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

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

Antti Karttunen, Table of n, a(n) for n = 0..1024

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

Crossrefs

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

Sequence in context: A060340 A241989 A246089 * A246100 A247250 A017909

Adjacent sequences: A078507 A078508 A078509 * A078511 A078512 A078513

Keyword

nonn

Author

Neil Fernandez, Jan 05 2003

Status

approved