A135231 Row sums of triangle A135230.

1, 2, 4, 6, 12, 22, 44, 86, 172, 342, 684, 1366, 2732, 5462, 10924, 21846, 43692, 87382, 174764, 349526, 699052, 1398102, 2796204, 5592406, 11184812, 22369622, 44739244, 89478486, 178956972, 357913942, 715827884, 1431655766, 2863311532, 5726623062, 11453246124, 22906492246, 45812984492

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Formula

a(2*n+1) = A005578(n+1) if n is odd.

Conjectures from Chai Wah Wu, Aug 31 2023: (Start)

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n > 3.

G.f.: (-2*x^3 - x^2 + 1)/((x - 1)*(x + 1)*(2*x - 1)). (End)

Example

a(3) = 6 = sum of row 4 terms of triangle A135230; (1 + 2 + 2 + 1).
a(5) = 22 = A005578(6).
a(6) = 44 = A005578(7) + 1.

Maple

T:= proc(n, k) option remember;
      if k=n then 1
    elif k=0 then (3+(-1)^n)/2
    else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
      fi; end:
seq( add(T(n, j), j=0..n), n=0..40); # G. C. Greubel, Nov 20 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3+(-1)^n)/2, Sum[Binomial[n-1 - 2*j, k-1], {j, 0, Floor[(n-1)/2]}]]]; Table[Sum[T[n, j], {j, 0, n}], {n, 0, 40}] (* G. C. Greubel, Nov 20 2019 *)

Prog

(PARI) T(n, k) = if(k==n, 1, if(k==0, (3+(-1)^n)/2, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) )); \\ G. C. Greubel, Nov 20 2019
(Magma)
function T(n, k)
  if k eq n then return 1;
  elif k eq 0 then return (3+(-1)^n)/2;
  else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
  end if; return T; end function;
[(&+[T(n, j): j in [0..n]]): n in [0..40]]; // G. C. Greubel, Nov 20 2019
(Sage)
@CachedFunction
def T(n, k):
    if (k==n): return 1
    elif (k==0): return (3+(-1)^n)/2
    else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
[sum(T(n, j) for j in (0..n)) for n in (0..40)] # G. C. Greubel, Nov 20 2019

Crossrefs

Cf. A005578, A135230.

Sequence in context: A341582 A370648 A326114 * A326489 A217356 A030793

Adjacent sequences: A135228 A135229 A135230 * A135232 A135233 A135234

Keyword

nonn

Author

Gary W. Adamson, Nov 23 2007

Extensions

Terms a(16) onward added and offset changed by G. C. Greubel, Nov 20 2019

Status

approved