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%I #27 Dec 28 2022 01:55:55
%S 1,3,4,8,9,11,12,21,22,24,25,29,30,32,33,55,56,58,59,63,64,66,67,76,
%T 77,79,80,84,85,87,88,144,145,147,148,152,153,155,156,165,166,168,169,
%U 173,174,176,177,199,200,202,203,207,208,210,211,220,221,223,224,228,229
%N Integers expressible as sums of distinct even-subscripted Fibonacci numbers.
%C Number of partitions of a(n) into sums of distinct Fibonacci numbers is (n+1)st term of Stern's Diatomic series A002487. This sequence has A046815 as a subsequence.
%H Amiram Eldar, <a href="/A054204/b054204.txt">Table of n, a(n) for n = 1..10000</a>
%H Marjorie Bicknell-Johnson, <a href="http://www.fq.math.ca/Scanned/40-3/bicknell.pdf">The least integer having p Fibonacci representations (p prime)</a>, Fibonacci Quarterly 40 (2002), pp. 260-265.
%H Marjorie Bicknell-Johnson, <a href="http://www.fq.math.ca/Scanned/41-2/bicknell.pdf">Stern's Diatomic Array Applied to Fibonacci Representations</a>, Fibonacci Quarterly 41 (2003), pp. 169-180.
%H Sam Northshield, <a href="https://www.researchgate.net/profile/Sam-Northshield/publication/366321658_SOME_GENERALIZATIONS_OF_A_FORMULA_OF_REZNICK/">Some generalizations of a formula of Reznick</a>, SUNY Plattsburgh (2022).
%H <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%F Subscripts in Zeckendorf representation of a(n) are 2(e+1) where e is exponent used to write n as sum of powers of 2.
%e a(9)=22 since 9=2^3+2^0 and 22=F(2(3+1)) + F(2(0+1)) = F(8) + F(2).
%t fibEvenCount[n_] := Plus @@ (Reverse @ IntegerDigits[n, 2])[[2 ;; -1 ;; 2]]; evenIndexed = fibEvenCount /@ Select[Range[10000], BitAnd[#, 2 #] == 0 &]; Position[evenIndexed, _?(# == 0 &)] // Flatten (* _Amiram Eldar_, Jan 20 2020*)
%o (PARI) my(m=Mod('x,'x^2-3*'x+1)); a(n) = subst(lift(subst(Pol(binary(n)), 'x,m)), 'x,3); \\ _Kevin Ryde_, Nov 25 2020
%Y Cf. A002487, A014417, A046815.
%Y Cf. A022290, A062877 (odd-indexed Fibonaccis).
%Y Distinct additive closure of A001906.
%K nonn
%O 1,2
%A Marjorie Bicknell-Johnson (marjohnson(AT)earthlink.net), Apr 30 2000
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