cp's OEIS Frontend

A094768 Square spiral of sums of selected preceding terms, starting at 1 (a spiral Fibonacci-like sequence).

%I A094768 #17 Aug 13 2018 09:02:55
%S A094768 1,1,2,3,6,9,16,25,42,68,110,179,291,470,763,1236,2005,3241,5252,8502,
%T A094768 13770,22272,36058,58355,94455,152878,247333,400279,647722,1048180,
%U A094768 1696193,2744373,4440857,7185700,11627320,18814256,30443581,49257837
%N A094768 Square spiral of sums of selected preceding terms, starting at 1 (a spiral Fibonacci-like sequence).
%C A094768 Enter 1 into center position of the spiral. Repeat: Add to the number in the present position the numbers in all those already filled positions that are horizontally or vertically adjacent to it, go to next position of the spiral and enter the sum into it.
%C A094768 a(1) = 1, a(n) = a(n-1) + Sum_{i < n-1 and a(i) is adjacent to a(n-1)} a(i).
%C A094768 Here only four positions are considered adjacent, eight however in A094767.
%C A094768 Clockwise and counterclockwise construction of the spiral result in the same sequence.
%H A094768 Klaus Brockhaus, <a href="/A094768/b094768.txt">Table of n, a(n) for n = 1..729</a>
%e A094768 Clockwise constructed spiral begins
%e A094768 .
%e A094768   13770--22272--36058--58355--94455
%e A094768       |
%e A094768       |
%e A094768    8502     16-----25-----42-----68
%e A094768       |      |                    |
%e A094768       |      |                    |
%e A094768    5252      9      1------1    110
%e A094768       |      |             |      |
%e A094768       |      |             |      |
%e A094768    3241      6------3------2    179
%e A094768       |                           |
%e A094768       |                           |
%e A094768    2005---1236----763----470----291
%e A094768 .
%e A094768 where
%e A094768   a(2) = a(1) = 1,
%e A094768   a(3) = a(2) + a(1) = 2,
%e A094768   a(4) = a(3) + a(2) = 3,
%e A094768   a(5) = a(4) + a(3) + a(1) = 6,
%e A094768   a(6) = a(5) + a(4) = 9,
%e A094768   a(7) = a(6) + a(5) + a(1) = 16.
%o A094768 (PARI) {m=5; h=2*m-1; A=matrix(h, h); print1(A[m, m]=1, ","); pj=m; pk=m; T=[[1, 0], [0, -1], [ -1, 0], [0, 1]]; for(n=1, (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, 4, 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 A094768 Cf. A063826, A094767, A094769, A126937, A141481.
%K A094768 nonn
%O A094768 1,3
%A A094768 _Yasutoshi Kohmoto_, Jun 10 2004
%E A094768 Edited and extended beyond a(14) by _Klaus Brockhaus_, Aug 27 2008