A238275 - OEIS

%I #23 Feb 12 2025 16:45:25

%S 1,9,65,457,3201,22409,156865,1098057,7686401,53804809,376633665,

%T 2636435657,18455049601,129185347209,904297430465,6330082013257,

%U 44310574092801,310174018649609,2171218130547265,15198526913830857,106389688396816001,744727818777712009

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

%C 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; ...

%C 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

%H Vincenzo Librandi, <a href="/A238275/b238275.txt">Table of n, a(n) for n = 0..200</a>

%H A. Q. Baig, M. Imran, W. Khalid, and M. Naeem, <a href="https://doi.org/10.1139/cjc-2017-0083">Molecular description of carbon graphite and crystal cubic carbon structures</a>, Canadian J. Chem., 95, 674-686, 2017.

%H W. Gao, M. K. Siddiqui, M. Naeem and N. A. Rehman, <a href="https://doi.org/10.3390/molecules22091496">Topological characterization of carbon graphite and crystal cubic carbon structures</a>, Molecules, 22, 1496, 1-12, 2017.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-7).

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

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

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

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

%F E.g.f.: exp(x)*(4*exp(6*x) - 1)/3. - _Stefano Spezia_, Feb 12 2025

%e a(0) = 1;

%e a(1) = 1 + 7 + 1 = 9;

%e a(2) = 1 + 7 + 49 + 7 + 1 = 65;

%e a(3) = 1 + 7 + 49 + 343 + 49 + 7 + 1 = 457; etc.

%t Table[(4 7^n - 1)/3, {n, 0, 40}] (* _Vincenzo Librandi_, Feb 23 2014 *)

%o (Magma) [(4*7^n - 1)/3: n in [0..30]]; // _Vincenzo Librandi_, Feb 23 2014

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

%Y Cf. A112468, A112739.

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Feb 21 2014