A248337 -id:A248337 - cp's OEIS frontend

A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

1, 2, 6, 4, 12, 36, 8, 24, 72, 216, 16, 48, 144, 432, 1296, 32, 96, 288, 864, 2592, 7776, 64, 192, 576, 1728, 5184, 15552, 46656, 128, 384, 1152, 3456, 10368, 31104, 93312, 279936, 256, 768, 2304, 6912, 20736, 62208, 186624, 559872, 1679616, 512, 1536, 4608, 13824, 41472, 124416, 373248, 1119744, 3359232, 10077696

Offset: 0

Reinhard Zumkeller, Nov 20 2004

			From _Stefano Spezia_, Apr 28 2024: (Start)
Triangle begins:
   1;
   2,  6;
   4, 12,  36;
   8, 24,  72, 216;
  16, 48, 144, 432, 1296;
  32, 96, 288, 864, 2592, 7776;
  ...
(End)

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000079, A000400, A001021, A003586, A005010, A007283, A016129.
Cf. A016149, A051958, A053524, A100852, A248337.
Cf. A065333(T(n, k))=1.

A100851:= func< n,k | 2^n*3^k >;
[A100851(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 11 2024

A100851[n_, k_]= 2^n*3^k;
Table[A100851[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 11 2024 *)

def A100851(n,k): return 2^n*3^k
flatten([[A100851(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 11 2024

T(n,0) = A000079(n).
T(n,1) = A007283(n) for n>0.
T(n,2) = A005010(n) for n>1.
T(n,n) = A000400(n) = A100852(n,n).
Sum_{k=0..n} T(n, k) = A016129(n).
T(2*n, n) = A001021(n). - Reinhard Zumkeller, Mar 04 2006
G.f.: 1/((1 - 2*x)*(1 - 6*x*y)). - Stefano Spezia, Apr 28 2024
From G. C. Greubel, Nov 11 2024: (Start)
Sum_{k=0..n} (-1)^k*T(n, k) = A053524(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (1/2)*((1-(-1)^n)*A248337((n+1)/2) + (1 + (-1)^n)*A016149(n/2)).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n) *A051958((n+2)/2) + 2*(1-(-1)^n)*A051958((n+1)/2)). (End)
Sum_{n>=0, k=0..n} 1/T(n,k) = 12/5. - Amiram Eldar, May 12 2025

A248338 a(n) = 10^n - 4^n.

Original entry on oeis.org

0, 6, 84, 936, 9744, 98976, 995904, 9983616, 99934464, 999737856, 9998951424, 99995805696, 999983222784, 9999932891136, 99999731564544, 999998926258176, 9999995705032704, 99999982820130816, 999999931280523264, 9999999725122093056, 99999998900488372224

Offset: 0

Vincenzo Librandi, Oct 05 2014

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (14,-40).

Cf. A000079, A000302, A005057, A011557, A016157.

Cf. similar sequences listed in A248337.

Magma
```
[10^n-4^n: n in [0..30]];
```

Table[10^n - 4^n, {n, 0, 30}] (* or *)
CoefficientList[Series[(6 x)/((1-4 x)(1-10 x)), {x, 0, 30}], x]

vector(20,n,10^(n-1)-4^(n-1)) \\ Derek Orr, Oct 05 2014

def A248338(n): return pow(10,n) - pow(4,n)
print([A248338(n) for n in range(41)]) # G. C. Greubel, Nov 13 2024

G.f.: 6*x/((1-4*x)*(1-10*x)).
a(n) = 14*a(n-1) - 40*a(n-2).
a(n) = 2^n*(5^n - 2^n) = A000079(n) * A005057(n) = A011557(n) - A000302(n).
a(n+1) = 6*A016157(n). [Bruno Berselli, Oct 05 2014]
E.g.f.: 2*exp(7*x)*sinh(3*x). - G. C. Greubel, Nov 13 2024

cp's OEIS Frontend

A100851 Triangle read by rows: T(n,k) = 2^n * 3^k, 0 <= k <= n, n >= 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula