login
A078510
Spiro-Fibonacci numbers, a(n) = sum of two previous terms that are nearest when terms arranged in a spiral.
14
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
0,9
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
LINKS
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)
EXAMPLE
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)
PROG
(Scheme, with memoization-macro definec)
(definec (A078510 n) (if (< n 2) n (+ (A078510 (- n 1)) (A078510 (A265409 n)))))
;; Antti Karttunen, Dec 13 2015
KEYWORD
nonn
AUTHOR
Neil Fernandez, Jan 05 2003
STATUS
approved