A193911 Sums of the diagonals of the matrix formed by listing the h-Stohr sequences in increasing order.

1, 3, 7, 14, 25, 43, 69, 110, 167, 255, 375, 558, 805, 1179, 1681, 2438, 3451, 4975, 7011, 10070, 14153, 20283, 28461, 40734, 57103, 81663, 114415, 163550, 229069, 327355, 458409, 654998, 917123, 1310319, 1834587, 2620998, 3669553, 5242395, 7339525, 10485230

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Eric Weisstein's World of Mathematics, Stöhr Sequence.

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,1,4,-2).

Formula

All h-Stohr sequences have formula: h terms 1,2,..,2^(n-1),..,2^(h-1) and then continue (2^h-1)(n-h)+1. - Henry Bottomley, Feb 04 2000

So we get the sums from the piecewise function:

for odd n>=1, a(n)=2^((n+1)/2)-n+((n+1)/2)-2+Sum_{i=0..((n+1)/2)-1}(2*i+1)*(2^(((n+1)/2)-i) -1);

for even n>=2, a(n)=2^((n/2)+2)-n-4+Sum_{i=0..(n/2)-1}(2*i+1)*(2^((n/2)-i) -1). - Jeffrey R. Goodwin, Aug 09 2011

Let odd m>=3, then a(n)=a(m)-A000295(((m+1)/2)+1), where n>=2 is even. - Jeffrey R. Goodwin, Aug 09 2011

Let even m>=2, then a(n)=a(m)-A077802(m/2)=a(m)-A095151(m/2), where n>=1 is odd. - Jeffrey R. Goodwin, Aug 09 2011

G.f.:(1 + x - x^2)/((-1 + x)^3*(-1 - x + 2*x^2 + 2*x^3)). - Alexander R. Povolotsky, Aug 09 2011

a(n+4) = -2*a(n)+3*a(n+2)+n+5. - Alexander R. Povolotsky, Aug 09 2011

a(n) = 1/8*(2^(n/2+2)*((10-7*sqrt(2))*(-1)^n+10+7*sqrt(2))-(-1)^n-2*n*(n+12)-79). - _Alexander R. Povolotsky, Aug 09 2011

Example

Portion of the first three rows:
A033627, 2-Stohr  1  2  4  7
A026474, 3-Stohr  1  2  4  8
A051039, 4-Stohr  1  2  4  8
Thus a(1)=1, a(2)=2+1=3, and a(3)=4+2+1=7.

Mathematica

A193911=t={0, 1}; Do[AppendTo[t, t[[-2]]+t[[-1]]]; AppendTo[t, 2*t[[-2]]], {n, 41}]; Drop[Nest[Accumulate, t, 2], 1] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)
LinearRecurrence[{2, 2, -6, 1, 4, -2}, {1, 3, 7, 14, 25, 43}, 40] (* Harvey P. Dale, Jun 20 2015 *)

Crossrefs

Sequence in context: A051170 A011795 A265252 * A206417 A207381 A343017

Adjacent sequences: A193908 A193909 A193910 * A193912 A193913 A193914

Keyword

nonn,easy

Author

Jeffrey R. Goodwin, Aug 08 2011

Status

approved