This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(10^10) = 52010594272060810683. - _David A. Corneth_, Mar 26 2015
G := (x/((1-x)*(1+2*x))) * (1 + 2*x*mul(1+x^(2^k-1)+2*x^(2^k),k=0..20)); # N. J. A. Sloane, May 20 2009, Jun 05 2009 # From N. J. A. Sloane, Dec 25 2009: A139250 is T, A139251 is a. a:=[0,1,2,4]; T:=[0,1,3,7]; M:=10; for k from 1 to M do a:=[op(a),2^(k+1)]; T:=[op(T),T[nops(T)]+a[nops(a)]]; for j from 1 to 2^(k+1)-1 do a:=[op(a), 2*a[j+1]+a[j+2]]; T:=[op(T),T[nops(T)]+a[nops(a)]]; od: od: a; T;
CoefficientList[ Series[ (x/((1 - x)*(1 + 2x))) (1 + 2x*Product[1 + x^(2^k - 1) + 2*x^(2^k), {k, 0, 20}]), {x, 0, 53}], x] (* Robert G. Wilson v, Dec 06 2010 *)
a[0] = 0; a[n_] := a[n] = Module[{m, k}, m = 2^(Length[IntegerDigits[n, 2]] - 1); k = (2m^2+1)/3; If[n == m, k, k + 2 a[n - m] + a[n - m + 1] - 1]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 06 2018, after David A. Corneth *)
A139250(n,print_all=0)={my(p=[], /* set of "used" points. Points are written as complex numbers, c=x+iy. Toothpicks are of length 2 */ ee=[[0,1]], /* list of (exposed) endpoints. Exposed endpoints are listed as [c,d] where c=x+iy is the position of the endpoint, and d (unimodular) is the direction */ c,d,ne, cnt=1); print_all && print1("0,1"); n<2 && return(n); for(i=2,n, p=setunion(p, Set(Mat(ee~)[,1])); /* add endpoints (discard directions) from last move to "used" points */ ne=[]; /* new (exposed) endpoints */ for( k=1, #ee, /* add endpoints of new toothpicks if not among the used points */ setsearch(p, c=ee[k][1]+d=ee[k][2]*I) || ne=setunion(ne,Set([[c,d]])); setsearch(p, c-2*d) || ne=setunion(ne,Set([[c-2*d,-d]])); ); /* using Set() we have the points sorted, so it's easy to remove those which finally are not exposed because they touch a new toothpick */ forstep( k=#ee=eval(ne), 2, -1, ee[k][1]==ee[k-1][1] && k-- && ee=vecextract(ee,Str("^"k"..",k+1))); cnt+=#ee; /* each exposed endpoint will give a new toothpick */ print_all && print1(","cnt));cnt} \\ M. F. Hasler, Apr 14 2009
\\works for n > 0
a(n) = {my(k = (2*msb(n)^2 + 1) / 3); if(n==msb(n),k , k + 2*a(n-msb(n)) + a(n - msb(n) + 1) - 1)}
msb(n)=my(t=0);while(n>>t>0,t++);2^(t-1)\\ David A. Corneth, Mar 26 2015
def msb(n):
t = 0
while n>>t > 0:
t += 1
return 2**(t - 1)
def a(n):
k = (2 * msb(n)**2 + 1) / 3
return 0 if n == 0 else k if n == msb(n) else k + 2*a(n - msb(n)) + a(n - msb(n) + 1) - 1
[a(n) for n in range(101)] # Indranil Ghosh, Jul 01 2017, after David A. Corneth's PARI script
If we label the generations of cells turned ON by consecutive numbers we get a rosetta cell pattern: . . . . . . . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . 4 3 4 . . . . . . . . . . . . . 4 . 2 . 4 . . . . . . . . . . . 4 3 2 1 2 3 4 . . . . . . . . . . . 4 . 2 . 4 . . . . . . . . . . . . . 4 3 4 . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . . . . . . In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON four "2", leading to a(2)=a(1)+4=5. In the third generation, four "3" are turned ON, a(3)=a(2)+4=9. In the fourth generation, each of the four wings allows three 4's to be turned ON, a(4)=a(3)+4*3=21. From _Omar E. Pol_, Feb 18 2015: (Start) Also, written as an irregular triangle T(j,k), j>=0, k>=1, in which the row lengths are the terms of A011782: 1; 5; 9, 21; 25, 37, 49, 85; 89, 101,113,149,161,197,233,341; 345,357,369,405,417,453,489,597,609,645,681,789,825,933,1041,1365; ... The right border gives the positive terms of A002450. (End) It appears that T(j,k) = A162795(j,k) = A169707(j,k), if k is a power of 2, for example: it appears that the three mentioned triangles only share the elements from the columns 1, 2, 4, 8, 16, ... - _Omar E. Pol_, Feb 20 2015
Since this is the partial sum sequence of A147582, it is most easily obtained using the Maple code given in A147582. # [x,y] coordinates of cells on Lse := [[0,0]] ; # enclosing rectangle of the cells on (that is, minima and maxima in Lse) xmin := 0 ; xmax := 0 ; ymin := 0 ; ymax := 0 ; # count neighbors of x,y which are on; return 0 if [x,y] is in L cntnei := proc(x,y,L) local a,p,xpt,ypt; a := 0 ; if not [x,y] in L then for p in Lse do xpt := op(1,p) ; ypt := op(2,p) ; if ( abs(xpt-x) = 1 and ypt=y ) or ( x=xpt and abs(ypt-y) = 1) then a := a+1 ; fi; od: fi: RETURN(a) ; end: # loop over generations/steps for stp from 1 to 10 do Lnew := [] ; for x from xmin-1 to xmax+1 do for y from ymin-1 to ymax+1 do if cntnei(x,y,Lse) = 1 then Lnew := [op(Lnew),[x,y]] ; fi; od: od: for p in Lnew do xpt := op(1,p) ; ypt := op(2,p) ; xmin := min(xmin,xpt) ; xmax := max(xmax,xpt) ; ymin := min(ymin,ypt) ; ymax := max(ymax,ypt) ; od: Lse := [op(Lse),op(Lnew)] ; print(nops(Lse)) ;
Join[{0},Map[Function[Apply[Plus,Flatten[ #1]]],CellularAutomaton[{686,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},200]]] (* Nadia Heninger and N. J. A. Sloane, Aug 11 2009; modified by Paolo Xausa, Aug 12 2022 to include the a(0) term *)
ArrayPlot /@ CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 16] (* N. J. A. Sloane, Nov 08 2014 *)
A147562list[nmax_]:=Accumulate[Join[{0,1},4*3^(DigitCount[Range[nmax-1],2,1]-1)]];A147562list[100] (* Paolo Xausa, May 21 2023 *)
a(n) = if (n, 1 + 4*sum(k=1, n-1, 3^(hammingweight(k)-1)), 0); \\ Michel Marcus, Jul 05 2022
Illustration of initial terms: . o o . \ / . o o o . | | / . o o - o o o - o . / / \ / \ . o o - o o - o o . | / | . o o o . / \ . o o . . 1 5 13 .
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