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

1,4

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.

Links

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A351016 A068911 A243543 * A068018 A358650 A357546

Adjacent sequences: A094766 A094767 A094768 * A094770 A094771 A094772

Keyword

nonn

Author

Yasutoshi Kohmoto, Jun 10 2004

Extensions

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

Status

approved