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A249452 Numbers k such that A249441(k) = 3.

%I A249452 #50 Aug 03 2025 21:04:57
%S A249452 15,31,47,63,95,127,191,255,383,511,767,1023,1535,2047,3071,4095,6143,
%T A249452 8191,12287,16383,24575,32767,49151,65535,98303,131071,196607,262143,
%U A249452 393215,524287,786431,1048575,1572863,2097151,3145727,4194303,6291455,8388607,12582911
%N A249452 Numbers k such that A249441(k) = 3.
%C A249452 Or k for which none of entries in the k-th row of Pascal's triangle (A007318) is divisible by 4 (cf. comment in A249441).
%C A249452 Using the Kummer carries theorem, one can prove that, for n>=2, a(n) has the form of either 1...1 or 101...1 in base 2.
%C A249452 The sequence is a subset of so-called binomial coefficient predictors (BCP) in base 2 (see Shevelev link, Th. 6 and Cor. 8), which were found also using Kummer theorem and have a very close binary structure.
%H A249452 E. E. Kummer, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0044&amp;IDDOC=266967">Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen</a>, J. Reine Angew. Math. 44 (1852), 93-146.
%H A249452 V. Shevelev, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Shevelev/shevelev14.html">Binomial Coefficient Predictors</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8
%H A249452 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2).
%F A249452 a(n) has either form 2^k - 1 or 3*2^m-1, k, m >= 4 (cf. A000225, A055010). Since, for k>=5, 2^k-1<3*2^(k-1)-1<2^(k+1)-1, we have that, for n>=1, a(2*n) = 2^(n+4)-1; a(2*n+1) = 3*2^(n+3)-1. - _Vladimir Shevelev_, Oct 29 2014, Nov 06 2014
%F A249452 a(1) = 15, and for n>1, a(n) = A052955(n+6). [Follows from above] - _Antti Karttunen_, Nov 03 2014
%F A249452 G.f.: (15+16*x-14*x^2-16*x^3)/(1-x-2*x^2+2*x^3); a(n) = 16*A029744(n)-1. - _Peter J. C. Moses_, Oct 30 2014
%t A249452 CoefficientList[Series[(15 + 16 x - 14 x^2 - 16 x^3)/(1 - x -2 x^2 + 2 x^3), {x, 0, 70}], x] (* _Vincenzo Librandi_, Oct 30 2014 *)
%t A249452 LinearRecurrence[{1,2,-2},{15,31,47,63},40] (* _Harvey P. Dale_, Apr 01 2019 *)
%o A249452 (PARI) a(n)=if(n==1, 15, (n%2+2)<<(n\2+3)-1) \\ _Charles R Greathouse IV_, Nov 06 2014
%o A249452 (PARI) is(n)=(n+1)>>valuation(n+1, 2)<5 && !setsearch([1, 2, 3, 5, 7, 11, 23], n) \\ _Charles R Greathouse IV_, Nov 06 2014
%Y A249452 Cf. A000225, A029744, A048278, A052955, A055010, A089633, A249441.
%K A249452 nonn,easy
%O A249452 1,1
%A A249452 _Vladimir Shevelev_, Oct 29 2014
%E A249452 More terms from _Peter J. C. Moses_, Oct 29 2014