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Search: id:A102376
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| 1, 4, 4, 16, 4, 16, 16, 64, 4, 16, 16, 64, 16, 64, 64, 256, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 4, 16, 16, 64, 16, 64, 64, 256, 16, 64, 64, 256, 64, 256, 256, 1024, 16, 64, 64, 256, 64, 256, 256, 1024, 64, 256, 256, 1024, 256, 1024, 1024
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Consider a simple cellular automaton, a grid of binary cells c(i,j), where the next state of the grid is calculated by applying the following rule to each cell: c(i,j) = ( c(i+1,j-1) + c(i+1,j+1) + c(i-1,j-1) + c(i-1,j+1) ) mod 2 If we start with a single cell having the value 1 and all the others 0, then the aggregate values of the subsequent states of the grid will be the terms in this sequence. - Andras Erszegi (erszegi.andras(AT)chello.hu), Mar 31 2006
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FORMULA
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G.f.: product{k>=0, 1 + 4x^(2^k)}; a(n)=product{k=0..log_2(n), 4^b(n, k)}, b(n, k)=coefficient of 2^k in binary expansion of n; a(n)=sum{k=0..n, (C(n, k) mod 2)*3^A000120(n-k)}. (Formulas due to Paul D. Hanna).
a(n)=sum{k=0..n, mod(C(n, k), 2)*sum{j=0..k, mod(C(k, j), 2)*sum{i=0..j, mod(C(j, i), 2)}}} - Paul Barry (pbarry(AT)wit.ie), Apr 01 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u^2 - 2*u*v + 5*v^2) - 4*v^3. - Michael Somos May 29 2008
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EXAMPLE
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1 + 4*x + 4*x^2 + 16*x^3 + 4*x^4 + 16*x^5 + 16*x^6 + 64*x^7 + 4*x^8 + ...
Contribution from Omar E. Pol (info(AT)polprimos.com), Jun 07 2009: (Start)
Triangle begins:
1;
4;
4,16;
4,16,16,64;
4,16,16,64,16,64,64,256;
4,16,16,64,16,64,64,256,16,64,64,256,64,256,256,1024;
4,16,16,64,16,64,64,256,16,64,64,256,64,256,256,1024,16,64,64,256,64,256,...
(End)
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PROGRAM
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(PARI) {a(n) = if( n<0, 0, 4^subst( Pol( binary(n)), x, 1))} /* Michael Somos May 29 2008 */
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CROSSREFS
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For generating functions Prod_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
Cf. A001316, A048883.
Cf. A000079. [From Omar E. Pol (info(AT)polprimos.com), Jun 07 2009]
Sequence in context: A076821 A165825 A056959 this_sequence A091278 A127473 A079611
Adjacent sequences: A102373 A102374 A102375 this_sequence A102377 A102378 A102379
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Jan 05 2005
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