A059721 -id:A059721 - cp's OEIS frontend

A059722 a(n) = n(2n^2 - 2*n + 1).

0, 1, 10, 39, 100, 205, 366, 595, 904, 1305, 1810, 2431, 3180, 4069, 5110, 6315, 7696, 9265, 11034, 13015, 15220, 17661, 20350, 23299, 26520, 30025, 33826, 37935, 42364, 47125, 52230, 57691, 63520, 69729, 76330, 83335, 90756, 98605, 106894, 115635, 124840

Offset: 0

Henry Bottomley, Feb 07 2001

Mean of the first four nonnegative powers of 2n+1, i.e., ((2n+1)^0 + (2n+1)^1 + (2n+1)^2 + (2n+1)^3)/4. E.g., a(2) = (1 + 3 + 9 + 27)/4 = 10.
Equatorial structured meta-diamond numbers, the n-th number from an equatorial structured n-gonal diamond number sequence. There are no 1- or 2-gonal diamonds, so 1 and (n+2) are used as the first and second terms since all the sequences begin as such. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Starting with offset 1 = row sums of triangle A143803. - Gary W. Adamson, Sep 01 2008
Form an array from the antidiagonals containing the terms in A002061 to give antidiagonals 1; 3,3; 7,4,7; 13,8,8,13; 21,14,9,14,21; and so on. The difference between the sum of the terms in n+1 X n+1 matrices and those in n X n matrices is a(n) for n>0. - J. M. Bergot, Jul 08 2013
Sum of the numbers from (n-1)^2 to n^2. - Wesley Ivan Hurt, Sep 08 2014

Harry J. Smith, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of sequences with the formula m*(m+1)^2 - (k+2)*m^2 (table includes this sequence for k = 2-m, m >= 0).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000330, A005900, A081436, A100178, A100179, A059722: "equatorial" structured diamonds; A000447: "polar" structured meta-diamond; A006484 for other structured meta numbers; and A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Cf. A000217, A000290, A001844, A002061, A002414, A053698, A059721, A059723, A136392, A143803.

[n*(2*n^2 - 2*n + 1) : n in [0..50]]; // Wesley Ivan Hurt, Sep 08 2014

A059722:=n->n*(2*n^2 - 2*n + 1): seq(A059722(n), n=0..50); # Wesley Ivan Hurt, Sep 08 2014

Table[n (2 n^2 - 2 n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Sep 08 2014 *)

a(n) = { n*(2*n^2 - 2*n + 1) } \\ Harry J. Smith, Jun 28 2009

a(n) = A053698(2*n-1)/4.
a(n) = Sum_{j=1..n} ((n+j-1)^2-j^2+1). - Zerinvary Lajos, Sep 13 2006
From R. J. Mathar, Sep 02 2008: (Start)
G.f.: x*(1 + x)*(1 + 5*x)/(1 - x)^4.
a(n) = A002414(n-1) + A002414(n).
a(n+1) - a(n) = A136392(n+1). (End)
a(n) = (A000290(n) + A000290(n+1)) * (A000217(n+1) - A000217(n)). - J. M. Bergot, Nov 02 2012
a(n) = n * A001844(n-1). - Doug Bell, Aug 18 2015
a(n) = A000217(n^2) - A000217(n^2-2*n). - Bruno Berselli, Jun 26 2018
E.g.f.: exp(x)*x*(1 + 4*x + 2*x^2). - Stefano Spezia, Jun 20 2021

Edited with new definition by N. J. A. Sloane, Aug 29 2008

A059723 Mean of the first 2^n nonnegative powers of 3, i.e., (1 + 3 + 3^2 + ... + 3^(2^n-1))/2^n.

Original entry on oeis.org

1, 2, 10, 410, 1345210, 28953440450810, 26825654846035253786389446010, 46055408506791340513753409614892651037805514032327504332410

Offset: 0

Henry Bottomley, Feb 07 2001

nonn

			a(3) = (1 + 3 + 9 + 27 + 81 + 243 + 729 + 2187)/8 = 3280/8 = 410.

Harry J. Smith, Table of n, a(n) for n = 0..11
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A059721, A059722.

{ for (n=0, 11, write("b059723.txt", n, " ", (3^(2^n) - 1)/2^(n + 1)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = (3^(2^n) - 1)/2^(n+1).

A108495 a(n) = (n^7 - n)/6.

Original entry on oeis.org

0, 0, 21, 364, 2730, 13020, 46655, 137256, 349524, 797160, 1666665, 3247860, 5971966, 10458084, 17568915, 28476560, 44739240, 68389776, 102036669, 148978620, 213333330, 300181420, 415726311, 567470904, 764411900, 1017252600

Offset: 0

Henry Bottomley, Jun 06 2005

Also integer sequences for (n^2-n)/1 (A002378 offset), (n^3-n)/2 (A027480 offset), (n^43-n)/42 (A108496) and (n^1807-n)/1806.

			a(2) = (2^7 - 2)/6 = 126/6 = 21.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Zagier, Problems posed at the St Andrews Colloquium, 1996.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A014117, A108497, A108499.

[(n^7-n)/6: n in [0..40]]; // Vincenzo Librandi, May 02 2011

Table[(n^7-n)/6,{n,0,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{0,0,21,364,2730,13020,46655,137256},30] (* Harvey P. Dale, Apr 16 2014 *)

[(n**7-n)//6 for n in range(41)] # David Radcliffe, Jun 06 2025

a(n) = (n-1)*A059721(n) = -A024004(n)*n/6.

G.f.: 7*x^2*(3 + 28*x + 58*x^2 + 28*x^3 + 3*x^4)/(1-x)^8. [Colin Barker, May 08 2012]

A108498 Triangle read by rows: T(n,k) = sum_i{1<=i<=n} k^i mod n, showing 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 4, 5, 6, 0, 0, 6, 0, 4, 0, 2, 0, 0, 0, 5, 3, 0, 2, 6, 0, 8, 0, 0, 6, 2, 0, 0, 0, 6, 2, 0, 0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 0, 6, 12, 6, 2, 0, 0, 0, 6, 12, 6, 2

Offset: 1

Henry Bottomley, Jun 06 2005

			Rows start: 0; 0,0; 0,2,0; 0,2,0,0; 0,2,3,4,0; 0,0,0,0,0,0; 0,2,3,4,5,6,0; etc.
T(7,3) = 3+9+27+81+243+729+2187 mod 7 = 3279 mod 7 = 3.

Cf. A014117, A059721, A108048, A108497, A108499.

T(n, k+n)=T(n, k). T(n, 0)=T(n, 1)=T(n, n)=T(1, k)=T(2, k)=T(6, k)=T(42, k)=T(1806, k)=0. T(p, k)=k for p prime and 1

A108048 Mean of first 42 positive powers of n, i.e., (n + n^2 + n^3 + ... + n^42)/42.

Original entry on oeis.org

0, 1, 209430786243, 3907821040411155686, 614057559169335453819010, 6767073673905716055915469215, 13749422954239269326919955861621, 8665930063459510313925842272832668

Offset: 0

Henry Bottomley, Jun 01 2005

nonn

Similar sequences can be constructed for mean of first powers, giving A001477, for mean of first two powers, giving A000217, and for mean of first six powers, giving A059721. Next similar sequence seems to be for mean of first 1806 powers.

			a(2) = (2 + 4 + 8 + ... + 4398046511104)/42 = 8796093022206/42 = 209430786243.

Cf. A001477, A000217, A059721.

Table[Mean[n^Range[42]],{n,0,10}] (* Harvey P. Dale, Jun 19 2018 *)

a(n) = if (n<2, n, n*(n^42-1)/(42*(n-1))); \\ Michel Marcus, Jul 26 2017

a(n) = n*(n^42-1)/(42*(n-1)).

A228290 a(n) = n^6 + n^5 + n^4 + n^3 + n^2 + n.

Original entry on oeis.org

0, 6, 126, 1092, 5460, 19530, 55986, 137256, 299592, 597870, 1111110, 1948716, 3257436, 5229042, 8108730, 12204240, 17895696, 25646166, 36012942, 49659540, 67368420, 90054426, 118778946, 154764792, 199411800, 254313150, 321272406, 402321276, 499738092

Offset: 0

Alois P. Heinz, Aug 19 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Column k=6 of A228275.

a:= n-> (1+(1+(1+(1+(1+n)*n)*n)*n)*n)*n:
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 6*n,
      (n+1)*(n^2+n+1)*a(n-1)/((n-1)*(n^2-3*n+3)))
    end:
seq(a(n), n=0..30);
# third Maple program:
a:= n-> `if`(n=1, 6, (n^7-n)/(n-1)):
seq(a(n), n=0..30);

a(n) = n^6 + n^5 + n^4 + n^3 + n^2 + n; \\ Joerg Arndt, Sep 03 2013

a <- c(0, 6, 126, 1092, 5460, 19530, 55986)
for(n in (length(a)+1):30) a[n] <- 7*a[n-1] -21*a[n-2] +35*a[n-3] -35*a[n-4] +21*a[n-5] -7*a[n-6] +a[n-7]
a
[Yosu Yurramendi, Sep 03 2013]

G.f.: -6*x*(7*x^4+42*x^3+56*x^2+14*x+1)/(x-1)^7.
a(n) = (n+1)*(n^2+n+1)*a(n-1)/((n-1)*(n^2-3*n+3)) for n>1.
a(1) = 6, else a(n) = (n^7-n)/(n-1).
a(n) = 6*A059721(n) = n*(n+1)*(1+n+n^2)*(1-n+n^2). - R. J. Mathar, Aug 21 2013
a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7) for n>6, a(0)=0, a(1)=6, a(2)=126, a(3)=1092, a(4)=5460, a(5)=19530, a(6)=55986. - Yosu Yurramendi, Sep 03 2013

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cp's OEIS Frontend

A059722 a(n) = n*(2*n^2 - 2*n + 1).

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A059723 Mean of the first 2^n nonnegative powers of 3, i.e., (1 + 3 + 3^2 + ... + 3^(2^n-1))/2^n.

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A108495 a(n) = (n^7 - n)/6.

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A108498 Triangle read by rows: T(n,k) = sum_i{1<=i<=n} k^i mod n, showing 1<=k<=n.

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A108048 Mean of first 42 positive powers of n, i.e., (n + n^2 + n^3 + ... + n^42)/42.

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A228290 a(n) = n^6 + n^5 + n^4 + n^3 + n^2 + n.

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A059722 a(n) = n(2n^2 - 2*n + 1).