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A272398
The union of hexagonal numbers (A000384) and centered 9-gonal numbers (A060544).
2
1, 6, 10, 15, 28, 45, 55, 66, 91, 120, 136, 153, 190, 231, 253, 276, 325, 378, 406, 435, 496, 561, 595, 630, 703, 780, 820, 861, 946, 1035, 1081, 1128, 1225, 1326, 1378, 1431, 1540, 1653, 1711, 1770, 1891, 2016, 2080, 2145, 2278, 2415, 2485, 2556, 2701, 2850
OFFSET
1,2
COMMENTS
The construction of the g.f. works basically as follows every third entry of A000384 equals every second entry of A060544, A000384(3n+1) = A060544(2n+1) = (3*n+1)*(6*n+1), which is an immediate consequence of their polynomial representations. So the sequence is the union of A000384 and the bisection 10, 55, 136, 253,... of A060544. Following Section 4.3 of Riordan's book "Combinatorial identities", subsampling and "aering" are done by replacing the independent variable of the g.f. by roots of the independent variable. So this sequence has rational g.f. because it is derived by regular interlacing of the two original sequences which also have rational g.f.'s. - R. J. Mathar, Jul 15 2016
LINKS
FORMULA
a(4*n-3) = A272399(n).
Conjectures:
a(n) = (-1+(-1)^n-6*((-i)^n+i^n)*n+18*n^2)/16 where i is the imaginary unit.
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-2*a(n-5)+2*a(n-6)-2*a(n-7)+a(n-8) for n>8.
G.f.: x*(1+4*x+5*x^3+6*x^4+x^5+x^6) / ((1-x)^3*(1+x)*(1+x^2)^2).
CROSSREFS
Cf. A000384, A060544, A272399 (intersection).
Sequence in context: A315287 A315288 A238047 * A020159 A357529 A048017
KEYWORD
nonn
AUTHOR
Colin Barker, Apr 28 2016
STATUS
approved