, 41, 42, 43, 43, 43, 44, 45, 46, 47, 48, 49, 50, 50, 50, 51, 52, 53, 54, 55, 56, 57, 57, 57, 58, 59, 60, 61, 62, 63, 64, 65, 65
Offset: 1

Views

Author

Antti Karttunen, Dec 09 2015

Keywords

Comments

By convention: a(1) = 0 because as 1 is a starting point of such spirals, it has no "inner neighbors" for itself.
Each n occurs A265411(n) times.

Examples

			We arrange natural numbers as a counterclockwise spiral into the square grid in the following manner (here A stands for 10, B for 11 and C for 12). The first square corresponds with n, and the second square with the value of a(n):
                    55433
            543C    51113C
            612B    61012B
            789A    71122A
                    7789AA
-
For each n > 1, we look for the nearest horizontally or vertically adjacent neighbor of n towards the center that is not n-1, which will then be value of a(n) [e.g., it is 1 for 4, 6 and 8, while it is 2 for 9 and 11 and 3 for 12] unless n is in the corner (one of the terms of A033638), in which case the value is the nearest diagonally adjacent neighbor towards the center.

Links

Crossrefs

One more than A265409(n-1).
Cf. A000267, A033638, A240025.
Cf. A265400 (a variant).
Cf. A265411, A265412, A265413 (positions of records, i.e., where value increases).

Formula

a(1) = 0; for 1 < n < 8, a(n) = 1 and for n >= 8: if either A240025(n-1) or A240025(n-2) is not zero [when n or n-1 is in A033638], then a(n) = a(n-1), otherwise, a(n) = 1 + a(n-1).
a(1) = 0; for 1 < n < 8, a(n) = 1 and for n >= 8: a(n) = a(n-1) + (1-A240025(n-1))*(1-A240025(n-2)). [The same formula in a more compact form.]
Other identities. For all n >= 0:
a(A265413(n)) = n. [Sequence is the least monotonic left inverse of A265413.]
a(A265412(n)) = n. [Also inverse of A265412.]
Showing 1-5 of 5 results.

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