A238375 Row sums of triangle in A152719.

1, 2, 4, 6, 11, 16, 28, 40, 69, 98, 168, 238, 407, 576, 984, 1392, 2377, 3362, 5740, 8118, 13859, 19600, 33460, 47320, 80781, 114242, 195024, 275806, 470831, 665856, 1136688, 1607520, 2744209, 3880898, 6625108, 9369318, 15994427, 22619536, 38613964, 54608392

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = Sum_{k=0..n} A152719(n,k).

G.f.: (1+x)/((1-2*x^2-x^4)*(1-x)).

a(2*n) = A005409(n+2).

a(2*n+1) = 2*A048739(n).

a(n) = (-4 + 2*(1+(-1)^n)*Pell((n+4)/2) + (1-(-1)^n)*Q((n+3)/2))/4, where Pell(n) = A000129(n) and Q(n) = A002203(n). - G. C. Greubel, May 21 2021

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)+a(n-4)-a(n-5). - Wesley Ivan Hurt, May 22 2021

Example

Triangle A152719 and row sums:
  1;  ............................. sum =  1
  1, 1;  .......................... sum =  2
  1, 2, 1;  ....................... sum =  4
  1, 2, 2,  1;  ................... sum =  6
  1, 2, 5,  2,  1;  ............... sum = 11
  1, 2, 5,  5,  2, 1;  ............ sum = 16
  1, 2, 5, 12,  5, 2, 1;  ......... sum = 28
  1, 2, 5, 12, 12, 5, 2, 1;  ...... sum = 40

Mathematica

Table[Sum[Fibonacci[1+Min[k, n-k], 2], {k, 0, n}], {n, 0, 45}] (* G. C. Greubel, May 21 2021 *)

Prog

(Sage)
def Pell(n): return n if (n<2) else 2*Pell(n-1) + Pell(n-2)
def a(n): return sum(Pell(1+min(k, n-k)) for k  in (0..n))
[a(n) for n in (0..45)] # G. C. Greubel, May 21 2021
(PARI) my(x='x+O('x^44)); Vec((1+x)/((1-2*x^2-x^4)*(1-x))) \\ Joerg Arndt, May 22 2021

Crossrefs

Cf. A000129, A002203, A005409, A048739, A135153 (first differences), A152719.

Sequence in context: A103692 A114921 A103442 * A056342 A094719 A294811

Adjacent sequences: A238372 A238373 A238374 * A238376 A238377 A238378

Keyword

easy,nonn

Author

Philippe Deléham, Feb 25 2014

Status

approved