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URL: https://oeis.org/A330987

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A330987
Alternatively add and half-multiply pairs of the nonnegative integers.
2
1, 3, 9, 21, 17, 55, 25, 105, 33, 171, 41, 253, 49, 351, 57, 465, 65, 595, 73, 741, 81, 903, 89, 1081, 97, 1275, 105, 1485, 113, 1711, 121, 1953, 129, 2211, 137, 2485, 145, 2775, 153, 3081, 161, 3403, 169, 3741, 177, 4095, 185, 4465, 193, 4851, 201, 5253, 209
OFFSET
1,2
COMMENTS
In groups of two, add and half-multiply the integers: 0+1, (2*3)/2, 4+5, (6*7)/2, ....
From Bernard Schott, Jan 06 2020: (Start)
The bisection of this sequence gives:
For n odd = 2*k+1, k >= 0: a(2*k+1) = 8*k+1 = A017077(k),
For n even = 2*k, k >= 1: a(2*k) = T(4*k-2) = A000217(4*k-2) = (2*k-1)*(4*k-1) = A033567(k) where T(j) is the j-th triangular number. (End)
FORMULA
From Colin Barker, Jan 05 2020: (Start)
G.f.: x*(1 + 3*x + 6*x^2 + 12*x^3 - 7*x^4 + x^5) / ((1 - x)^3*(1 + x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
a(n) = -1 + 2*(-1)^n - (1/2)*(-1+7*(-1)^n)*n + (1+(-1)^n)*n^2.
(End)
E.g.f.: (1 + 4*x + 2*x^2)*cosh(x) - (3 + x)*sinh(x) - 1. - Stefano Spezia, Jan 05 2020 after Colin Barker
MATHEMATICA
a[n_]:=If[OddQ[n], 4n-3, (n-1)(2n-1)]; Array[a, 53] (* Stefano Spezia, Jan 05 2020 *)
PROG
(PARI) Vec(x*(1 + 3*x + 6*x^2 + 12*x^3 - 7*x^4 + x^5) / ((1 - x)^3*(1 + x)^3) + O(x^50)) \\ Colin Barker, Jan 06 2020
CROSSREFS
Cf. A330983.
Interspersion of A017077 and A033567 (excluding first term). - Michel Marcus, Jan 06 2020
Sequence in context: A377947 A006077 A291898 * A365919 A109612 A032668
KEYWORD
nonn,easy
AUTHOR
George E. Antoniou, Jan 05 2020
STATUS
approved