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

1,2

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.

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

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.

Example

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

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*)

Prog

(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

Crossrefs

Cf. A002487, A014417, A046815.

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

Distinct additive closure of A001906.

Sequence in context: A193532 A068056 A006520 * A050003 A002156 A285033

Adjacent sequences: A054201 A054202 A054203 * A054205 A054206 A054207

Keyword

nonn,changed

Author

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

Status

approved