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A094769 Square spiral of sums of selected preceding terms, starting at 0 followed by 1 (a spiral Fibonacci-like sequence).

%I A094769 #17 Aug 13 2018 09:03:02
%S A094769 0,1,1,2,4,6,12,18,37,56,94,189,285,475,952,1434,2392,3830,7666,11518,
%T A094769 19202,30732,61482,92281,153874,246248,400178,800450,1200967,2001985,
%U A094769 3203426,5205696,10411867,15619275,26034003,41658056,67695885,109356333
%N A094769 Square spiral of sums of selected preceding terms, starting at 0 followed by 1 (a spiral Fibonacci-like sequence).
%C A094769 Enter 0 into center position and 1 into next position of the spiral. Repeat: Add to the number in the present position the numbers in all those already filled positions that are horizontally, vertically or diagonally adjacent to it, go to next position of the spiral and enter the sum into it.
%C A094769 a(1) = 0, a(2) = 1, a(n) = a(n-1) + Sum_{i < n-1 and a(i) is adjacent to a(n-1)} a(i).
%C A094769 As in A094767 eight positions are considered adjacent here.
%C A094769 Clockwise and counterclockwise construction of the spiral result in the same sequence.
%H A094769 Klaus Brockhaus, <a href="/A094769/b094769.txt">Table of n, a(n) for n = 1..729</a>
%e A094769 Clockwise constructed spiral begins
%e A094769 .
%e A094769   19202--30732--61482--92281-153874
%e A094769       |
%e A094769       |
%e A094769   11518     12-----18-----37-----56
%e A094769       |      |                    |
%e A094769       |      |                    |
%e A094769    7666      6      0------1     94
%e A094769       |      |             |      |
%e A094769       |      |             |      |
%e A094769    3830      4------2------1    189
%e A094769       |                           |
%e A094769       |                           |
%e A094769    2392---1434----952----475----285
%e A094769 .
%e A094769 where
%e A094769   a(1) = 0,
%e A094769   a(2) = 1,
%e A094769   a(3) = a(2) + a(1) = 1,
%e A094769   a(4) = a(3) + a(2) + a(1) = 2,
%e A094769   a(5) = a(4) + a(3) + a(2) + a(1) = 4,
%e A094769   a(6) = a(5) + a(4) + a(1) = 6,
%e A094769   a(7) = a(6) + a(5) + a(4) + a(1) = 12.
%o A094769 (PARI) {m=5; h=2*m-1; A=matrix(h, h); print1(A[m, m]=0, ","); print1(A[m, m+1]=1, ","); pj=m; pk=m+1; T=[[1, 0], [1, -1], [0, -1], [ -1, -1], [ -1, 0], [ -1, 1], [0, 1], [1, 1]]; for(n=2, (h-2)^2-1, g=sqrtint(n); r=(g+g%2)\2; q=4*r^2; d=n-q; if(n<=q-2*r, j=d+3*r; k=r, if(n<=q, j=r; k=-d-r, if(n<=q+2*r, j=r-d; k=-r, j=-r; k=d-3*r))); j=j+m; k=k+m; s=A[pj, pk]; for(c=1, 8, v=[pj, pk]; v+=T[c]; s=s+A[v[1], v[2]]); A[j, k]=s; print1(s, ","); pj=j; pk=k)} \\ _Klaus Brockhaus_, Aug 27 2008
%Y A094769 Cf. A063826, A094767, A094768, A126937, A141481.
%K A094769 nonn
%O A094769 1,4
%A A094769 _Yasutoshi Kohmoto_, Jun 10 2004
%E A094769 Edited and extended beyond a(12) by _Klaus Brockhaus_, Aug 27 2008