A195142 Concentric 10-gonal numbers.

0, 1, 10, 21, 40, 61, 90, 121, 160, 201, 250, 301, 360, 421, 490, 561, 640, 721, 810, 901, 1000, 1101, 1210, 1321, 1440, 1561, 1690, 1821, 1960, 2101, 2250, 2401, 2560, 2721, 2890, 3061, 3240, 3421, 3610, 3801, 4000, 4201, 4410, 4621, 4840, 5061, 5290

Offset

0,3

Comments

Also concentric decagonal numbers. Also sequence found by reading the line from 0, in the direction 0, 10, ..., and the same line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Main axis, perpendicular to A028895 in the same spiral.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

G.f.: -x*(1+8*x+x^2) / ( (1+x)*(x-1)^3 ). - R. J. Mathar, Sep 18 2011

a(n) = -a(n-1) + 5*n^2 - 5*n + 1, a(0)=0. - Vincenzo Librandi, Sep 27 2011

From Bruno Berselli, Sep 27 2011: (Start)

a(n) = a(-n) = (10*n^2 + 3*(-1)^n - 3)/4.

a(n) = a(n-2) + 10*(n-1). (End)

a(n) = 2*a(n-1) + 0*a(n-2) - 2*a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=10, a(3)=21. - Harvey P. Dale, Sep 29 2011

Sum_{n>=1} 1/a(n) = Pi^2/60 + tan(sqrt(3/5)*Pi/2)*Pi/(2*sqrt(15)). - Amiram Eldar, Jan 16 2023

Mathematica

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-2]+10(n-1)}, a[n], {n, 50}] (* or *) LinearRecurrence[{2, 0, -2, 1}, {0, 1, 10, 21}, 50] (* Harvey P. Dale, Sep 29 2011 *)

Prog

(Magma) [(10*n^2+3*(-1)^n-3)/4: n in [0..50]]; // Vincenzo Librandi, Sep 27 2011
(Haskell)
a195142 n = a195142_list !! n
a195142_list = scanl (+) 0 a090771_list
-- Reinhard Zumkeller, Jan 07 2012

Crossrefs

A033583 and A069133 interleaved.

Cf. A028895, A032527, A032528, A077221, A085787, A195042, A195143, A195145, A195146, A195147, A195148, A195149.

Cf. A090771 (first differences).

Column 10 of A195040. - Omar E. Pol, Sep 28 2011

Sequence in context: A240536 A060852 A345962 * A256884 A089584 A192743

Adjacent sequences: A195139 A195140 A195141 * A195143 A195144 A195145

Keyword

nonn,easy

Author

Omar E. Pol, Sep 17 2011

Status

approved