A009947 Sequence of nonnegative integers, but insert n/2 after every even number n.

0, 0, 1, 2, 1, 3, 4, 2, 5, 6, 3, 7, 8, 4, 9, 10, 5, 11, 12, 6, 13, 14, 7, 15, 16, 8, 17, 18, 9, 19, 20, 10, 21, 22, 11, 23, 24, 12, 25, 26, 13, 27, 28, 14, 29, 30, 15, 31, 32, 16, 33, 34, 17, 35, 36, 18, 37, 38, 19, 39, 40, 20, 41, 42, 21, 43, 44, 22, 45, 46

Offset: 0

Author

Bill Gosper

Keywords

• nonn
• easy

Comments

Coefficients in expansion of e/3 = Sum_{n>=1} a(n)/n!, using greedy algorithm.

Numerators of Peirce sequence of order 2.

References

• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 151.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A071281, A214090 (parity), A001477.

Cf. A166711 (signed).

Programs

Haskell

a009947 n = a009947_list !! n
a009947_list = concatMap (\x -> [2 * x, x, 2 * x + 1]) [0..]
-- Reinhard Zumkeller, Jul 06 2012

Maple

A009947 := proc(a,n) local i,b,c; b := a; c := [ floor(b) ]; for i from 1 to n-1 do b := b-c[ i ]/i!; c := [ op(c), floor(b*(i+1)!) ]; od; c; end:

Mathematica

Flatten[Table[If[EvenQ[n],{n,n/2},n],{n,0,40}]] (* Harvey P. Dale, Feb 17 2016 *)

PARI

a(n)=if(n%3==1, n\3, n\3*2+!!(n%3)) \\ Charles R Greathouse IV, Sep 02 2015

PARI

concat(vector(2), Vec(x^2*(x^3+x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2) + O(x^100))) \\ Colin Barker, Mar 29 2017

Formula

G.f.: x^2*(x^3+x^2+2*x+1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Aug 31 2013

a(n) = (n^2-n+floor(n/3)*(18*floor(n/3)^2-3*(4*n-7)*floor(n/3)+2*n^2-10*n+7))/2. - Luce ETIENNE, Mar 29 2017

Sum_{n>=2} (-1)^n/a(n) = Pi/4 + log(2)/2. - Amiram Eldar, Jan 11 2023