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

1, 1, 2, 3, 6, 9, 16, 25, 42, 68, 110, 179, 291, 470, 763, 1236, 2005, 3241, 5252, 8502, 13770, 22272, 36058, 58355, 94455, 152878, 247333, 400279, 647722, 1048180, 1696193, 2744373, 4440857, 7185700, 11627320, 18814256, 30443581, 49257837

Offset: 1

Author

Yasutoshi Kohmoto, Jun 10 2004

Keywords

• nonn

Comments

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.

a(1) = 1, a(n) = a(n-1) + Sum_{i < n-1 and a(i) is adjacent to a(n-1)} a(i).

Here only four positions are considered adjacent, eight however in A094767.

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

Examples

			Clockwise constructed spiral begins
.
  13770--22272--36058--58355--94455
      |
      |
   8502     16-----25-----42-----68
      |      |                    |
      |      |                    |
   5252      9      1------1    110
      |      |             |      |
      |      |             |      |
   3241      6------3------2    179
      |                           |
      |                           |
   2005---1236----763----470----291
.
where
  a(2) = a(1) = 1,
  a(3) = a(2) + a(1) = 2,
  a(4) = a(3) + a(2) = 3,
  a(5) = a(4) + a(3) + a(1) = 6,
  a(6) = a(5) + a(4) = 9,
  a(7) = a(6) + a(5) + a(1) = 16.
		

Links

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

Crossrefs

Cf. A063826, A094767, A094769, A126937, A141481.

Programs

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

Extensions

Edited and extended beyond a(14) by Klaus Brockhaus, Aug 27 2008