A178748 Total number of '1' bits in the terms of 'rows' of A178746.

1, 2, 7, 14, 37, 80, 187, 410, 913, 1988, 4327, 9326, 20029, 42776, 91027, 192962, 407785, 859244, 1805887, 3786518, 7922581, 16544192, 34486507, 71769194, 149130817, 309446420, 641262487, 1327264190, 2744006893, 5666970728, 11691855427, 24099538706, 49630733209

Offset: 0

Author

David Scambler, Jun 09 2010

Keywords

• nonn
• easy

Comments

Sum of adjacent terms equals the difference of adjacent terms in A127981. - David Scambler, Jun 10 2010

Examples

			a(0) = bitcount(1) = 1.
a(1) = bitcount(3) = 2.
a(2) = bitcount(6) + bitcount(6) + bitcount(7) = 2 + 2 + 3 = 7.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,3,-4,-4).

Crossrefs

Cf. A178747 (sum of terms in rows of A178746).

Cf. A127981.

Programs

Mathematica

LinearRecurrence[{2,3,-4,-4},{1,2,7,14},40] (* Harvey P. Dale, Aug 27 2021 *)

PARI

seq(n)={my(a=vector(n+1), f=0, p=0, k=1, s=0); while(k<=#a, my(b=bitxor(p+1,p)); f=bitxor(f,b); p=bitxor(p, bitand(b,f)); if(p>2^k, a[k]=s; k++; s=0); s+=hammingweight(p)); a} \\ Andrew Howroyd, Mar 03 2020

PARI

a(n) = {(2^n*(3*n+8) + (3*n+1)*(-1)^n)/9} \\ Andrew Howroyd, Mar 03 2020

PARI

Vec((1 - 2*x^3) / ((1 + x)^2*(1 - 2*x)^2) + O(x^30)) \\ Colin Barker, Mar 04 2020

Formula

G.f: (1/2)*x^3 - 1/4 + (x^4 + x^3 - (3/4)*x^2 - (1/2)*x + 1/4)*F(x) = 0. [From GUESSS]

From David Scambler, Jun 10 2010: (Start)

a(n) = (2^n*(3*n+8) + (3*n+1)*(-1)^n)/9.

a(n) + a(n-1) = A127981(n+1) - A127981(n).

(End)

From Colin Barker, Mar 04 2020: (Start)

G.f.: (1 - 2*x^3) / ((1 + x)^2*(1 - 2*x)^2).

a(n) = 2*a(n-1) + 3*a(n-2) - 4*a(n-3) - 4*a(n-4) for n>3.

(End)

Extensions

Terms a(16) and beyond from Andrew Howroyd, Mar 03 2020