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 A081437 Diagonal in array of n-gonal numbers A081422. 15
 1, 10, 33, 76, 145, 246, 385, 568, 801, 1090, 1441, 1860, 2353, 2926, 3585, 4336, 5185, 6138, 7201, 8380, 9681, 11110, 12673, 14376, 16225, 18226, 20385, 22708, 25201, 27870, 30721, 33760, 36993, 40426, 44065, 47916, 51985, 56278, 60801, 65560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS One of a family of sequences with palindromic generators. 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 Also the cyclomatic number (= circuit rank) of the n+1 X n+1 rook graph. - Eric W. Weisstein, Jun 20 2017 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Eric Weisstein's World of Mathematics, Circuit Rank Eric Weisstein's World of Mathematics, Rook Graph Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA a(n) = n^3 + 4*n^2 + 4*n + 1. G.f.: (1 +5*x -7*x^2 +x^3)/(1-x)^5. 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 E.g.f.: (1 +9*x +7*x^2 +x^3)*exp(x). - G. C. Greubel, Aug 14 2019 MAPLE 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 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 MATHEMATICA 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 *) CoefficientList[Series[(1 + 5 x - 7 x^2 + x^3)/(1 - x)^5, {x, 0, 60}], x] (* Vincenzo Librandi, Aug 08 2013 *) PROG (MAGMA) [n^3+4*n^2+4*n+1: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013 (PARI) vector(40, n, n--; (n+1)^3+n*(n+1)) \\ G. C. Greubel, Aug 14 2019 (Sage) [(n+1)^3+n*(n+1) for n in (0..40)] # G. C. Greubel, Aug 14 2019 (GAP) List([0..40], n-> (n+1)^3+n*(n+1)); # G. C. Greubel, Aug 14 2019 CROSSREFS Cf. A081435, A081436, A081437, A081438. Equals A027620(n-1) + 1. Sequence in context: A065149 A299287 A299285 * A085490 A162433 A003012 Adjacent sequences:  A081434 A081435 A081436 * A081438 A081439 A081440 KEYWORD nonn,easy AUTHOR Paul Barry, Mar 21 2003 STATUS approved

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Last modified February 16 15:17 EST 2020. Contains 331961 sequences. (Running on oeis4.)