A116555 -id:A116555 - cp's OEIS frontend

A116528 a(0)=0, a(1)=1, and for n>=2, a(2n) = a(n), a(2n+1) = 2*a(n) + a(n+1).

0, 1, 1, 3, 1, 5, 3, 7, 1, 7, 5, 13, 3, 13, 7, 15, 1, 9, 7, 19, 5, 23, 13, 29, 3, 19, 13, 33, 7, 29, 15, 31, 1, 11, 9, 25, 7, 33, 19, 43, 5, 33, 23, 59, 13, 55, 29, 61, 3, 25, 19, 51, 13, 59, 33, 73, 7, 43, 29, 73, 15, 61, 31, 63, 1, 13

Offset: 0

Roger L. Bagula, Mar 15 2006

Equals row 2 of the array in A178239, an infinite set of sequences of the form a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1). - Gary W. Adamson, May 23 2010

Given an infinite lower triangular matrix M with (1, 1, 2, 0, 0, 0, ...) in every column, shifted down twice for columns k>1; lim_{n->infinity} M^n = A116528, the left-shifted vector considered as a sequence with offset 1. - Gary W. Adamson, May 05 2010

G. C. Greubel, Table of n, a(n) for n = 0..2500
H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.
Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

Cf. A006046, A116529, A116552, A116553, A116554, A116555.

a:=func< n | n lt 2 select n else ((n mod 2) eq 0) select Self(Round((n+1)/2)) else (2*Self(Round(n/2)) + Self(Round((n+2)/2))) >;
[a(n): n in [0..70]]; // G. C. Greubel, Jul 07 2019

A116528 := proc(n)
   option remember;
   if n <= 1 then
      n;
   elif type(n,'even') then
      procname(n/2) ;
   else
      2* procname((n-1)/2)+procname((n+1)/2) ;
   end if;
end proc:
seq(A116528(n),n=0..70) ; # R. J. Mathar, Nov 16 2011

b[0]:= 0; b[1]:= 1; b[n_?EvenQ]:= b[n] = b[n/2]; b[n_?OddQ]:= b[n] = 2*b[(n-1)/2] + b[(n+1)/2]; a = Table[b[n], {n, 1, 70}]

a(n) = if(n<2, n, if(n%2==0, a(n/2), 2*a((n-1)/2) + a((n+1)/2))); \\ G. C. Greubel, Jul 07 2019

def a(n):
    if (n<2): return n
    elif (mod(n,2)==0): return a(n/2)
    else: return 2*a((n-1)/2) + a((n+1)/2)
[a(n) for n in (0..70)] # G. C. Greubel, Jul 07 2019

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 2*x^(2^(k+1))). - Ilya Gutkovskiy, Jul 07 2019

Edited by G. C. Greubel, Oct 30 2016

A116529 a(2n + 1) = a(n), a(2n + 2) = 2*a(n) + a(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 7, 5, 12, 1, 7, 4, 9, 3, 10, 7, 17, 2, 11, 7, 16, 5, 17, 12, 29, 1, 14, 7, 15, 4, 15, 9, 22, 3, 15, 10, 23, 7, 24, 17, 41, 2, 21, 11, 24, 7, 25, 16, 39, 5, 26, 17, 39, 12, 41, 29, 70, 1, 31, 14, 29, 7, 28, 15

Offset: 0

Roger L. Bagula, Mar 15 2006

G. C. Greubel, Table of n, a(n) for n = 0..2500
Kevin Ryde, PARI/GP Code

Cf. A116528, A116552, A116553, A116554, A116555.

gg:= 1:
for iter from 1 to 7 do
  gg:= convert(series(1+(x^4+2*x^2+x)*eval(gg,x=x^2), x, 2^iter+1),polynom)
od:
seq(coeff(gg,x,n),n=0..2^7); # Robert Israel, Nov 13 2017

b[0] := 0; b[1] := 1;
b[n_?EvenQ] := b[n] = b[n/2];
b[n_?OddQ] := b[n] = 2*b[(n - 1)/2] + b[(n - 3)/2];
Table[b[n], {n, 1, 70}]

PARI
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\\ See links.
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From G. C. Greubel, Oct 30 2016: (Start)
a(2*n + 1) = a(n), n>=1.
a(2*n + 2) = 2*a(n) + a(n-1), n>=1. (End)
G.f. g(x) satisfies g(x) = 1 + (x^4+2*x^2+x)*g(x^2). - Robert Israel, Nov 13 2017

New name using formula, Joerg Arndt, Dec 17 2022

A116552 a(2n+1) = 3a(n), a(2n+2) = 4a(n) + a(n-1).

Original entry on oeis.org

1, 3, 4, 9, 13, 12, 19, 27, 40, 39, 61, 36, 61, 57, 88, 81, 127, 120, 187, 117, 196, 183, 283, 108, 205, 183, 280, 171, 289, 264, 409, 243, 412, 381, 589, 360, 607, 561, 868, 351, 655, 588, 901, 549, 928, 849, 1315, 324, 715, 615, 928, 549, 937, 840, 1303, 513

Offset: 0

Roger L. Bagula, Mar 15 2006

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A116528, A116529, A116553, A116554, A116555.

b[0] := 0; b[1] := 1;
b[n_?EvenQ] := b[n] = 3*b[n/2];
b[n_?OddQ] := b[n] = 4*b[(n - 1)/2] + b[(n - 3)/2];
Table[b[n], {n, 1, 70}]

From G. C. Greubel, Oct 30 2016: (Start)
a(2*n+1) = 3*a(n).
a(2*n+2) = 4*a(n) + a(n-1). (End)

New name using formula, Joerg Arndt, Dec 17 2022

A116553 a(2n+1) = 5a(n), a(2n+2) = 6a(n) + a(n-1).

Original entry on oeis.org

1, 5, 6, 25, 31, 30, 41, 125, 156, 155, 211, 150, 211, 205, 276, 625, 791, 780, 1061, 775, 1086, 1055, 1421, 750, 1111, 1055, 1416, 1025, 1441, 1380, 1861, 3125, 4026, 3955, 5371, 3900, 5471, 5305, 7146, 3875, 5711, 5430, 7291, 5275, 7416, 7105, 9581, 3750

Offset: 0

Roger L. Bagula, Mar 15 2006

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A116528, A116529, A116552, A116554, A116555.

b[0] := 0; b[1] := 1;
b[n_?EvenQ] := b[n] = 5*b[n/2];
b[n_?OddQ] := b[n] = 6*b[(n - 1)/2] + b[(n - 3)/2];
Table[b[n], {n, 1, 70}]

From G. C. Greubel, Oct 30 2016: (Start)
a(2*n+1) = 5*a(n).
a(2*n+2) = 6*a(n) + a(n-1). (End)

New name using formula, Joerg Arndt, Dec 17 2022

A116554 a(2n+1) = 7a(n), a(2n+2) = 8a(n) + a(n-1).

Original entry on oeis.org

1, 7, 8, 49, 57, 56, 71, 343, 400, 399, 505, 392, 505, 497, 624, 2401, 2815, 2800, 3543, 2793, 3592, 3535, 4439, 2744, 3641, 3535, 4432, 3479, 4481, 4368, 5489, 16807, 19832, 19705, 24921, 19600, 25215, 24801, 31144, 19551, 25887, 25144, 31529

Offset: 0

Roger L. Bagula, Mar 15 2006

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A116528, A116529, A116552, A116553, A116555.

b[0] := 0; b[1] := 1;
b[n_?EvenQ] := b[n] = 7*b[n/2];
b[n_?OddQ] := b[n] = 8*b[(n - 1)/2] + b[(n - 3)/2];
Table[b[n], {n, 1, 70}]

From G. C. Greubel, Oct 30 2016: (Start)
a(2*n+1) = 7*a(n).
a(2*n+2) = 8*a(n) + a(n-1). (End)

Edited by N. J. A. Sloane, Feb 11 2007

New name using formula, Joerg Arndt, Dec 17 2022

cp's OEIS Frontend

A116528 a(0)=0, a(1)=1, and for n>=2, a(2*n) = a(n), a(2*n+1) = 2*a(n) + a(n+1).

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A116529 a(2*n + 1) = a(n), a(2*n + 2) = 2*a(n) + a(n-1).

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A116552 a(2*n+1) = 3*a(n), a(2*n+2) = 4*a(n) + a(n-1).

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Mathematica

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A116553 a(2*n+1) = 5*a(n), a(2*n+2) = 6*a(n) + a(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A116554 a(2*n+1) = 7*a(n), a(2*n+2) = 8*a(n) + a(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A116528 a(0)=0, a(1)=1, and for n>=2, a(2n) = a(n), a(2n+1) = 2*a(n) + a(n+1).

A116529 a(2n + 1) = a(n), a(2n + 2) = 2*a(n) + a(n-1).

A116552 a(2n+1) = 3a(n), a(2n+2) = 4a(n) + a(n-1).

A116553 a(2n+1) = 5a(n), a(2n+2) = 6a(n) + a(n-1).

A116554 a(2n+1) = 7a(n), a(2n+2) = 8a(n) + a(n-1).