bibkey: barker2014a226857 authors: Colin Barker; Alonso del Arte year: 2014 title: “OEIS A226857, Numbers that are both the sum of two Fibonacci numbers and the product of two Fibonacci numbers” doi: null url: https://oeis.org/A226857 claim: “%N Numbers that are both the sum of two Fibonacci numbers and the product of two Fibonacci numbers. %F Conjecture: a(n) = a(n-3)+a(n-6) for n>12. - Colin Barker, Nov 09 2014 %F Empirical g.f.: -x^2*(x^10 +x^9 +x^8 +2x^7 +3x^6 +3x^5 +3x^4 +3x^3 +3x^2 +2*x +1) / (x^6 +x^3 -1). - Colin Barker, Nov 09 2014 %C All Fibonacci numbers are in the sequence. The only prime numbers in this sequence are prime Fibonacci numbers. %A Alonso del Arte, Jun 19 2013” strata_touched:

• D5/S1/Recurrence/BarkerFibonacciSumProductRecurrence license: citation-only triage: anchor

OEIS A226857

The entry defines the numbers that are simultaneously sums and products of two Fibonacci numbers. Colin Barker’s 2014 comment gives the recurrence conjecture quoted in the claim.

The formal sequence uses Fibonacci indexing with F_0 = 0 and the OEIS offset one. Equal summands and equal factors are allowed, as in the entry’s %t program. The recurrence is proved for every n > 12 by classifying the member set as {F_k, 2F_k, 3F_k} and identifying its increasing enumeration. The empirical generating-function line is not separately claimed.

Verified locator

• URL: https://oeis.org/A226857
• Locator: FORMULA, “Conjecture: a(n) = a(n-3)+a(n-6) for n>12. - Colin Barker, Nov 09 2014”

The entry has offset one, while the formal definition counts F_0 = 0 and uses zero-based Nat.nth; equal summands and equal factors are permitted by the %t program.

The recurrence is proved for every n > 12 via the classification of the member set as {F_k, 2F_k, 3F_k}. The empirical generating-function line is not separately claimed.