bibkey: hanna2012a204061 authors: Paul D. Hanna year: 2012 title: “OEIS A204061, G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n )” doi: null url: https://oeis.org/A204061 claim: “G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers. a(n) == 1 (mod 5) iff n has no 2’s in its base 5 expansion (A023729), otherwise a(n) == 0 (mod 5); this is a conjecture needing proof.” strata_touched:

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

OEIS A204061

Paul D. Hanna’s entry defines a Pell-square exponential generating series and conjectures that its coefficients reduce modulo five to the indicator of indices with no digit two in base five. The formal statement reads the exponential in the 5-adic field and proves 5-adic integrality and this residue law; integrality over the ordinary integers is outside its scope.

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• URL: https://oeis.org/A204061
• The following entry lines are quoted verbatim from the supplied locator.

%N G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.
%C a(n) == 1 (mod 5) iff n has no 2's in its base 5 expansion (A023729), otherwise a(n) == 0 (mod 5); this is a conjecture needing proof.
%F G.f.: 1 / ( sqrt(1+x) * (1-6*x+x^2)^(1/4) ).
%A _Paul D. Hanna_, Jan 10 2012