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A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers. 6
1, 1, 5, 21, 101, 501, 2561, 13345, 70561, 377281, 2035285, 11059205, 60454005, 332138405, 1832677185, 10150115201, 56398558081, 314273655745, 1755700634981, 9830544087221, 55155558312901, 310027473436821, 1745567243959105, 9843160519978401, 55582528404717601 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..625

FORMULA

G.f.: 1 / ( sqrt(1+x) * (1-6*x+x^2)^(1/4) ).

Self-convolution yields A026933: Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers.

a(n) ~ 2^(1/8) * GAMMA(3/4) * (3+2*sqrt(2))^(n+1/2) / (4 * Pi * n^(3/4)). - Vaclav Kotesovec, Oct 30 2014

EXAMPLE

G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 101*x^4 + 501*x^5 + 2561*x^6 +...

where log(A(x)) = x + 3^2*x^2/2 + 7^2*x^3/3 + 17^2*x^4/4 + 41^2*x^5/5 + 99^2*x^6/6 + 239^2*x^7/7 +...+ A001333(n)^2*x^n/n +...

The last digit of the terms in this sequence seems to be either a '1' or a '5':

by conjecture, a(n) == 0 (mod 5) whenever n has a 2 in its base 5 expansion;

if true, terms a(2*5^k) through a(3*5^k - 1) all end with digit '5' for k>=0.

PROG

(PARI) {A001333(n)=polcoeff((1-x)/(1-2*x-x^2+x*O(x^n)), n)}

{a(n)=polcoeff(exp(sum(k=1, n, A001333(k)^2*x^k/k)+x*O(x^n)), n)}

(PARI) {a(n)=polcoeff(1/(sqrt(1+x+x*O(x^n))*(1-6*x+x^2+x*O(x^n))^(1/4)), n)}

CROSSREFS

Cf. A204062, A026933, A001333, A023729.

Sequence in context: A017969 A050897 A199215 * A046633 A280623 A203154

Adjacent sequences:  A204058 A204059 A204060 * A204062 A204063 A204064

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 10 2012

STATUS

approved

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Last modified September 27 16:20 EDT 2020. Contains 337383 sequences. (Running on oeis4.)