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A195142
Concentric 10-gonal numbers.
14
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.
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. A090771 (first differences).
Column 10 of A195040. - Omar E. Pol, Sep 28 2011
Sequence in context: A240536 A060852 A345962 * A256884 A089584 A192743
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Sep 17 2011
STATUS
approved