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A084990 a(n) = n*(n^2+3*n-1)/3. 22
0, 1, 6, 17, 36, 65, 106, 161, 232, 321, 430, 561, 716, 897, 1106, 1345, 1616, 1921, 2262, 2641, 3060, 3521, 4026, 4577, 5176, 5825, 6526, 7281, 8092, 8961, 9890, 10881, 11936, 13057, 14246, 15505, 16836, 18241, 19722, 21281, 22920, 24641, 26446, 28337, 30316 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Row sums of triangle A131782 starting (1, 6, 17, 36, 65, 106,...). - Gary W. Adamson, Jul 14 2007

a(n) is the number of triples (x,y,z) in {1,2,..,n}^3 with x <= y <= z or x >= y >= z. - Jack Kennedy, Mar 14 2009

a(2*n) is the difference between numbers of nonnegative multiples of 2*n+1  with even  and odd digit sum in base 2*n in interval [0, 16*n^4). - Vladimir Shevelev, May 18 2012

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

S. Chaiken, C. R. H. Hanusa, T. Zaslavsky, A q-queens problem III. Partial queens, February 19, 2014. See Conjecture 4.4.

V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n) = 2 * A000292(n-1) - 1 (notice offset=-1 in A000292!).

a(n) = (n-1)*(n+1)*(n+3)/3 + 1. - Reinhard Zumkeller, Aug 20 2007

a(n) = A077415(n+1) + 1 for n>0; a(n) = A000290(n) + A007290(n); a(n+1) = Sum(A028387(k): 0<=k<=n). - Reinhard Zumkeller, Aug 20 2007

a(2*n) = sum {i=0,...,16*n^4, i==0 mod 2*n+1}(-1)^s_(2*n)(i), where s_k(n) is the digit sum of n in the base k. - Vladimir Shevelev, May 18 2012

a(2*n) = 2/(2*n+1)*sum{i=1,...,n} tan^4(pi*i/(2*n+1)). - Vladimir Shevelev, May 23 2012

a(n) = sum_{i=1..n} i*(i+1)-1. - Wesley Ivan Hurt, Oct 19 2013

G.f.: x*(1+2*x-x^2)/(1-x)^4. - Vincenzo Librandi, Mar 28 2014

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - Vincenzo Librandi, Mar 28 2014

a(n) = A064043(n)/3. - Alois P. Heinz, Jul 21 2017

EXAMPLE

Let n=2. Consider nonnegative multiples of 5 up to 16*2^4-1=255. There are 52 such numbers and from them only 8 (namely, 35,50,55,115,140,200,205, 220) have odd digit sum in base 4. Therefore, a(4)=(52-8)-8=36. - Vladimir Shevelev, May 18 2012

MAPLE

A084990:=n->n*(n^2+3*n-1)/3; seq(A084990(k), k=0..100); # Wesley Ivan Hurt, Oct 19 2013

MATHEMATICA

Table[n*(n^2+3*n-1)/3, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2010 *)

CoefficientList[Series[(x + 2 x^2 - x^3)/((1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 28 2014 *)

LinearRecurrence[{4, -6, 4, -1}, {0, 1, 6, 17}, 50] (* Harvey P. Dale, Aug 18 2015 *)

PROG

(MAGMA) I:=[0, 1, 6, 17]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Mar 28 2014

(PARI) a(n) = n*(n^2+3*n-1)/3 \\ Charles R Greathouse IV, Sep 24 2015

CROSSREFS

Cf. A064043, A131782.

Sequence in context: A083045 A012277 A307502 * A024181 A023663 A048208

Adjacent sequences:  A084987 A084988 A084989 * A084991 A084992 A084993

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson, Jul 16 2003

STATUS

approved

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Last modified April 22 06:01 EDT 2021. Contains 343161 sequences. (Running on oeis4.)