A081437 - OEIS

%I #44 Feb 16 2025 08:32:48

%S 1,10,33,76,145,246,385,568,801,1090,1441,1860,2353,2926,3585,4336,

%T 5185,6138,7201,8380,9681,11110,12673,14376,16225,18226,20385,22708,

%U 25201,27870,30721,33760,36993,40426,44065,47916,51985,56278,60801,65560

%N Diagonal in array of n-gonal numbers A081422.

%C One of a family of sequences with palindromic generators.

%C For q a prime power, a(q-1) = q^3 + q^2 - q is the number of pairs of commuting nilpotent 2*2 matrices with coefficients in GF(q). (Proof: the zero matrix commutes with all q^2 nilpotent matrices, there are q^2-1 nonzero nilpotent matrices, all conjugate, each commuting with q nilpotent matrices.) - _Mark Wildon_, Jun 20 2017

%C Also the cyclomatic number (= circuit rank) of the n+1 X n+1 rook graph. - _Eric W. Weisstein_, Jun 20 2017

%H Vincenzo Librandi, <a href="/A081437/b081437.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CircuitRank.html">Circuit Rank</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = n^3 + 4*n^2 + 4*n + 1.

%F G.f.: (1 +5*x -7*x^2 +x^3)/(1-x)^5.

%F a(0)=1, a(1)=10, a(2)=33, a(3)=76; for n>3, a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - _Harvey P. Dale_, Jan 24 2012

%F E.g.f.: (1 +9*x +7*x^2 +x^3)*exp(x). - _G. C. Greubel_, Aug 14 2019

%p a:=n->sum(n*k, k=0..n):seq(a(n)+sum(n*k, k=2..n), n=1..40); # _Zerinvary Lajos_, Jun 10 2008

%p a:=n->sum(-2+sum(2+sum(2, j=1..n),j=1..n),j=1..n):seq(a(n)/2,n=1..40); # _Zerinvary Lajos_, Dec 06 2008

%t Table[n^3 + 4 n^2 + 4n + 1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 10, 33, 76}, 40] (* _Harvey P. Dale_, Jan 24 2012 *)

%t CoefficientList[Series[(1 + 5 x - 7 x^2 + x^3)/(1 - x)^5, {x, 0, 60}], x] (* _Vincenzo Librandi_, Aug 08 2013 *)

%o (Magma) [n^3+4*n^2+4*n+1: n in [0..50]]; // _Vincenzo Librandi_, Aug 08 2013

%o (PARI) vector(40, n, n--; (n+1)^3+n*(n+1)) \\ _G. C. Greubel_, Aug 14 2019

%o (Sage) [(n+1)^3+n*(n+1) for n in (0..40)] # _G. C. Greubel_, Aug 14 2019

%o (GAP) List([0..40], n-> (n+1)^3+n*(n+1)); # _G. C. Greubel_, Aug 14 2019

%Y Cf. A081435, A081436, A081437, A081438.

%Y Equals A027620(n-1) + 1.

%K nonn,easy,changed

%O 0,2

%A _Paul Barry_, Mar 21 2003