A123203 a(n) = 2^(n+1) - 3*n.

1, 2, 7, 20, 49, 110, 235, 488, 997, 2018, 4063, 8156, 16345, 32726, 65491, 131024, 262093, 524234, 1048519, 2097092, 4194241, 8388542, 16777147, 33554360, 67108789, 134217650, 268435375, 536870828, 1073741737, 2147483558

Offset: 1

Author

Gary W. Adamson, Jun 13 2007

Keywords

• nonn
• easy

Comments

An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 186, leads to this sequence. For the central square this vector leads to the companion sequence A036563. - Johannes W. Meijer, Aug 15 2010

Examples

			a(4) = 20, row sums of 4th row of triangle A131062: (1, 9, 9, 1).
a(4) = 20 = (1, 3, 3, 1) dot (1, 1, 4, 4) = (1 + 3 + 12 + 4).
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Joseph Breen and Emma Copeland, Non-orientable Nurikabe, arXiv:2506.12612 [math.CO], 2025. See pp. 1, 4.
• Tamas Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015) 73-94.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A000079, A000295, A109128, A131060, A131061, A131063, A131064, A131065, A131066.

Programs

Magma

[2^(n+1) -3*n: n in [1..40]]; // G. C. Greubel, Sep 14 2024

Mathematica

Table[2^(n+1) - 3*n, {n,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
LinearRecurrence[{4,-5,2},{1,2,7},40] (* Harvey P. Dale, Mar 30 2024 *)

SageMath

def A123203(n): return 2^(n+1) -3*n
[A123203(n) for n in range(1,41)] # G. C. Greubel, Sep 14 2024

Formula

Binomial transform of [1, 1, 4, 4, 4, ...].

Equals row sums of triangle A131061.

From Johannes W. Meijer, Aug 15 2010; corrected by Colin Barker, Jul 28 2012: (Start)

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

a(n) = 3*A000295(n-1) + A000079(n-1).

(End)

G.f.: x*(1 - 2*x + 4*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Jul 28 2012

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Colin Barker, Jul 29 2012

E.g.f.: 2*exp(2*x) - 3*x*exp(x) - 2. - G. C. Greubel, Sep 14 2024

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Nov 15 2008

Title changed by G. C. Greubel, Sep 14 2024