A079824 Sum of numbers in n-th upward diagonal of triangle in A079823.

1, 2, 7, 12, 25, 37, 62, 84, 125, 160, 221, 272, 357, 427, 540, 632, 777, 894, 1075, 1220, 1441, 1617, 1882, 2092, 2405, 2652, 3017, 3304, 3725, 4055, 4536, 4912, 5457, 5882, 6495, 6972, 7657, 8189, 8950, 9540, 10381, 11032, 11957, 12672, 13685, 14467, 15572

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

From _Philippe Deléham_, Feb 16 2004: (Start)

a(2*n) = (n/6)*(7*n^2 + 3*n + 2);

a(2*n-1) = (n/6)*(7*n^2 - 6*n + 5). (End)

G.f.: x*(1+x+2*x^2+2*x^3+x^4) / ( (1+x)^3*(1-x)^4 ). - _R. J. Mathar_, Aug 23 2012

From _Richard Peterson_, Aug 19 2020: (Start)

a(2*n) - a(2*n-1) = A000326(n).

a(2*n+1) - a(2*n) = n^2 + (n+1)^2. (End)

a(n) = (15 + 25*n + 15*n^2 + 14*n^3 - 3*(-1)^n*(5 + n*(3 + n)))/96. - _Torlach Rush_, Aug 14 2022

E.g.f.: (1/48)*( x*(33 + 27*x + 7*x^2)*cosh(x) + (15 + 21*x + 30*x^2 + 7*x^3)*sinh(x) ). - _G. C. Greubel_, Dec 08 2023

Maple

A079824aux := proc(n, k)
    A000124(n)+k ;
end proc:
A079824 := proc(n)
    local a, k, n0 ;
    n0 := n-1 ;
    a := 0 ;
    for k from 0 to floor(n0/2) do
        a := a+A079824aux(n0-k, k) ;
    end do:
    a ;
end proc: # _R. J. Mathar_, Aug 23 2012

Mathematica

LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {1, 2, 7, 12, 25, 37, 62}, 60] (* _Harvey P. Dale_, May 06 2014 *)

Prog

(Python)
def a(n): return (15 + 25*n + 15*(n**2) + 14*(n**3) - 3*(((-1)**n))*(5 + n*(3 + n))) // 96 # _Torlach Rush_, Aug 14 2022
(Magma) [(15+25*n+15*n^2+14*n^3 -3*(-1)^n*(5+3*n+n^2))/96: n in [1..60]]; // _G. C. Greubel_, Dec 08 2023
(SageMath) [(15+25*n+15*n^2+14*n^3 -3*(-1)^n*(5+3*n+n^2))/96 for n in range(1, 61)] # _G. C. Greubel_, Dec 08 2023

Crossrefs

Cf. A000326, A079823, A185787 and A185788 (bisections).

Sequence in context: A288656 A180804 A122264 * A059329 A242201 A350093

Adjacent sequences: A079821 A079822 A079823 * A079825 A079826 A079827

Keyword

base,easy,nonn

Author

_Amarnath Murthy_, Feb 11 2003

Extensions

More terms from Jason D. W. Taff (jtaff(AT)jburroughs.org), Oct 31 2003

More terms from _Philippe Deléham_, Feb 16 2004

Typo corrected by _Kevin Ryde_, Aug 23 2012

Status

approved