A071281 -id:A071281 - cp's OEIS frontend

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

Bill Gosper

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

Numerators of Peirce sequence of order 2.

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

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).

Cf. A071281, A214090 (parity), A001477.

Cf. A166711 (signed).

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

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:

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

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

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

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

A071288 Denominators of Peirce sequence of order 6.

Original entry on oeis.org

2, 4, 6, 5, 3, 1, 6, 5, 4, 6, 3, 5, 4, 6, 2, 5, 6, 3, 4, 5, 6, 2, 4, 6, 5, 3, 1, 6, 5, 4, 6, 3, 5, 4, 6, 2, 5, 6, 3, 4, 5, 6, 2, 4, 6, 5, 3, 1, 6, 5, 4, 6, 3, 5, 4, 6, 2, 5, 6, 3, 4, 5, 6, 2, 4, 6, 5, 3, 1, 6, 5, 4, 6, 3, 5, 4, 6, 2, 5, 6

Offset: 0

N. J. A. Sloane, Jun 11 2002

			The Peirce sequences of orders 1, 2, 3, 4, 5 begin:
0/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 ...
0/2 0/1 1/2 2/2 1/1 3/2 4/2 2/1 ... (numerators are A009947)
0/2 0/3 0/1 1/3 1/2 2/3 2/2 3/3 ...
0/2 0/4 0/3 0/1 1/4 1/3 2/4 1/2 ...
0/2 0/4 0/5 0/3 0/1 1/5 1/4 1/3 ...

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

Cf. A071281-A071287.

Conjectures from Colin Barker, Mar 29 2017: (Start)
G.f.: (6*x^20 + 5*x^19 + 4*x^18 + 3*x^17 + 6*x^16 + 5*x^15 + 2*x^14 + 6*x^13 + 4*x^12 + 5*x^11 + 3*x^10 + 6*x^9 + 4*x^8 + 5*x^7 + 6*x^6 + x^5 + 3*x^4 + 5*x^3 + 6*x^2 + 4*x + 2)/(1 - x^21).
a(n) = a(n-21) for n>20.
(End)

More terms from Reiner Martin, Oct 15 2002

A071282 Denominators of Peirce sequence of order 3.

Original entry on oeis.org

2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3, 1, 3, 2, 3, 2, 3

Offset: 0

N. J. A. Sloane, Jun 11 2002

			The Peirce sequences of orders 1, 2, 3, 4, 5 begin:
0/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 ...
0/2 0/1 1/2 2/2 1/1 3/2 4/2 2/1 ... (numerators are A009947)
0/2 0/3 0/1 1/3 1/2 2/3 2/2 3/3 ...
0/2 0/4 0/3 0/1 1/4 1/3 2/4 1/2 ...
0/2 0/4 0/5 0/3 0/1 1/5 1/4 1/3 ...

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

Cf. A071281-A071288.

Conjectures from Colin Barker, Mar 29 2017: (Start)
G.f.: (3*x^5 + 2*x^4 + 3*x^3 + x^2 + 3*x + 2)/(1 - x^6).
a(n) = a(n-6) for n>5.
(End)

More terms from Reiner Martin, Oct 15 2002

A071287 Numerators of Peirce sequence of order 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 2, 3, 1, 3, 4, 2, 3, 4, 5, 2, 4, 6, 5, 3, 1, 7, 6, 5, 8, 4, 7, 6, 9, 3, 8, 10, 5, 7, 9, 11, 4, 8, 12, 10, 6, 2, 13, 11, 9, 14, 7, 12, 10, 15, 5, 13, 16, 8, 11, 14, 17, 6, 12, 18, 15, 9, 3, 19, 16, 13, 20, 10, 17, 14, 21, 7, 18, 22

Offset: 0

N. J. A. Sloane, Jun 11 2002

			The Peirce sequences of orders 1, 2, 3, 4, 5 begin:
0/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 ...
0/2 0/1 1/2 2/2 1/1 3/2 4/2 2/1 ... (numerators are A009947)
0/2 0/3 0/1 1/3 1/2 2/3 2/2 3/3 ...
0/2 0/4 0/3 0/1 1/4 1/3 2/4 1/2 ...
0/2 0/4 0/5 0/3 0/1 1/5 1/4 1/3 ...

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

Cf. A071281-A071288.

Conjectures from Colin Barker, Mar 29 2017: (Start)
G.f.: x^6*(x^41 + x^40 + x^39 + x^38 + 2*x^37 + 2*x^36 + x^35 + 3*x^34 + 2*x^33 + 3*x^32 + 2*x^31 + 4*x^30 + 3*x^29 + 4*x^28 + 5*x^27 + x^26 + 3*x^25 + 5*x^24 + 6*x^23 + 4*x^22 + 2*x^21 + 5*x^20 + 4*x^19 + 3*x^18 + 2*x^17 + 4*x^16 + 3*x^15 + x^14 + 3*x^13 + 2*x^12 + 2*x^11 + x^10 + 2*x^9 + x^8 + x^7 + x^6)/(x^42 - 2*x^21 + 1).
a(n) = 2*a(n-21) - a(n-42) for n>41.
(End)

More terms from Reiner Martin, Oct 15 2002

A071283 Numerators of Peirce sequence of order 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 2, 3, 2, 4, 3, 1, 5, 4, 6, 3, 5, 7, 4, 8, 6, 2, 9, 7, 10, 5, 8, 11, 6, 12, 9, 3, 13, 10, 14, 7, 11, 15, 8, 16, 12, 4, 17, 13, 18, 9, 14, 19, 10, 20, 15, 5, 21, 16, 22, 11, 17, 23, 12, 24, 18, 6, 25, 19, 26, 13, 20, 27, 14, 28, 21, 7, 29, 22, 30, 15, 23, 31

Offset: 0

N. J. A. Sloane, Jun 11 2002

			The Peirce sequences of orders 1, 2, 3, 4, 5 begin:
0/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 ...
0/2 0/1 1/2 2/2 1/1 3/2 4/2 2/1 ... (numerators are A009947)
0/2 0/3 0/1 1/3 1/2 2/3 2/2 3/3 ...
0/2 0/4 0/3 0/1 1/4 1/3 2/4 1/2 ...
0/2 0/4 0/5 0/3 0/1 1/5 1/4 1/3 ...

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

Cf. A071281-A071288.

Conjectures from Colin Barker, Mar 29 2017: (Start)
G.f.: x^4*(x^19 + x^18 + x^17 + 2*x^16 + 2*x^15 + 3*x^14 + x^13 + 3*x^12 + 4*x^11 + 2*x^10 + 3*x^9 + 2*x^8 + x^7 + 2*x^6 + x^5 + x^4)/(x^20 - 2*x^10 + 1).
a(n) = 2*a(n-10) - a(n-20) for n>19.
(End)

More terms from Reiner Martin, Oct 15 2002

A071284 Denominators of Peirce sequence of order 4.

Original entry on oeis.org

2, 4, 3, 1, 4, 3, 4, 2, 3, 4, 2, 4, 3, 1, 4, 3, 4, 2, 3, 4, 2, 4, 3, 1, 4, 3, 4, 2, 3, 4, 2, 4, 3, 1, 4, 3, 4, 2, 3, 4, 2, 4, 3, 1, 4, 3, 4, 2, 3, 4, 2, 4, 3, 1, 4, 3, 4, 2, 3, 4, 2, 4, 3, 1, 4, 3, 4, 2, 3, 4, 2, 4, 3, 1, 4, 3, 4, 2, 3, 4

Offset: 0

N. J. A. Sloane, Jun 11 2002

			The Peirce sequences of orders 1, 2, 3, 4, 5 begin:
0/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 ...
0/2 0/1 1/2 2/2 1/1 3/2 4/2 2/1 ... (numerators are A009947)
0/2 0/3 0/1 1/3 1/2 2/3 2/2 3/3 ...
0/2 0/4 0/3 0/1 1/4 1/3 2/4 1/2 ...
0/2 0/4 0/5 0/3 0/1 1/5 1/4 1/3 ...

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

Cf. A071281-A071288.

Conjectures from Colin Barker, Mar 29 2017: (Start)
G.f.: (4*x^9 + 3*x^8 + 2*x^7 + 4*x^6 + 3*x^5 + 4*x^4 + x^3 + 3*x^2 + 4*x + 2)/(1 - x^10).
a(n) = a(n-10) for n>9.
(End)

More terms from Reiner Martin, Oct 15 2002

A071285 Numerators of Peirce sequence of order 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 1, 3, 2, 3, 4, 2, 4, 5, 3, 1, 6, 5, 4, 7, 6, 3, 8, 5, 7, 9, 4, 8, 10, 6, 2, 11, 9, 7, 12, 10, 5, 13, 8, 11, 14, 6, 12, 15, 9, 3, 16, 13, 10, 17, 14, 7, 18, 11, 15, 19, 8, 16, 20, 12, 4, 21, 17, 13, 22, 18, 9, 23, 14, 19, 24, 10, 20, 25, 15, 5

Offset: 0

N. J. A. Sloane, Jun 11 2002

			The Peirce sequences of orders 1, 2, 3, 4, 5 begin:
0/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 ...
0/2 0/1 1/2 2/2 1/1 3/2 4/2 2/1 ... (numerators are A009947)
0/2 0/3 0/1 1/3 1/2 2/3 2/2 3/3 ...
0/2 0/4 0/3 0/1 1/4 1/3 2/4 1/2 ...
0/2 0/4 0/5 0/3 0/1 1/5 1/4 1/3 ...

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

Cf. A071281-A071288.

Conjectures from Colin Barker, Mar 29 2017: (Start)
G.f.: x^5*(x^29 + x^28 + x^27 + 2*x^26 + x^25 + 2*x^24 + 3*x^23 + 2*x^22 + 3*x^21 + 4*x^20 + x^19 + 3*x^18 + 5*x^17 + 4*x^16 + 2*x^15 + 4*x^14 + 3*x^13 + 2*x^12 + 3*x^11 + x^10 + 2*x^9 + 2*x^8 + x^7 + x^6 + x^5)/(x^30 - 2*x^15 + 1).
a(n) = 2*a(n-15) - a(n-30) for n>29.
(End)

Corrected and extended by Reiner Martin, Oct 15 2002

A071286 Denominators of Peirce sequence of order 5.

Original entry on oeis.org

2, 4, 5, 3, 1, 5, 4, 3, 5, 4, 2, 5, 3, 4, 5, 2, 4, 5, 3, 1, 5, 4, 3, 5, 4, 2, 5, 3, 4, 5, 2, 4, 5, 3, 1, 5, 4, 3, 5, 4, 2, 5, 3, 4, 5, 2, 4, 5, 3, 1, 5, 4, 3, 5, 4, 2, 5, 3, 4, 5, 2, 4, 5, 3, 1, 5, 4, 3, 5, 4, 2, 5, 3, 4, 5, 2, 4, 5, 3, 1

Offset: 0

N. J. A. Sloane, Jun 11 2002

			The Peirce sequences of orders 1, 2, 3, 4, 5 begin:
0/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 ...
0/2 0/1 1/2 2/2 1/1 3/2 4/2 2/1 ... (numerators are A009947)
0/2 0/3 0/1 1/3 1/2 2/3 2/2 3/3 ...
0/2 0/4 0/3 0/1 1/4 1/3 2/4 1/2 ...
0/2 0/4 0/5 0/3 0/1 1/5 1/4 1/3 ...

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

Cf. A071281-A071288.

Conjectures from Colin Barker, Mar 29 2017: (Start)
G.f.: (5*x^14 + 4*x^13 + 3*x^12 + 5*x^11 + 2*x^10 + 4*x^9 + 5*x^8 + 3*x^7 + 4*x^6 + 5*x^5 + x^4 + 3*x^3 + 5*x^2 + 4*x + 2)/(1 - x^15).
a(n) = a(n-15) for n>14.
(End)

More terms from Reiner Martin, Oct 15 2002

cp's OEIS Frontend

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

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A071288 Denominators of Peirce sequence of order 6.

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A071282 Denominators of Peirce sequence of order 3.

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A071287 Numerators of Peirce sequence of order 6.

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A071283 Numerators of Peirce sequence of order 4.

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A071284 Denominators of Peirce sequence of order 4.

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A071285 Numerators of Peirce sequence of order 5.

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A071286 Denominators of Peirce sequence of order 5.

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