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A147875 Second heptagonal numbers: a(n) = n*(5*n+3)/2.

%I A147875 #87 Mar 13 2025 13:28:40
%S A147875 0,4,13,27,46,70,99,133,172,216,265,319,378,442,511,585,664,748,837,
%T A147875 931,1030,1134,1243,1357,1476,1600,1729,1863,2002,2146,2295,2449,2608,
%U A147875 2772,2941,3115,3294,3478,3667,3861,4060,4264,4473,4687,4906,5130,5359,5593
%N A147875 Second heptagonal numbers: a(n) = n*(5*n+3)/2.
%C A147875 Zero followed by partial sums of A016897.
%C A147875 Apparently = every 2nd term of A111710 and A085787.
%C A147875 Bisection of A085787. Sequence found by reading the line from 0, in the direction 0, 13, ... and the line from 4, in the direction 4, 27, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - _Omar E. Pol_, Jul 18 2012
%C A147875 Numbers of the form m^2 + k*m*(m+1)/2: in this case is k=3. See also A254963. - _Bruno Berselli_, Feb 11 2015
%H A147875 Harvey P. Dale, <a href="/A147875/b147875.txt">Table of n, a(n) for n = 0..1000</a>
%H A147875 Leo Tavares, <a href="/A147875/a147875.jpg">Illustration: Sliced Hexagons</a>
%H A147875 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A147875 G.f.: x*(4+x)/(1-x)^3.
%F A147875 a(n) = Sum_{k=0..n-1} A016897(k).
%F A147875 a(n) - a(n-1) = 5*n -1. - _Vincenzo Librandi_, Nov 26 2010
%F A147875 G.f.: U(0) where U(k) = 1 + 2*(2*k+3)/(k + 2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3) + (2*k+2)*(2*k+3)/U(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Nov 14 2012
%F A147875 E.g.f.: U(0) where U(k) = 1 + 2*(2*k+3)/(k + 2 - 2*x*(k+2)^2*(k+3)/(2*x*(k+2)*(k+3) + (2*k+2)^2*(2*k+3)/U(k+1))); (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Nov 14 2012
%F A147875 a(n) = A130520(5n+3). - _Philippe Deléham_, Mar 26 2013
%F A147875 a(n) = A131242(10n+7)/2. - _Philippe Deléham_, Mar 27 2013
%F A147875 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=4, a(2)=13. - _Harvey P. Dale_, May 15 2013
%F A147875 Sum_{n>=1} 1/a(n) = 10/9 + sqrt(1 - 2/sqrt(5))*Pi/3 - 5*log(5)/6 + sqrt(5)*log((1 + sqrt(5))/2)/3 = 0.4688420784500060750083432... . - _Vaclav Kotesovec_, Apr 27 2016
%F A147875 a(n) = A000217(n) + A000217(2*n). - _Bruno Berselli_, Jul 01 2016
%F A147875 From _Ilya Gutkovskiy_, Jul 01 2016: (Start)
%F A147875 E.g.f.: x*(8 + 5*x)*exp(x)/2.
%F A147875 Dirichlet g.f.: (5*zeta(s-2) + 3*zeta(s-1))/2. (End)
%F A147875 a(n) = A000566(-n) for all n in Z. - _Michael Somos_, Jan 25 2019
%F A147875 From _Leo Tavares_, Feb 14 2022: (Start)
%F A147875 a(n) = A003215(n) - A000217(n+1). See Sliced Hexagons illustration in links.
%F A147875 a(n) = A000096(n) + 2*A000290(n). (End)
%e A147875 G.f. = 4*x + 13*x^2 + 27*x^3 + 46*x^4 + 70*x^5 + 99*x^6 + 133*x^7 + ... - _Michael Somos_, Jan 25 2019
%t A147875 Table[(n(5n+3))/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 4, 13}, 50] (* _Harvey P. Dale_, May 15 2013 *)
%o A147875 (PARI) a(n)=n*(5*n+3)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A147875 (Magma) [n*(5*n+3)/2: n in [0..50]]; // _G. C. Greubel_, Jul 04 2019
%o A147875 (Sage) [n*(5*n+3)/2 for n in (0..50)] # _G. C. Greubel_, Jul 04 2019
%o A147875 (GAP) List([0..50], n-> n*(5*n+3)/2); # _G. C. Greubel_, Jul 04 2019
%Y A147875 Cf. A016897, A111710, A000217, A085787, A224419 (positions of squares).
%Y A147875 Second n-gonal numbers: A005449, A014105, A045944, A179986, A033954, A062728, A135705.
%Y A147875 Cf. A000566.
%Y A147875 Cf. A003215, A000096, A000290.
%K A147875 nonn,easy
%O A147875 0,2
%A A147875 _Vladimir Joseph Stephan Orlovsky_, Nov 16 2008
%E A147875 Edited by _Klaus Brockhaus_ and _R. J. Mathar_, Nov 20 2008
%E A147875 New name from _Bruno Berselli_, Jan 13 2011