A238275 a(n) = (4*7^n - 1)/3.

1, 9, 65, 457, 3201, 22409, 156865, 1098057, 7686401, 53804809, 376633665, 2636435657, 18455049601, 129185347209, 904297430465, 6330082013257, 44310574092801, 310174018649609, 2171218130547265, 15198526913830857, 106389688396816001, 744727818777712009

Offset: 0

Author

Philippe Deléham, Feb 21 2014

Keywords

• nonn
• easy

Comments

Sum of n-th row of triangle of powers of 7: 1; 1 7 1; 1 7 49 7 1; 1 7 49 343 49 7 1; ...

Number of cubes in the crystal structure cubic carbon CCC(n+1), defined in the Baig et al. and in the Gao et al. references. - Emeric Deutsch, May 28 2018

Examples

			a(0) = 1;
a(1) = 1 + 7 + 1 = 9;
a(2) = 1 + 7 + 49 + 7 + 1 = 65;
a(3) = 1 + 7 + 49 + 343 + 49 + 7 + 1 = 457; etc.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• A. Q. Baig, M. Imran, W. Khalid, and M. Naeem, Molecular description of carbon graphite and crystal cubic carbon structures, Canadian J. Chem., 95, 674-686, 2017.
• W. Gao, M. K. Siddiqui, M. Naeem and N. A. Rehman, Topological characterization of carbon graphite and crystal cubic carbon structures, Molecules, 22, 1496, 1-12, 2017.
• Index entries for linear recurrences with constant coefficients, signature (8,-7).

Crossrefs

Cf. Similar sequences: A151575, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, this sequence, A238276, A138894, A090843, A199023.

Cf. A112468, A112739.

Programs

Magma

[(4*7^n - 1)/3: n in [0..30]]; // Vincenzo Librandi, Feb 23 2014

Mathematica

Table[(4 7^n - 1)/3, {n, 0, 40}] (* Vincenzo Librandi, Feb 23 2014 *)

Formula

G.f.: (1+x)/((1-x)*(1-7*x)).

a(n) = 7*a(n-1) + 2, a(0) = 1.

a(n) = 8*a(n-1) - 7*a(n-2), a(0) = 1, a(1) = 9.

a(n) = Sum_{k=0..n} A112468(n,k)*8^k.

E.g.f.: exp(x)*(4*exp(6*x) - 1)/3. - Stefano Spezia, Feb 12 2025