A119725 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k).

1, 1, 1, 1, 5, 1, 1, 13, 17, 1, 1, 29, 73, 53, 1, 1, 61, 233, 325, 161, 1, 1, 125, 649, 1349, 1297, 485, 1, 1, 253, 1673, 4645, 6641, 4861, 1457, 1, 1, 509, 4105, 14309, 27217, 29645, 17497, 4373, 1, 1, 1021, 9737, 40933, 97361, 140941, 123929, 61237, 13121, 1

Offset: 1

Author

Zerinvary Lajos, Jun 14 2006

Keywords

• easy
• nonn
• tabl

Comments

Second column is like A036563.

Second diagonal is A048473.

Examples

			Triangle begins:
  1;
  1,    1;
  1,    5,    1;
  1,   13,   17,     1;
  1,   29,   73,    53,     1;
  1,   61,  233,   325,   161,      1;
  1,  125,  649,  1349,  1297,    485,      1;
  1,  253, 1673,  4645,  6641,   4861,   1457,     1;
  1,  509, 4105, 14309, 27217,  29645,  17497,  4373,     1;
  1, 1021, 9737, 40933, 97361, 140941, 123929, 61237, 13121, 1;
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened
• Termeszet Vilaga A XI. Természet-Tudomány Diákpályázat díjnyertesei 133.EVF. 6.SZ. jun. 2002. Vegh Lea (and Vegh Erika): Pascal-tipusu haromszogek

Crossrefs

Cf. A007318, A036563, A048473, A119726, A119727.

Programs

Magma

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return 3*T(n-1,k-1) + 2*T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 18 2019

Maple

T:= proc(n, k) option remember;
      if k=1 and k=n then 1
    else 3*T(n-1, k-1) + 2*T(n-1, k)
      fi
    end:
seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Nov 18 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 3*T[n-1, k-1] + 2*T[n-1, k]]; Table[T[n,k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

PARI

T(n,k) = if(k==1 || k==n, 1, 3*T(n-1,k-1) + 2*T(n-1,k)); \\ G. C. Greubel, Nov 18 2019

Sage

@CachedFunction
def T(n, k):
    if (k==1 or k==n): return 1
    else: return 3*T(n-1, k-1) + 2*T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 18 2019

Extensions

Edited by Don Reble, Jul 24 2006