A154221 -id:A154221 - cp's OEIS frontend

A154222 Row sums of number triangle A154221.

1, 2, 4, 8, 17, 38, 87, 200, 457, 1034, 2315, 5132, 11277, 24590, 53263, 114704, 245777, 524306, 1114131, 2359316, 4980757, 10485782, 22020119, 46137368, 96469017, 201326618, 419430427, 872415260, 1811939357, 3758096414, 7784628255, 16106127392, 33285996577

Offset: 0

Paul Barry, Jan 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Cf. A045618, A154221.

[(1/4)*(4*(n+1)+(n-1)*2^n+0^n): n in [0..35]]; // Vincenzo Librandi, Sep 07 2016

Join[{1},LinearRecurrence[{6, -13, 12, -4}, {2, 4, 8,17}, 25]] (* or *) Table[(1/4)*( 4*(n+1) + (n-1)*2^n + 0^n), {n,0,25}] (* G. C. Greubel, Sep 06 2016 *)

Vec((x^4-2*x^3+5*x^2-4*x+1)/((x-1)^2*(2*x-1)^2) + O(x^100)) \\ Colin Barker, Oct 11 2014

a(n) = (1/4)*( 4*(n+1) + (n-1)*2^n + 0^n).
From Colin Barker, Oct 11 2014: (Start)
a(n) = A045618(n-4) + 2^n for n>3.
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4) for n>4.
a(n) = (4 - 2^n + (4+2^n)*n)/4 for n>0.
G.f.: (x^4 - 2*x^3 + 5*x^2 - 4*x + 1) / ((x-1)^2*(2*x-1)^2).
(End)
E.g.f.: (1/4)*(1 + 4*(1 + x)*exp(x) + (2*x - 1)*exp(2*x)). - G. C. Greubel, Sep 06 2016

More terms and xrefs from Colin Barker, Oct 11 2014

A154223 Diagonal sums of number triangle A154221.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 16, 32, 61, 125, 246, 502, 999, 2023, 4040, 8136, 16265, 32649, 65290, 130826, 261643, 523787, 1047564, 2096140, 4192269, 8386573, 16773134, 33550350, 67100687, 134209551, 268419088, 536854544, 1073709073, 2147450897, 4294901778, 8589869074, 17179738131, 34359607315, 68719214612, 137438691348

Offset: 0

Paul Barry, Jan 05 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-9,4,6,-4).

I:=[1,1,2,3,5,9,16]; [n le 7 select I[n] else 3*Self(n-1)+Self(n-2)-9*Self(n-3)+4*Self(n-4)+6*Self(n-5)-4*Self(n-6): n in [1..40]]; // Vincenzo Librandi, Sep 07 2016

A154223 := proc(n)
    a := 0 ;
    for npr from n by -1 do
        k := n-npr ;
        if k <= npr then
            a := a+A154221(npr,k) ;
        else
            return a;
        end if;
    end do:
end proc: # R. J. Mathar, Feb 05 2015

Join[{1}, LinearRecurrence[{3, 1, -9, 4, 6, -4}, {1, 2, 3, 5, 9, 16}, 25]] (* G. C. Greubel, Sep 06 2016 *)

Vec((1-x-x^2)*(1-x-2*x^2+2*x^3-x^4)/((1-x)^2*(1+x)*(1-2*x)*(1-2*x^2)) + O(x^40)) \\ Colin Barker, Sep 07 2016

G.f.: (1 - x - x^2)*(1 - x - 2*x^2 + 2*x^3 - x^4) / ((1-x)^2*(1+x)*(1-2*x)*(1-2*x^2)). - Colin Barker, Sep 07 2016

cp's OEIS Frontend

A154222 Row sums of number triangle A154221.

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