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Alternatively add and half-multiply pairs of the nonnegative integers.
2

%I #30 Feb 28 2020 23:19:14

%S 1,3,9,21,17,55,25,105,33,171,41,253,49,351,57,465,65,595,73,741,81,

%T 903,89,1081,97,1275,105,1485,113,1711,121,1953,129,2211,137,2485,145,

%U 2775,153,3081,161,3403,169,3741,177,4095,185,4465,193,4851,201,5253,209

%N Alternatively add and half-multiply pairs of the nonnegative integers.

%C In groups of two, add and half-multiply the integers: 0+1, (2*3)/2, 4+5, (6*7)/2, ....

%C From _Bernard Schott_, Jan 06 2020: (Start)

%C The bisection of this sequence gives:

%C For n odd = 2*k+1, k >= 0: a(2*k+1) = 8*k+1 = A017077(k),

%C 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)

%H Colin Barker, <a href="/A330987/b330987.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).

%F From _Colin Barker_, Jan 05 2020: (Start)

%F G.f.: x*(1 + 3*x + 6*x^2 + 12*x^3 - 7*x^4 + x^5) / ((1 - x)^3*(1 + x)^3).

%F a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.

%F a(n) = -1 + 2*(-1)^n - (1/2)*(-1+7*(-1)^n)*n + (1+(-1)^n)*n^2.

%F (End)

%F E.g.f.: (1 + 4*x + 2*x^2)*cosh(x) - (3 + x)*sinh(x) - 1. - _Stefano Spezia_, Jan 05 2020 after _Colin Barker_

%t a[n_]:=If[OddQ[n],4n-3,(n-1)(2n-1)]; Array[a,53] (* _Stefano Spezia_, Jan 05 2020 *)

%o (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

%Y Cf. A330983.

%Y Interspersion of A017077 and A033567 (excluding first term). - _Michel Marcus_, Jan 06 2020

%K nonn,easy

%O 1,2

%A _George E. Antoniou_, Jan 05 2020