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

%I A078510 #19 Dec 18 2015 11:19:24
%S A078510 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,
%T A078510 36,42,48,54,61,69,78,88,98,108,119,131,144,158,172,186,201,217,235,
%U A078510 256,280,304,328,355,386,422,464,512,560,608,662,723,792,870,958,1056
%N A078510 Spiro-Fibonacci numbers, a(n) = sum of two previous terms that are nearest when terms arranged in a spiral.
%C A078510 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
%H A078510 Antti Karttunen, <a href="/A078510/b078510.txt">Table of n, a(n) for n = 0..1024</a>
%F A078510 From _Antti Karttunen_, Dec 13 2015: (Start)
%F A078510 a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) + a(A265409(n)).
%F A078510 equally, for n > 1, a(n) = a(n-1) + a(n - A265359(n)).
%F A078510 a(n) = A001222(A265408(n)).
%F A078510 (End)
%e A078510 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.
%e A078510 From _Antti Karttunen_, Dec 17 2015: (Start)
%e A078510 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:
%e A078510 +--------+--------+--------+--------+
%e A078510 |a(15)   |a(14)   |a(13)   |a(12)   |
%e A078510 | = a(14)| = a(13)| = a(12)| = a(11)|
%e A078510 | + a(4) | + a(3) | + a(2) | + a(2) |
%e A078510 | = 9    | = 8    | = 7    | = 6    |
%e A078510 +--------+--------+--------+--------+
%e A078510 |a(4)    |a(3)    |a(2)    |a(11)   |
%e A078510 | = a(3) | = a(2) | = a(1) | = a(10)|
%e A078510 | + a(0) | + a(0) | + a(0) | + a(2) |
%e A078510 | = 1    | = 1    | = 1    | = 5    |
%e A078510 +--------+--------+--------+--------+
%e A078510 |a(5)    | START  |   ^    |a(10)   |
%e A078510 | = a(4) | a(0)=0 | a(1)=1 | = a(9) |
%e A078510 | + a(0) |   -->  |        | + a(1) |
%e A078510 | = 1    |        |        | = 4    |
%e A078510 +--------+--------+--------+--------+
%e A078510 |a(6)    |a(7)    |a(8)    |a(9)    |
%e A078510 | = a(5) | = a(6) | = a(7) | = a(8) |
%e A078510 | + a(0) | + a(0) | + a(1) | + a(1) |
%e A078510 | = 1    | = 1    | = 2    | = 3    |
%e A078510 +--------+--------+--------+--------+
%e A078510 (End)
%o A078510 (Scheme, with memoization-macro definec)
%o A078510 (definec (A078510 n) (if (< n 2) n (+ (A078510 (- n 1)) (A078510 (A265409 n)))))
%o A078510 ;; _Antti Karttunen_, Dec 13 2015
%Y A078510 Cf. A000045, A001222, A033951, A063826, A265407, A265408, A265409, A265359, A265370, A265404.
%K A078510 nonn
%O A078510 0,9
%A A078510 _Neil Fernandez_, Jan 05 2003