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A129696 Antidiagonal sums of triangular array T defined in A014430: T(j,k) = binomial(j+1, k)-1 for 1 <= k <= j. 11
1, 2, 5, 9, 17, 29, 50, 83, 138, 226, 370, 602, 979, 1588, 2575, 4171, 6755, 10935, 17700, 28645, 46356, 75012, 121380, 196404, 317797, 514214, 832025, 1346253, 2178293, 3524561, 5702870, 9227447, 14930334, 24157798, 39088150, 63245966 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

If T is construed as a lower triangular matrix M over the rational field, the inverse M^-1 is a lower triangular matrix containing fractions. Its row sums are the Bernoulli numbers. First column of M^-1 is 1, -1, 2/3, -1/4, -1/30, 1/12, 1/42, -1/12, ... . Multiplied by j! this gives 1, -2, 4, -6, -4, 60, 120, -3660, ... .

The Kn22 sums, see A180662 for the definition of these sums, of the 'Races with Ties' triangle A035317 lead to this sequence. - Johannes W. Meijer, Jul 20 2011

This sequence is the convolution of (1,1,2,3,5,8,13,21,...) and (1,1,2,2,3,3,4,4,5,5,...), i.e., the Fibonacci numbers (A000045) and A008619. - Clark Kimberling, May 28 2012

REFERENCES

P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note no. 12 du Centre de Calcul Scientifique de l'Armement, 1969.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

FORMULA

From Paul Barry, Jan 18 2009: (Start)

a(n) = Sum_{k=0..floor(n/2)} A000071(n-2*k+3).

a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-2*k} Fibonacci(j+1)). (End)

a(n+1) = a(n-1) + a(n) + 1 + floor(n/2) for n>1, a(0)=1, a(1)=2. - Alex Ratushnyak, Jul 30 2012

From R. J. Mathar, Jul 25 2013: (Start)

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

a(n) + a(n+1) = A001924(n+1). (End)

a(n) = Fibonacci(n+3) - 2 - floor(n/2). - Emeric Deutsch, Nov 22 2014

a(n) = (-5/4 - (-1)^n/4 + (2^(-n)*((1 - t)^n*(-2 + t) + (1 + t)^n*(2 + t)))/t + (-1 - n)/2), where t=sqrt(5). - Colin Barker, Feb 09 2017

EXAMPLE

First seven rows of T are

[ 1 ]

[ 2, 2 ]

[ 3, 5, 3 ]

[ 4, 9, 9, 4 ]

[ 5, 14, 19, 14, 5 ]

[ 6, 20, 34, 34, 20, 6 ]

[ 7, 27, 55, 69, 55, 27, 7 ].

1=1. 2=2. 3+2=5. 4+5=9. 5+9+3=17. 6+14+9=29.

MAPLE

with(combinat): a := proc (n) options operator, arrow: fibonacci(n+3)-2-floor((1/2)*n) end proc: seq(a(n), n = 1 .. 34); # Emeric Deutsch, Nov 22 2014

MATHEMATICA

a[n_] := a[n] = a[n-1] + a[n-2] + (n + Mod[n, 2])/2; a[1] = 1; a[2] = 2; Table[a[n], {n, 1, 36}] (* Jean-Fran├žois Alcover, Mar 04 2013 *)

Table[Fibonacci[n + 3] - 2 - Floor[n/2], {n, 100}] (* Vincenzo Librandi, Nov 23 2014 *)

PROG

(MAGMA) m:=36; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do for k:=1 to j do M[j, k]:=Binomial(j+1, k)-1; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; // Klaus Brockhaus, Jun 11 2007

(Python)

prpr = 1

prev = 2

for n in range(2, 100):

    print prpr,

    curr = prpr+prev + 1 + n//2

    prpr = prev

    prev = curr

# Alex Ratushnyak, Jul 30 2012

(MAGMA) [Fibonacci(n+3)-2-Floor(n/2): n in [1..40]]; // Vincenzo Librandi, Nov 23 2014

CROSSREFS

Cf. A014430, A052952 (first differences), A027641, A000045.

Sequence in context: A034329 A230441 A133470 * A082281 A285458 A000569

Adjacent sequences:  A129693 A129694 A129695 * A129697 A129698 A129699

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Jun 01 2007

EXTENSIONS

Edited and extended by Klaus Brockhaus, Jun 11 2007

STATUS

approved

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Last modified February 22 09:40 EST 2019. Contains 320390 sequences. (Running on oeis4.)