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

0, 1, 1, 2, 4, 6, 12, 18, 37, 56, 94, 189, 285, 475, 952, 1434, 2392, 3830, 7666, 11518, 19202, 30732, 61482, 92281, 153874, 246248, 400178, 800450, 1200967, 2001985, 3203426, 5205696, 10411867, 15619275, 26034003, 41658056, 67695885, 109356333

Offset: 1

Author

Yasutoshi Kohmoto, Jun 10 2004

Keywords

• nonn

Comments

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.

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).

As in A094767 eight positions are considered adjacent here.

Clockwise and counterclockwise construction of the spiral result in the same sequence.

Examples

			Clockwise constructed spiral begins
.
  19202--30732--61482--92281-153874
      |
      |
  11518     12-----18-----37-----56
      |      |                    |
      |      |                    |
   7666      6      0------1     94
      |      |             |      |
      |      |             |      |
   3830      4------2------1    189
      |                           |
      |                           |
   2392---1434----952----475----285
.
where
  a(1) = 0,
  a(2) = 1,
  a(3) = a(2) + a(1) = 1,
  a(4) = a(3) + a(2) + a(1) = 2,
  a(5) = a(4) + a(3) + a(2) + a(1) = 4,
  a(6) = a(5) + a(4) + a(1) = 6,
  a(7) = a(6) + a(5) + a(4) + a(1) = 12.
		

Links

• Klaus Brockhaus, Table of n, a(n) for n = 1..729

Crossrefs

Cf. A063826, A094767, A094768, A126937, A141481.

Programs

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

Extensions

Edited and extended beyond a(12) by Klaus Brockhaus, Aug 27 2008