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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.)