A006578 Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)). (Formerly M3329)

0, 1, 4, 8, 14, 21, 30, 40, 52, 65, 80, 96, 114, 133, 154, 176, 200, 225, 252, 280, 310, 341, 374, 408, 444, 481, 520, 560, 602, 645, 690, 736, 784, 833, 884, 936, 990, 1045, 1102, 1160, 1220, 1281, 1344, 1408, 1474, 1541, 1610, 1680, 1752, 1825, 1900, 1976, 2054

Offset

0,3

Comments

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

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

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

Number of pairs (x,y) with x in {0,...,n}, y odd in {0,...,2n}, and x < y. - Clark Kimberling, Jul 02 2012

Also A049451 and positives A000567 interleaved. - Omar E. Pol, Aug 03 2012

Similar to A001082. Members of this family are A093005, A210977, this sequence, A210978, A181995, A210981, A210982. - Omar E. Pol, Aug 09 2012

References

Marc LeBrun, personal communication.

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Bruno Berselli, Hexagonal spiral containing the initial terms.

Marc Le Brun, Email to N. J. A. Sloane, Jul 1991

Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche (2021) Vol. 76, No. 1, see p. 283.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

Expansion of x*(1+2*x) / ((1-x)^2*(1-x^2)). - Simon Plouffe in his 1992 dissertation

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

Partial sums of A032766. - Paul Barry, May 30 2003

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

a(n) = (6*n^2 + 4*n - 1 + (-1)^n)/8. - Paul Barry, May 30 2003

a(n) = A001859(-1-n) for all n in Z. - Michael Somos, May 10 2006

a(n) = (A002378(n)/2 + A035608(n))/2. - Reinhard Zumkeller, Feb 07 2010

a(n) = (3*n^2 + 2*n - (n mod 2))/4. - Ctibor O. Zizka, Mar 11 2012

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

a(n) = 3*n*(n+1)/2 - A001859(n). - Clark Kimberling, Jul 02 2012

a(n) = Sum_{i=1..n} (n - i + 1) * 2^( (i+1) mod 2 ). - Wesley Ivan Hurt, Mar 30 2014

a(n) = A002717(n) - A002717(n-1). - Michael Somos, Jun 09 2014

a(n) = Sum_{k=1..n} floor((n+k+1)/2). - Wesley Ivan Hurt, Mar 31 2017

a(n) = A002620(n+1)+2*A002620(n). - R. J. Mathar, Apr 28 2017

Sum_{n>=1} 1/a(n) = 3 - Pi/(4*sqrt(3)) - 3*log(3)/4. - Amiram Eldar, May 28 2022

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

Example

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 + ...

Maple

with (combinat): seq(count(Partition((3*n+1)), size=3), n=0..52); # Zerinvary Lajos, Mar 28 2008
# 2nd program
A006578 := proc(n)
    (6*n^2 + 4*n - 1 + (-1)^n)/8 ;
end proc: # R. J. Mathar, Apr 28 2017

Mathematica

Accumulate[LinearRecurrence[{1, 1, -1}, {0, 1, 3}, 100]] (* Harvey P. Dale, Sep 29 2013 *)
a[ n_] := Quotient[n + 1, 2] (Quotient[n, 2] 3 + 1); (* Michael Somos, Jun 09 2014 *)
a[ n_] := Quotient[3 (n + 1)^2 + 1, 4] - (n + 1); (* Michael Somos, Jun 10 2015 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 1, 4, 8}, 53] (* Ray Chandler, Aug 03 2015 *)

Prog

(PARI) {a(n) = (3*(n+1)^2 + 1)\4 - n - 1}; /* Michael Somos, Mar 10 2006 */
(Magma) [(6*n^2+4*n-1+(-1)^n)/8: n in [0..50] ]; // Vincenzo Librandi, Aug 20 2011

Crossrefs

Cf. A001859, A077043, A002620, A002378.

Row sums of A104567.

Cf. A000034, A032766, A002717, A070893. - Richard Choulet, Jan 28 2010

Cf. A051125.

Sequence in context: A088804 A344012 A027924 * A122224 A183955 A299896

Adjacent sequences: A006575 A006576 A006577 * A006579 A006580 A006581

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Offset and description changed by N. J. A. Sloane, Nov 30 2006

Status

approved