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 A087221 Number of compositions (ordered partitions) of n into powers of 4. 4
 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, 36, 50, 69, 96, 133, 184, 254, 352, 488, 676, 935, 1294, 1792, 2482, 3436, 4756, 6584, 9116, 12621, 17473, 24190, 33490, 46365, 64190, 88868, 123034, 170334, 235818, 326478, 451994, 625764, 866338, 1199400, 1660510 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Series trisections have a common ratio: sum(k>=0, a(3k+1)*x^k) / sum(k>=0, a(3k)*x^k) = sum(k>=0, a(3k+2)*x^k) / sum(k>=0, a(3k+1)*x^k) = sum(k>=0, a(3k+3)*x^k) / sum(k>=0, a(3k+2)*x^k) = sum(k>=0, x^((4^n-1)/3) ) = (1 + x + x^5 + x^21 + x^85 + x^341 +...). LINKS T. D. Noe, Table of n, a(n) for n=0..500 FORMULA G.f.: 1/( 1 - sum(k>=0, x^(4^k) ) ). [Joerg Arndt, Oct 21 2012] G.f. satisfies A(x) = A(x^4)/(1 - x*A(x^4)), A(0) = 1. a(n) ~ c * d^n, where d=1.384450093664460722709070772652942206959424183007359023442195..., c=0.526605891697738213614083414993893445498621299371909641096106... - Vaclav Kotesovec, May 01 2014 EXAMPLE A(x) = A(x^4) + x*A(x^4)^2 + x^2*A(x^4)^3 + x^3*A(x^4)^4 + ... = 1 +x + x^2 +x^3 +2x^4 +3x^5 +5x^6 +7x^7 + 10x^8 +... MAPLE a:= proc(n) option remember;       `if`(n=0, 1, add(a(n-4^i), i=0..ilog[4](n)))     end: seq(a(n), n=0..50);  # Alois P. Heinz, Jan 11 2014 MATHEMATICA a[n_] := a[n] = If[n==0, 1, Sum[a[n-4^i], {i, 0, Log[4, n]}]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *) PROG (PARI) a(n)=local(A, m); if(n<1, n==0, m=1; A=1+O(x); while(m<=n, m*=4; A=1/(1/subst(A, x, x^4)-x)); polcoeff(A, n)) (PARI) N=66; x='x+O('x^N); Vec( 1/( 1 - sum(k=0, ceil(log(N)/log(4)), x^(4^k)) ) ) /* Joerg Arndt, Oct 21 2012 */ CROSSREFS Cf. A078932, A087222, A087232, A087224. Different from A003269. Sequence in context: A247084 A017898 A003269 * A295072 A206739 A107586 Adjacent sequences:  A087218 A087219 A087220 * A087222 A087223 A087224 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 27 2003 STATUS approved

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Last modified April 10 15:47 EDT 2021. Contains 342845 sequences. (Running on oeis4.)