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A050488 a(n) = 3*(2^n-1) - 2*n. 23
0, 1, 5, 15, 37, 83, 177, 367, 749, 1515, 3049, 6119, 12261, 24547, 49121, 98271, 196573, 393179, 786393, 1572823, 3145685, 6291411, 12582865, 25165775, 50331597, 100663243, 201326537, 402653127, 805306309, 1610612675, 3221225409 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of words of length n+1 where first element is from {0,1,2}, other elements are from {0,1} and sequence does not decrease (for n=2 there are 3*2^2 sequences, but 000, 100, 110, 111, 200, 210, 211 decrease, so a(2) = 12-7 = 5).

Number of subgroups of C_(2^n) X C_(2^n) (see A060724).

Starting with 1 = row sums of triangle A054582. - Gary W. Adamson, Jun 23 2008

Starting with "1" equals the eigensequence of a triangle with integer squares (1, 4, 9, 16,...) as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010

(1 + 2x + 2x^2 + 2x^3 + ...)*(1 + 3x + 7x^2 + 15x^3 + ...) = (1 + 5x + 15x^2 + 37x^3 + ...). - Gary W. Adamson, Mar 14 2012

The partial sums of A033484. - J. M. Bergot, Oct 03 2012

Binomial transform is 0, 1, 7, 33, ... (shifted A066810); inverse binomial transform is 0, 1, 3, 3, ... (3 repeated). - R. J. Mathar, Oct 05 2012

Define a triangle by T(n,0) = n*(n+1) + 1, T(n,n) = n + 1, and T(r,c) = T(r-1,c-1) + T(r-1,c) otherwise; then a(n+1) is the sum of the terms of row n. - J. M. Bergot, Mar 30 2013

Starting with "1" are also the antidiagonal sums of the array formed by partial sums of integer squares (1, 4, 9, 16,...). - Luciano Ancora, Apr 24 2015

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Tamas Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015) 73-94.

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

FORMULA

Row sums of A125165: (1, 5, 15, 37...). Binomial transform of [1, 4, 6, 6, 6...] = [1, 5, 15, 37,...]. 4th diagonal from the right of A126777 = (1, 5, 15,...). - Gary W. Adamson, Dec 23 2006

a(n) = 2*a(n-1) + (2n-1). - Gary W. Adamson, Sep 30 2007

From Johannes W. Meijer, Feb 20 2009: (Start)

a(n+1) = A156920(n+1,1).

a(n+1) = A156919(n+1,1)/2^n.

a(n+1) = A142963(n+2,1)/2.

a(n) = 4a(n-1) - 5a(n-2) + 2a(n-3) for n>2 with a(0) = 0, a(1) = 1, a(2) = 5.

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

(End)

a(n) = 2*n + 2*a(n-1) - 1 (with a(0)=0). - Vincenzo Librandi, Aug 06 2010

a(n+1) = sum_{k=0..n} A000079(k) * A005408(n-k), convolution of the powers of 2 with the odd numbers. - Reinhard Zumkeller, Mar 08 2012

MATHEMATICA

s=0; lst={s}; Do[s+=n+=s-1; AppendTo[lst, s], {n, 2, 5!, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 07 2008 *)

Table[3(2^n-1)-2n, {n, 0, 30}] (* or *) LinearRecurrence[{4, -5, 2}, {0, 1, 5}, 40] (* Harvey P. Dale, Apr 09 2018 *)

PROG

(Haskell)

a050488 n = sum $ zipWith (*) a000079_list (reverse $ take n a005408_list)

-- Reinhard Zumkeller, Jul 24 2015

(PARI) a(n)=3*(2^n-1)-2*n \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

A050487(2^m-1).

Equals (1/2) A051667.

Cf. A000225, A054852, A126277, A125165, A156925, A000079, A005408.

Sequence in context: A146797 A213487 A005491 * A142964 A188282 A014316

Adjacent sequences:  A050485 A050486 A050487 * A050489 A050490 A050491

KEYWORD

nonn,easy

AUTHOR

James A. Sellers, Dec 26 1999

STATUS

approved

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Last modified May 24 22:50 EDT 2018. Contains 304537 sequences. (Running on oeis4.)