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A074378 Even triangular numbers halved. 28
0, 3, 5, 14, 18, 33, 39, 60, 68, 95, 105, 138, 150, 189, 203, 248, 264, 315, 333, 390, 410, 473, 495, 564, 588, 663, 689, 770, 798, 885, 915, 1008, 1040, 1139, 1173, 1278, 1314, 1425, 1463, 1580, 1620, 1743, 1785, 1914, 1958, 2093, 2139, 2280, 2328, 2475 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is also the exact set of integers a(n) such that a(n)+1+2+3+4+...x=3a(n), where x is sufficiently large. For example a(15)=203 because 203+(1+2+3+4+...+28)=609 and 609=3*203. [Gil Broussard, Sep 01 2008]

a(n) is the set of all n such that 16*n+1 is a perfect square. [Gary Detlefs, Feb 21 2010]

Also integers of the form sum(k=0..n, k/2). [Arkadiusz Wesolowski, Feb 07 2012]

A074378 (n) = A074378 (n even) + A074378 (n odd)

A074378 (n even) = (n^2-n/2)*((1+(-1)^n))/2 transformed from A033991

A074378 (n odd) = ((n-1)^2+((n-1)/2))*((1-(-1)^n))/2 transformed from A007742

So, the final expression:

A074378 (n)=(n^2-n/2)*((1+(-1)^n))/2+((n-1)^2+((n-1)/2))*((1-(-1)^n))/2.

LINKS

Table of n, a(n) for n=0..49.

Neville Holmes, More Geometric Integer Sequences

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

FORMULA

Sum_{n>=0} q^a(n) = (Prod_{n>0} (1-q^n))*(Sum_{n>=0} A035294(n)*q^n).

n*(n+1)/4 where n*(n+1)/2 is even.

G.f.: x*(3+2*x+3*x^2)/((1-x)*(1-x^2)^2).

a(n) = (2*n+1)*floor((n+1)/2); a(2*k) = k(4*k+1); a(2*k+1) = (k+1)(4*k+3). [Benoit Jubin, Feb 05 2009]

a(2*n) = A007742(n), a(2*n-1) = A033991(n). [Arkadiusz Wesolowski, Jul 20 2012]

MATHEMATICA

lst = {}; Do[a = 4*n^2; x = {a - n, a + n}; AppendTo[lst, x], {n, 0, 25}]; Rest@Flatten[lst] (* Arkadiusz Wesolowski, Jul 20 2012 *) 1/2 * Select[Accumulate@ Range[0, 100], EvenQ] (* or *)

1/2 * Select[PolygonalNumber@ Range[0, 100], EvenQ] (* Michael De Vlieger, Jun 01 2017, Version 10.4 *)

PROG

(PARI) a(n)=(2*n+1)*(n-n\2)

(MAGMA) f:=func<i | i*(4*i+1)>; [0] cat [f(n*m): m in [-1, 1], n in [1..25]]; // Bruno Berselli, Nov 13 2012

CROSSREFS

Cf. A011848, A014493, A074377, A035294.

Cf. A010709, A047522 [From Vincenzo Librandi, Feb 14 2009]

Cf. A266883: numbers n such that 16*n-15 is a square.

Sequence in context: A028942 A278314 A179213 * A185301 A179304 A026645

Adjacent sequences:  A074375 A074376 A074377 * A074379 A074380 A074381

KEYWORD

nonn,easy

AUTHOR

W. Neville Holmes, Sep 04 2002

STATUS

approved

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Last modified June 28 04:43 EDT 2017. Contains 288813 sequences.