1,2

S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962.

D. Singmaster, On the cellular automaton of Ulam and Warburton, unpublished manuscript, 2003.

David Applegate, The movie version

O. E. Pol, Illustration of initial terms (One-step rook) [From Omar E. Pol (info(AT)polprimos.com), Nov 02 2009]

O. E. Pol, Illustration of initial terms (One-step bishop) [From Omar E. Pol (info(AT)polprimos.com), Nov 02 2009]

O. E. Pol, Illustration of initial terms (Overlapping squares) [From Omar E. Pol (info(AT)polprimos.com), Nov 02 2009]

O. E. Pol, Illustration of initial terms (Overlapping X-toothpicks) [From Omar E. Pol (info(AT)polprimos.com), Nov 12 2009]

O. E. Pol, Illustration of initial terms of A139251, A160121, A147582 (Overlapping figures) [From Omar E. Pol (info(AT)polprimos.com), Nov 12 2009]

a(1)=1; for n > 1, a(n)=4*3^{wt(n-1)-1} - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 30 2009. This formula is (essentially) given by Singmaster - N. J. A. Sloane, Aug 06 2009.

G.f.: x + 4*x*(Prod_{k>=0} (1+3*x^(2^k)) - 1)/3. - N. J. A. Sloane, Jun 10 2009

Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 14 2009: (Start)

When written as a triangle:

.1;

.4;

.4,12;

.4,12,12,36;

.4,12,12,36,12,36,36,108;

.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324;

.4,12,12,36,12,36,36,108,12,36,36,108,36,108,108,324,12,36,36,108,36,108,...

The rows converge to A161411. (End)

A000120 := proc(n) local a, d; a := 0 ; for d from 0 to ilog2(n) do a := a+ ( floor(n/2^d) mod 2) ; od: a ; end: A048881 := proc(n) A000120(n+1)-1 ; end: A147582 := proc(n) 4*3^A048881(n) ; end: seq(A147582(n), n=0..100) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 30 2009]

wt := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:

f:=d->[seq((2*d)*(2*d-1)^(wt(n-1)-1), n=2..120)];

f2:=d->[1, op(f(d))];

f2(2);

Cf. A147562, A147610 (the sequence divided by 4), A048881.

Cf. A000079, A161411, A151779, A139250.

Cf. A048883, A139251, A160121, A162349. [From Omar E. Pol (info(AT)polprimos.com), Nov 02 2009]

Sequence in context: A014012 A160721 A151836 this_sequence A162793 A109045 A079315

Adjacent sequences: A147579 A147580 A147581 this_sequence A147583 A147584 A147585

nonn,new

N. J. A. Sloane (njas(AT)research.att.com), Apr 29 2009

Extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 30 2009