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%I #53 Sep 08 2022 08:45:38
%S 0,3,16,39,72,115,168,231,304,387,480,583,696,819,952,1095,1248,1411,
%T 1584,1767,1960,2163,2376,2599,2832,3075,3328,3591,3864,4147,4440,
%U 4743,5056,5379,5712,6055,6408,6771,7144,7527,7920,8323,8736,9159,9592,10035
%N a(n) = (5*n-7)*(n-1).
%C Zero followed by partial sums of A017305.
%C Appears to be related to various other sequences: a(n) = A036666(2*n-2) for n>1; a(n) = A115006(2*n-3) for n>1; a(n) = A118015(5*n-6) for n>1; a(n) = A008738(5*n-7) for n>1.
%C Even dodecagonal numbers divided by 4. - _Omar E. Pol_, Aug 19 2011
%H Vincenzo Librandi, <a href="/A147874/b147874.txt">Table of n, a(n) for n = 1..2000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = Sum_{k=0..n-2} 10*k+3 = Sum_{k=0..n-2} A017305(k).
%F G.f.: x*(3 + 7*x)/(1-x)^3.
%F a(n) = 10*(n-2) + 3 + a(n-1).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F a(n) = A193872(n-1)/4. - _Omar E. Pol_, Aug 19 2011
%F a(n+1) = A131242(10n+2). - _Philippe Deléham_, Mar 27 2013
%F E.g.f.: -7 + (7 - 7*x + 5*x^2)*exp(x). - _G. C. Greubel_, Jul 30 2019
%F Sum_{n>=2} 1/a(n) = A294830. - _Amiram Eldar_, Nov 15 2020
%t s=0;lst={s};Do[s+=n++ +3;AppendTo[lst,s],{n,0,6!,10}];lst
%t Table[5n^2-12n+7,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{0,3,16},50] (* or *) PolygonalNumber[12,Range[0,100,2]]/4 (* _Harvey P. Dale_, Aug 08 2021 *)
%o (Magma) [ 0 ] cat [ &+[ 10*k+3: k in [0..n-1] ]: n in [1..50] ]; // _Klaus Brockhaus_, Nov 17 2008
%o (Magma) [ 5*n^2-2*n: n in [0..50] ];
%o (PARI) {m=50; a=7; for(n=0, m, print1(a=a+10*(n-1)+3, ","))} \\ _Klaus Brockhaus_, Nov 17 2008
%o (Sage) [(5*n-7)*(n-1) for n in (1..50)] # _G. C. Greubel_, Jul 30 2019
%o (GAP) List([1..50], n-> (5*n-7)*(n-1)); # _G. C. Greubel_, Jul 30 2019
%Y Cf. A017305 (10n+3), A036666, A115006, A118015 (floor(n^2/5)), A008738 (floor((n^2+1)/5)), A294830.
%Y Cf. A051624, A193872. - _Omar E. Pol_, Aug 19 2011
%K nonn,easy
%O 1,2
%A _Vladimir Joseph Stephan Orlovsky_, Nov 16 2008
%E Edited by _R. J. Mathar_ and _Klaus Brockhaus_, Nov 17 2008, Nov 20 2008
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