A102305 a(n) = n^2 + 2*n + 3.
6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027, 2118, 2211, 2306, 2403
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Programs
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Magma
[(n+1)^2+2: n in [1..60]]; // G. C. Greubel, Feb 03 2024
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Maple
A102305:=n->n^2+2*n+3: seq(A102305(n), n=1..100); # Wesley Ivan Hurt, Jan 22 2017
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Mathematica
Table[n^2+2n+3,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{6,11,18},50] (* Harvey P. Dale, Aug 05 2015 *) -
PARI
a(n)=n^2+2*n+3 \\ Charles R Greathouse IV, Oct 16 2015
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SageMath
[n^2+2*n+3 for n in range(1,61)] # G. C. Greubel, Feb 03 2024
Formula
a(n) = (1/5) * A027578(n-1).
a(n) = 2*n + a(n-1) + 1 (with a(1)=6). - Vincenzo Librandi, Nov 16 2010
a(n) = A059100(n+1). - Reinhard Zumkeller, Mar 21 2008
a(n) = A010000(n+1) for n >= 1. - Georg Fischer, Nov 02 2018
From Amiram Eldar, Sep 14 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi * coth(sqrt(2)*Pi)/(2*sqrt(2)) - 7/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = cosech(sqrt(2)*Pi)*Pi/(2*sqrt(2)) + 1/12. (End)
From G. C. Greubel, Feb 03 2024: (Start)
G.f.: (3 - 3*x + 2*x^2)/(1-x)^3.
E.g.f.: (3 + 3*x + x^2)*exp(x). (End)
Comments