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A006578
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Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e. A000217(n) + A002620(n)).
(Formerly M3329)
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25
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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
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OFFSET
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0,3
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COMMENTS
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Row sums of triangle A104567 = (1, 4, 8, 14, 21,...). - Gary W. Adamson, May 05 2007
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 another terms v_p=sum(u_k,k=0..p) 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
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REFERENCES
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Marc LeBrun, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..10000
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-2,1).
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FORMULA
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Expansion of x*(1+2*x) / ((1-x)^2*(1-x^2)).
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) = A002378(n)-A002620(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(-1-n) = A001859(n). - Michael Somos, May 10 2006
a(n) = (A002378(n)/2 + A035608(n))/2. - Reinhard Zumkeller, Feb 07 2010
a(n) = (3*n*n+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, July 2 2012]
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MAPLE
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A006578:=-(1+2*z)/(1+z)/(z-1)**3; [Simon Plouffe in his 1992 dissertation.]
with (combinat):seq(count(Partition((3*n+1)), size=3), n=0..52); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2008
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PROG
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(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
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CROSSREFS
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Cf. A001859, A077043.
A006578 + A002620 = A002378 = n(n+1).
Cf. A104567.
Cf. A000034, A032766, A002717, A070893. [From Richard Choulet, Jan 28 2010]
Cf. A051125.
Sequence in context: A183857 A088804 A027924 * A122224 A183955 A004797
Adjacent sequences: A006575 A006576 A006577 * A006579 A006580 A006581
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Mar 20 2000
Offset and description changed by N. J. A. Sloane, Nov 30 2006
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STATUS
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approved
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