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%I #43 Mar 22 2024 08:01:13
%S 0,2,17,45,86,140,207,287,380,486,605,737,882,1040,1211,1395,1592,
%T 1802,2025,2261,2510,2772,3047,3335,3636,3950,4277,4617,4970,5336,
%U 5715,6107,6512,6930,7361,7805,8262,8732,9215,9711,10220,10742,11277,11825,12386,12960
%N a(n) = n*(13*n - 9)/2.
%C Sum of n-th octagonal number and n-th 9-gonal (nonagonal) number.
%C Sum of reciprocals of a(n), for n>0: 0.629618994194109711163742089971688...
%H Bruno Berselli, <a href="/A226488/b226488.txt">Table of n, a(n) for n = 0..1000</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*(2+11*x)/(1-x)^3.
%F a(n) + a(-n) = A152742(n).
%F a(0)=0, a(1)=2, a(2)=17; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Jun 19 2013
%F E.g.f.: x*(4 + 13*x)*exp(x)/2. - _G. C. Greubel_, Aug 30 2019
%F a(n) = A000567(n) + A001106(n). - _Michel Marcus_, Aug 31 2019
%p A226488:=n->n*(13*n - 9)/2; seq(A226488(n), n=0..50); # _Wesley Ivan Hurt_, Feb 25 2014
%t Table[n(13n-9)/2, {n, 0, 50}]
%t LinearRecurrence[{3, -3, 1}, {0, 2, 17}, 50] (* _Harvey P. Dale_, Jun 19 2013 *)
%t CoefficientList[Series[x(2+11x)/(1-x)^3, {x, 0, 45}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%o (Magma) [n*(13*n-9)/2: n in [0..50]];
%o (Magma) I:=[0,2,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2) +Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Aug 18 2013
%o (PARI) a(n)=n*(13*n-9)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Sage) [n*(13*n-9)/2 for n in (0..50)] # _G. C. Greubel_, Aug 30 2019
%o (GAP) List([0..50], n-> n*(13*n-9)/2); # _G. C. Greubel_, Aug 30 2019
%Y Cf. A000567, A001106, A153080 (first differences).
%Y Cf. numbers of the form n*(n*k-k+4)/2 listed in A005843 (k=0), A000096 (k=1), A002378 (k=2), A005449 (k=3), A001105 (k=4), A005476 (k=5), A049450 (k=6), A218471 (k=7), A002939 (k=8), A062708 (k=9), A135706 (k=10), A180223 (k=11), A139267 (n=12), this sequence (k=13), A139268 (k=14), A226489 (k=15), A139271 (k=16), A180232 (k=17), A152995 (k=18), A226490 (k=19), A152965 (k=20), A226491 (k=21), A152997 (k=22).
%K nonn,easy
%O 0,2
%A _Bruno Berselli_, Jun 09 2013
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