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%I #22 Sep 08 2022 08:46:01
%S 0,10,31,63,106,160,225,301,388,486,595,715,846,988,1141,1305,1480,
%T 1666,1863,2071,2290,2520,2761,3013,3276,3550,3835,4131,4438,4756,
%U 5085,5425,5776,6138,6511,6895,7290,7696,8113,8541,8980,9430,9891,10363
%N Second 13-gonal numbers: a(n) = n*(11*n+9)/2.
%C Sequence found by reading the line from 0, in the direction 0, 31... and the line from 10, in the direction 10, 63,..., in the square spiral whose vertices are the generalized 13-gonal numbers A195313.
%H G. C. Greubel, <a href="/A211013/b211013.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(10+x)/(1-x)^3. - _Philippe Deléham_, Mar 27 2013
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 10, a(2) = 31. - _Philippe Deléham_, Mar 27 2013
%F a(n) = A051865(n) + 9n = A180223(n) + 8n = A022268(n) + 5n = A022269(n) + 4n = A152740(n) - n. - _Philippe Deléham_, Mar 27 2013
%F a(n) = A218530(11n+9). - _Philippe Deléham_, Mar 27 2013
%F E.g.f.: x*(20 + 11*x)*exp(x)/2. - _G. C. Greubel_, Jul 04 2019
%t Table[n*(11*n+9)/2, {n,0,50}] (* _G. C. Greubel_, Jul 04 2019 *)
%o (PARI) a(n)=n*(11*n+9)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%o (Magma) [n*(11*n+9)/2: n in [0..50]]; // _G. C. Greubel_, Jul 04 2019
%o (Sage) [n*(11*n+9)/2 for n in (0..50)] # _G. C. Greubel_, Jul 04 2019
%o (GAP) List([0..50], n-> n*(11*n+9)/2) # _G. C. Greubel_, Jul 04 2019
%Y Bisection of A195313.
%Y Second k-gonal numbers (k=5..14): A005449, A014105, A147875, A045944, A179986, A033954, A062728, A135705, this sequence, A211014.
%Y Cf. A051865.
%K nonn,easy
%O 0,2
%A _Omar E. Pol_, Aug 04 2012
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