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A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n*(n+1)*(10*n-7)/6.

%I A007587 M4895 #60 Aug 04 2025 09:46:30
%S A007587 0,1,13,46,110,215,371,588,876,1245,1705,2266,2938,3731,4655,5720,
%T A007587 6936,8313,9861,11590,13510,15631,17963,20516,23300,26325,29601,33138,
%U A007587 36946,41035,45415,50096,55088,60401,66045,72030,78366,85063,92131,99580,107420,115661
%N A007587 12-gonal (or dodecagonal) pyramidal numbers: a(n) = n*(n+1)*(10*n-7)/6.
%C A007587 Binomial transform of [1, 12, 21, 10, 0, 0, 0, ...] = (1, 13, 46, 110, ...). - _Gary W. Adamson_, Nov 28 2007
%C A007587 This sequence is related to A000566 by a(n) = n*A000566(n) - Sum_{i=0..n-1} A000566(i) and this is the case d=5 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/6. - _Bruno Berselli_, Oct 18 2010
%D A007587 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
%D A007587 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
%D A007587 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007587 Vincenzo Librandi, <a href="/A007587/b007587.txt">Table of n, a(n) for n = 0..1000</a>
%H A007587 B. Berselli, A description of the recursive method in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian), 2008.
%H A007587 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A007587 a(n) = (10*n-7)*binomial(n+1, 2)/3.
%F A007587 G.f.: x*(1+9*x)/(1-x)^4.
%F A007587 a(n) = Sum_{k=0..n} k*(5*k-4). - _Klaus Brockhaus_, Nov 20 2008
%F A007587 a(n) = Sum_{i=0..n-1} (n-i)*(10*i+1), with a(0)=0. - _Bruno Berselli_, Feb 10 2014
%F A007587 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Wesley Ivan Hurt_, Oct 23 2014
%F A007587 E.g.f.: exp(x)*x*(6 + 33*x + 10*x^2)/6. - _Elmo R. Oliveira_, Aug 04 2025
%p A007587 A007587:=n->n*(n+1)*(10*n-7)/6: seq(A007587(n), n=0..50); # _Wesley Ivan Hurt_, Oct 23 2014
%t A007587 CoefficientList[Series[x(1+9x)/(1-x)^4, {x,0,45}], x] (* _Vincenzo Librandi_, Jun 20 2013 *)
%t A007587 Table[n(n+1)(10n-7)/6,{n,0,50}] (* _Harvey P. Dale_, Nov 13 2013 *)
%o A007587 (Magma) [ n eq 1 select 0 else Self(n-1)+(n-1)*(5*n-9): n in [1..45] ]; // _Klaus Brockhaus_, Nov 20 2008
%o A007587 (PARI) a(n)=if(n,([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,4,-6,4]^n*[0;1;13;46])[1,1],0) \\ _Charles R Greathouse IV_, Oct 07 2015
%o A007587 (PARI) vector(45, n, n*(n-1)*(10*n-17)/6) \\ _G. C. Greubel_, Aug 30 2019
%o A007587 (Sage) [n*(n+1)*(10*n-7)/6 for n in (0..45)] # _G. C. Greubel_, Aug 30 2019
%o A007587 (GAP) List([0..45], n-> n*(n+1)*(10*n-7)/6); # _G. C. Greubel_, Aug 30 2019
%Y A007587 Cf. A093645 ((10, 1) Pascal, column m=3). Partial sums of A051624.
%Y A007587 Cf. A000566.
%Y A007587 See similar sequences listed in A237616.
%K A007587 nonn,easy,nice
%O A007587 0,3
%A A007587 _N. J. A. Sloane_, _R. K. Guy_