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A152980
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First differences of toothpick corner sequence A153006.
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50
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1, 2, 3, 3, 4, 7, 8, 5, 4, 7, 9, 10, 15, 22, 20, 9, 4, 7, 9, 10, 15, 22, 21, 14, 15, 23, 28, 35, 52, 64, 48, 17, 4, 7, 9, 10, 15, 22, 21, 14, 15, 23, 28, 35, 52, 64, 49, 22, 15, 23, 28, 35, 52, 65, 56, 43, 53, 74, 91, 122, 168, 176, 112, 33, 4, 7, 9, 10, 15, 22, 21, 14, 15, 23, 28, 35, 52
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OFFSET
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0,2
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COMMENTS
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Rows of A152978 when written as a triangle converge to this sequence. - Omar E. Pol, Jul 19 2009
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LINKS
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FORMULA
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G.f.: (1 + x) * Prod_{ n >= 1} (1 + x^(2^n-1) + 2*x^(2^n)). - N. J. A. Sloane, May 20 2009, corrected May 21 2009
For formula see A147646 (or, better, see the Maple code below).
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EXAMPLE
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Triangle begins:
.1;
.2;
.3,3;
.4,7,8,5;
.4,7,9,10,15,22,20,9;
.4,7,9,10,15,22,21,14,15,23,28,35,52,64,48,17;
....
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MAPLE
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Maple code from N. J. A. Sloane, May 18 2009. First define old version with offset 1:
S:=proc(n) option remember; local i, j;
if n <= 0 then RETURN(0); fi;
if n <= 2 then RETURN(2^(n-1)); fi;
i:=floor(log(n)/log(2));
j:=n-2^i;
if j=0 then RETURN(n/2+1); fi;
if j<2^i-1 then RETURN(2*S(j)+S(j+1)); fi;
if j=2^i-1 then RETURN(2*S(j)+S(j+1)-1); fi;
-1;
end;
# Now change the offset:
T:=n->S(n+1);
G := (1 + x) * mul(1 + x^(2^k-1) + 2*x^(2^k), k=1..20);
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MATHEMATICA
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nmax = 78;
G = x*((1 + x)/(1 - x)) * Product[ (1 + x^(2^n - 1) + 2*x^(2^n)), {n, 1, Log2[nmax] // Ceiling}];
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CROSSREFS
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For generating functions of the form Prod_{k>=c} (1+a*x^(2^k-1)+b*x^2^k)) for the following values of (a,b,c) see: (1,1,0) A160573, (1,1,1) A151552, (1,1,2) A151692, (2,1,0) A151685, (2,1,1) A151691, (1,2,0) A151688 and A152980, (1,2,1) A151550, (2,2,0) A151693, (2,2,1) A151694.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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