%I M3329 #139 Aug 23 2023 08:35:12

%S 0,1,4,8,14,21,30,40,52,65,80,96,114,133,154,176,200,225,252,280,310,

%T 341,374,408,444,481,520,560,602,645,690,736,784,833,884,936,990,1045,

%U 1102,1160,1220,1281,1344,1408,1474,1541,1610,1680,1752,1825,1900,1976,2054

%N Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

%C Equals (1, 2, 3, 4, ...) convolved with (1, 2, 1, 2, ...). a(4) = 14 = (1, 2, 3, 4) dot (2, 1, 2, 1) = (2 + 2 + 6 + 4). - _Gary W. Adamson_, May 01 2009

%C We observe that is the transform of A032766 by the following transform T: T(u_0,u_1,u_2,u_3,...) = (u_0, u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In other words, v_p = Sum_{k=0..p} u_k and the g.f. phi_v of is given by phi_v = phi_u/(1-z). - _Richard Choulet_, Jan 28 2010

%C Equals row sums of a triangle with (1, 4, 7, 10, ...) in every column, shifted down twice for columns > 1. - _Gary W. Adamson_, Mar 03 2010

%C Number of pairs (x,y) with x in {0,...,n}, y odd in {0,...,2n}, and x < y. - _Clark Kimberling_, Jul 02 2012

%C Also A049451 and positives A000567 interleaved. - _Omar E. Pol_, Aug 03 2012

%C Similar to A001082. Members of this family are A093005, A210977, this sequence, A210978, A181995, A210981, A210982. - _Omar E. Pol_, Aug 09 2012

%D _Marc LeBrun_, personal communication.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A006578/b006578.txt">Table of n, a(n) for n = 0..10000</a>

%H Bruno Berselli, <a href="/A006578/a006578_1.jpg">Hexagonal spiral containing the initial terms</a>.

%H Marc Le Brun, <a href="/A006577/a006577.pdf">Email to N. J. A. Sloane, Jul 1991</a>

%H Emanuele Munarini, <a href="https://doi.org/10.4418/2021.76.1.14">Topological indices for the antiregular graphs</a>, Le Mathematiche (2021) Vol. 76, No. 1, see p. 283.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F Expansion of x*(1+2*x) / ((1-x)^2*(1-x^2)). - _Simon Plouffe_ in his 1992 dissertation

%F a(n) + A002620(n) = A002378(n) = n*(n+1).

%F Partial sums of A032766. - _Paul Barry_, May 30 2003

%F a(n) = a(n-1) + a(n-2) - a(n-3) + 3 = A002620(n) + A004526(n) = A001859(n) - A004526(n+1). - _Henry Bottomley_, Mar 08 2000

%F a(n) = (6*n^2 + 4*n - 1 + (-1)^n)/8. - _Paul Barry_, May 30 2003

%F a(n) = A001859(-1-n) for all n in Z. - _Michael Somos_, May 10 2006

%F a(n) = (A002378(n)/2 + A035608(n))/2. - _Reinhard Zumkeller_, Feb 07 2010

%F a(n) = (3*n^2 + 2*n - (n mod 2))/4. - _Ctibor O. Zizka_, Mar 11 2012

%F a(n) = Sum_{i=1..n} floor(3*i/2) = Sum_{i=0..n} (i + floor(i/2)). - _Enrique Pérez Herrero_, Apr 21 2012

%F a(n) = 3*n*(n+1)/2 - A001859(n). - _Clark Kimberling_, Jul 02 2012

%F a(n) = Sum_{i=1..n} (n - i + 1) * 2^( (i+1) mod 2 ). - _Wesley Ivan Hurt_, Mar 30 2014

%F a(n) = A002717(n) - A002717(n-1). - _Michael Somos_, Jun 09 2014

%F a(n) = Sum_{k=1..n} floor((n+k+1)/2). - _Wesley Ivan Hurt_, Mar 31 2017

%F a(n) = A002620(n+1)+2*A002620(n). - _R. J. Mathar_, Apr 28 2017

%F Sum_{n>=1} 1/a(n) = 3 - Pi/(4*sqrt(3)) - 3*log(3)/4. - _Amiram Eldar_, May 28 2022

%F E.g.f.: (x*(5 + 3*x)*cosh(x) - (1 - 5*x - 3*x^2)*sinh(x))/4. - _Stefano Spezia_, Aug 22 2023

%e G.f. = x + 4*x^2 + 8*x^3 + 14*x^4 + 21*x^5 + 30*x^6 + 40*x^7 + 52*x^8 + 65*x^9 + ...

%p with (combinat): seq(count(Partition((3*n+1)), size=3), n=0..52); # _Zerinvary Lajos_, Mar 28 2008

%p # 2nd program

%p A006578 := proc(n)

%p (6*n^2 + 4*n - 1 + (-1)^n)/8 ;

%p end proc: # _R. J. Mathar_, Apr 28 2017

%t Accumulate[LinearRecurrence[{1,1,-1}, {0,1,3}, 100]] (* _Harvey P. Dale_, Sep 29 2013 *)

%t a[ n_] := Quotient[n + 1, 2] (Quotient[n, 2] 3 + 1); (* _Michael Somos_, Jun 09 2014 *)

%t a[ n_] := Quotient[3 (n + 1)^2 + 1, 4] - (n + 1); (* _Michael Somos_, Jun 10 2015 *)

%t LinearRecurrence[{2, 0, -2, 1},{0, 1, 4, 8},53] (* _Ray Chandler_, Aug 03 2015 *)

%o (PARI) {a(n) = (3*(n+1)^2 + 1)\4 - n - 1}; /* _Michael Somos_, Mar 10 2006 */

%o (Magma) [(6*n^2+4*n-1+(-1)^n)/8: n in [0..50] ]; // _Vincenzo Librandi_, Aug 20 2011

%Y Cf. A001859, A077043, A002620, A002378.

%Y Row sums of A104567.

%Y Cf. A000034, A032766, A002717, A070893. - _Richard Choulet_, Jan 28 2010

%Y Cf. A051125.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

%E Offset and description changed by _N. J. A. Sloane_, Nov 30 2006