A054204 Integers expressible as sums of distinct even-subscripted Fibonacci numbers.

1, 3, 4, 8, 9, 11, 12, 21, 22, 24, 25, 29, 30, 32, 33, 55, 56, 58, 59, 63, 64, 66, 67, 76, 77, 79, 80, 84, 85, 87, 88, 144, 145, 147, 148, 152, 153, 155, 156, 165, 166, 168, 169, 173, 174, 176, 177, 199, 200, 202, 203, 207, 208, 210, 211, 220, 221, 223, 224, 228, 229

Offset: 1

Author

Marjorie Bicknell-Johnson (marjohnson(AT)earthlink.net), Apr 30 2000

Keywords

• nonn

Comments

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.

Examples

			a(9)=22 since 9=2^3+2^0 and 22=F(2(3+1)) + F(2(0+1)) = F(8) + F(2).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Marjorie Bicknell-Johnson, The least integer having p Fibonacci representations (p prime), Fibonacci Quarterly 40 (2002), pp. 260-265.
• Marjorie Bicknell-Johnson, Stern's Diatomic Array Applied to Fibonacci Representations, Fibonacci Quarterly 41 (2003), pp. 169-180.
• Sam Northshield, Some generalizations of a formula of Reznick, SUNY Plattsburgh (2022).
• Index entries for sequences related to Stern's sequences

Crossrefs

Cf. A002487, A014417, A046815.

Cf. A022290, A062877 (odd-indexed Fibonaccis).

Distinct additive closure of A001906.

Programs

Mathematica

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*)

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

Formula

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.