cp's OEIS Frontend

A094767 Square spiral of sums of selected preceding terms, starting at 1 (a spiral Fibonacci-like sequence).

%I A094767 #16 Aug 12 2018 21:27:16
%S A094767 1,1,2,4,8,13,26,40,81,123,205,412,620,1034,2072,3120,5204,8332,16677,
%T A094767 25056,41772,66854,133748,200749,334741,535694,870558,1741321,2612619,
%U A094767 4355177,6968828,11324625,22650284,33978635,56635145,90624176,147267645
%N A094767 Square spiral of sums of selected preceding terms, starting at 1 (a spiral Fibonacci-like sequence).
%C A094767 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, vertically or diagonally adjacent to it, go to next position of the spiral and enter the sum into it.
%C A094767 a(1) = 1, a(n) = a(n-1) + Sum_{i < n-1 and a(i) is adjacent to a(n-1)} a(i).
%C A094767 Here eight positions are considered adjacent, only four however in A094768.
%C A094767 Clockwise and counterclockwise construction of the spiral result in the same sequence.
%H A094767 Klaus Brockhaus, <a href="/A094767/b094767.txt">Table of n, a(n) for n = 1..729</a>
%e A094767 Clockwise constructed spiral begins
%e A094767 .
%e A094767   41772---66854--133748--200749--334741
%e A094767       |
%e A094767       |
%e A094767       |
%e A094767   25056      26------40------81-----123
%e A094767       |       |                       |
%e A094767       |       |                       |
%e A094767       |       |                       |
%e A094767   16677      13       1-------1     205
%e A094767       |       |               |       |
%e A094767       |       |               |       |
%e A094767       |       |               |       |
%e A094767    8332       8-------4-------2     412
%e A094767       |                               |
%e A094767       |                               |
%e A094767       |                               |
%e A094767    5204----3120----2072----1034-----620
%e A094767 .
%e A094767 where
%e A094767   a(2) = a(1) = 1,
%e A094767   a(3) = a(2) + a(1) = 2,
%e A094767   a(4) = a(3) + a(2) + a(1) = 4,
%e A094767   a(5) = a(4) + a(3) + a(2) + a(1) = 8,
%e A094767   a(6) = a(5) + a(4) + a(1) = 13,
%e A094767   a(7) = a(6) + a(5) + a(4) + a(1) = 26.
%o A094767 (PARI) {m=5; h=2*m-1; A=matrix(h, h); print1(A[m, m]=1, ","); pj=m; pk=m; T=[[1, 0], [1, -1], [0, -1], [ -1, -1], [ -1, 0], [ -1, 1], [0, 1], [1, 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, 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 A094767 Cf. A063826, A094768, A094769, A126937, A141481.
%K A094767 nonn
%O A094767 1,3
%A A094767 _Yasutoshi Kohmoto_, Jun 10 2004
%E A094767 Edited and extended beyond a(14) by _Klaus Brockhaus_, Aug 27 2008