A008590 -id:A008590 - cp's OEIS frontend

A047521 Numbers that are congruent to {0, 7} mod 8.

0, 7, 8, 15, 16, 23, 24, 31, 32, 39, 40, 47, 48, 55, 56, 63, 64, 71, 72, 79, 80, 87, 88, 95, 96, 103, 104, 111, 112, 119, 120, 127, 128, 135, 136, 143, 144, 151, 152, 159, 160, 167, 168, 175, 176, 183, 184, 191, 192, 199, 200, 207, 208, 215, 216, 223, 224, 231, 232

Offset: 1

N. J. A. Sloane

Numbers such that the n-th triangular number is divisible by 4. - Charles R Greathouse IV, Apr 07 2011

Except for 0, numbers whose binary reflected Gray code (A014550) ends with 00. - Amiram Eldar, May 17 2021

David Lovler, Table of n, a(n) for n = 1..10000
Lars Pos, Met kleine stapjes grote sprongen make, Pythagoras 61-4. Solutions of returning to the origin after steps of increasing width 1,2,3,.. in the 4 directions on a square grid (in Dutch).
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A008590 and A004771.

Cf. A014550, A030308, A274406.

{#,#+7}&/@(8*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{0,7,8},60] (* Harvey P. Dale, Oct 30 2016 *)

a(n) = 4*n - 5/2 + 3*(-1)^n/2; \\ David Lovler, Jul 25 2022

kmax <- 10 # by choice
a <- c(0,7)
for(k in 3:kmax) a <- c(a, a + 2^k)
a
# Yosu Yurramendi, Jan 18 2022

a(n) = 8*n - a(n-1) - 9 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 3*(-1)^n/2 - 5/2 + 4*n.
G.f.: x^2*(7+x) / ( (1+x)*(x-1)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=7 and b(k)=2^(k+2) for k > 0. - Philippe Deléham, Oct 17 2011
Sum_{n>=2} (-1)^n/a(n) = log(2)/2 + sqrt(2)*log(sqrt(2)+1)/8 - (sqrt(2)+1)*Pi/16. - Amiram Eldar, Dec 18 2021
E.g.f.: 1 + ((8*x -5)*exp(x) + 3*exp(-x))/2. David Lovler, Aug 22 2022

More terms from Vincenzo Librandi, Aug 06 2010

A085131 Multiples of 8 which are 7-smooth.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 96, 112, 120, 128, 144, 160, 168, 192, 200, 216, 224, 240, 256, 280, 288, 320, 336, 360, 384, 392, 400, 432, 448, 480, 504, 512, 560, 576, 600, 640, 648, 672, 720, 768, 784, 800, 840, 864, 896, 960, 1000, 1008, 1024

Offset: 1

Amarnath Murthy, Jul 06 2003

Equivalently, multiples of 8 with the largest prime divisor < 10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Intersection of A008590 and A002473.

Cf. A085125, A085126, A085127, A085128, A085129, A080194, A085132.

Select[8*Range[200],FactorInteger[#][[-1,1]]<10&] (* Harvey P. Dale, Oct 22 2017 *)

From Amiram Eldar, Sep 22 2024: (Start)
a(n) = 8*A002473(n).
Sum_{n>=1} 1/a(n) = 35/64. (End)

More terms from David Wasserman, Jan 28 2005

Offset changed by Andrew Howroyd, Sep 22 2024

A087410 Multiples of 8 with digits grouped in pairs and leading zeros omitted.

Original entry on oeis.org

81, 62, 43, 24, 4, 85, 66, 47, 28, 8, 89, 61, 4, 11, 21, 20, 12, 81, 36, 14, 41, 52, 16, 1, 68, 17, 61, 84, 19, 22, 0, 20, 82, 16, 22, 42, 32, 24, 2, 48, 25, 62, 64, 27, 22, 80, 28, 82, 96, 30, 43, 12, 32, 3, 28, 33, 63, 44, 35, 23, 60, 36, 83, 76, 38, 43, 92, 40, 4, 8, 41, 64, 24

Offset: 1

N. J. A. Sloane, Oct 19 2003

Cf. A008590, A059805, A087406, A060936, A087407, A087408, A087409, A087075, A087411.

FromDigits /@ Partition[ Flatten[ IntegerDigits[ Table[ 8n, {n, 1, 60}]]], 2] (* Robert G. Wilson v *)

More terms from Ray Chandler, Oct 20 2003

A144433 Multiples of 8 interleaved with the sequence of odd numbers >= 3.

Original entry on oeis.org

8, 3, 16, 5, 24, 7, 32, 9, 40, 11, 48, 13, 56, 15, 64, 17, 72, 19, 80, 21, 88, 23, 96, 25, 104, 27, 112, 29, 120, 31, 128, 33, 136, 35, 144, 37, 152, 39, 160, 41, 168, 43, 176, 45, 184, 47, 192, 49, 200, 51, 208, 53, 216, 55, 224, 57, 232, 59, 240, 61, 248, 63, 256, 65, 264

Offset: 1

Paul Curtz, Oct 04 2008

For n >= 2, these are the numerators of 1/n^2 - 1/(n+1)^2: A061037(4), A061039(5), A061041(6), A061043(7), A061045(8), A061047(9), A061049(10), etc.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A120070.

[5+3/2*(-1)^(n-1)*(n-1)+3*(-1)^(n-1)+5/2*(n-1): n in [1..70]]; // Vincenzo Librandi, Jul 30 2011

A144433:=n->(n+1)*4^(n mod 2); seq(A144433(n), n=1..100); # Wesley Ivan Hurt, Nov 27 2013

Table[(n + 1)* 4^Mod[n, 2], {n, 100}] (* Wesley Ivan Hurt, Nov 27 2013 *)

x='x+O('x^50); Vec( x*(8+3*x-x^3)/((1-x)^2*(1+x)^2)) \\ G. C. Greubel, Sep 19 2018

a(2*n+1) = A008590(n+1), a(2*n) = A005408(n).
a(2*n+1) + a(2*n+2) = A017281(n+1).
From R. J. Mathar, Apr 01 2009: (Start)
a(n) = 2*a(n-2) - a(n-4).
G.f.: x*(8+3*x-x^3)/((1-x)^2*(1+x)^2). (End)
a(n) = (n + 1) * 4^(n mod 2). - Wesley Ivan Hurt, Nov 27 2013

Edited by R. J. Mathar, Apr 01 2009

A225358 Partition numbers of the form 8k.

Original entry on oeis.org

56, 176, 792, 2323520, 4087968, 8118264, 92669720, 118114304, 150198136, 384276336, 541946240, 1188908248, 1844349560, 2291320912, 3163127352, 4351078600, 5371315400, 5964539504, 7346629512, 10015581680, 11097645016, 16670689208, 18440293320

Offset: 1

Omar E. Pol, May 05 2013

nonn

Intersection of A008590 and A000041.

			56 is in the sequence because 8*7 = 56 and 56 is a partition number: p(11) = A000041(11) = 56.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A000041, A008590, A052001, A072871, A087183, A225324, A225325, A225326, A225327, A225360, A127544, A225361.

Select[PartitionsP[Range[300]], Mod[#, 8] == 0 &]

a(n) = 8*A222178(n).

A098502 a(n) = 16*n - 4.

Original entry on oeis.org

12, 28, 44, 60, 76, 92, 108, 124, 140, 156, 172, 188, 204, 220, 236, 252, 268, 284, 300, 316, 332, 348, 364, 380, 396, 412, 428, 444, 460, 476, 492, 508, 524, 540, 556, 572, 588, 604, 620, 636, 652, 668, 684, 700, 716, 732, 748, 764, 780, 796, 812, 828, 844

Offset: 1

Ralf Stephan, Sep 15 2004

For n > 3, the number of squares on the infinite 4-column chessboard at <= n knight moves from any fixed start point.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences which agree for a long time but are different.

Cf. A004767, A008590, A017113, A017137, A017629, A098498, A158953.

[16*n - 4: n in [1..60]]; // Vincenzo Librandi, Jul 24 2011

Range[12, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

a(n)=16*n-4 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: 4*x*(3+x)/(1-x)^2. - Colin Barker, Jan 09 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + log(3 - 2*sqrt(2)))/(16*sqrt(2)). - Amiram Eldar, Sep 01 2024
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 4*(exp(x)*(4*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2.
a(n) = 4*A004767(n-1) = 2*A017137(n-1) = A017113(2*n-1). (End)

A164897 a(n) = 4n(n+1) + 3.

Original entry on oeis.org

3, 11, 27, 51, 83, 123, 171, 227, 291, 363, 443, 531, 627, 731, 843, 963, 1091, 1227, 1371, 1523, 1683, 1851, 2027, 2211, 2403, 2603, 2811, 3027, 3251, 3483, 3723, 3971, 4227, 4491, 4763, 5043, 5331, 5627, 5931, 6243, 6563, 6891, 7227, 7571, 7923, 8283, 8651, 9027, 9411

Offset: 0

Paul Curtz, Aug 30 2009

One-fourth the sum of the three terms produced by the division of complex numbers (2*n-3+(2*n-1)*i)/(2*n+1+(2*n+3)*i). For (b+c*i)/(d+e*i) the three terms in parentheses are ((b*d+c*e)+(c*d-b*e)*i)/(d^2+e^2). By substituting b=2*n-3, c=2*n-1, d=2*n+1, and e=2*n+3 one gets a(n). - J. M. Bergot, Sep 10 2015

The continued fraction expansion of sqrt(a(n)) is [2n+1; {2n+1, 4n+2}]. - Magus K. Chu, Sep 08 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..900
R. M. Green and Tianyuan Xu, 2-roots for simply laced Weyl groups, arXiv:2204.09765 [math.RT], 2022.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A008590, A010731, A016743, A069894.

Odd-indexed terms of A059100.

[4*n*(n+1)+3: n in [0..50]]; // Vincenzo Librandi, Apr 24 2011

A164897:=n->4*n*(n+1)+3: seq(A164897(n), n=0..100); # Wesley Ivan Hurt, Sep 10 2015

Table[4 n (n + 1) + 3, {n, 0, 50}] (* Harvey P. Dale, Jan 23 2011 *)

a(n)=4*n*(n+1)+3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000124(2*n) + A000124(2*n+1) = A069894(n)+1.
a(n+1) - a(n) = 8n+8 = A008590(n+1) (first differences).
a(n+1) - 2*a(n) + a(n-1) = 8 = A010731(n) (second differences).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n>2.
G.f.: (3+2*x+3*x^2) / (1-x)^3.
Sum_{k=n+1..2*n+1} a(k) - Sum_{k=0..n} a(k) = (2*n+2)^3. - Bruno Berselli, Jan 24 2011
E.g.f.: (4x^2 + 8x + 1)*exp(x). - G. C. Greubel, Sep 22 2015
a(n)^2 = A222465(n)*A222465(n+1) - 12. - Ezhilarasu Velayutham, Mar 18 2020
Sum_{n>=0} 1/a(n) = tanh(Pi/sqrt(2))*Pi/(4*sqrt(2)). - Amiram Eldar, Aug 21 2022
a(n) = A059100(2*n+1). - Dimitri Papadopoulos, Nov 21 2023

Definition simplified by R. J. Mathar, Sep 16 2009

A166911 a(n) = (9 + 14n + 12n^2 + 4*n^3)/3.

Original entry on oeis.org

3, 13, 39, 89, 171, 293, 463, 689, 979, 1341, 1783, 2313, 2939, 3669, 4511, 5473, 6563, 7789, 9159, 10681, 12363, 14213, 16239, 18449, 20851, 23453, 26263, 29289, 32539, 36021, 39743, 43713, 47939, 52429, 57191, 62233, 67563, 73189, 79119, 85361, 91923

Offset: 0

Paul Curtz, Oct 23 2009

The inverse binomial transform yields the quasi-finite sequence 3,10,16,8,0,.. (0 continued).
These are the bottom-left numbers in the blocks (each with 2 rows) shown in A172002, the
atomic number of the leftmost element in the 2nd, 4th, 6th etc. row of the Janet table.

Charles Janet, La structure du noyau de l'atome .., Nov 1927, page 15.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

[(9+14*n+12*n^2+4*n^3)/3: n in [0..40] ]; // Vincenzo Librandi, Aug 06 2011

LinearRecurrence[{4,-6,4,-1}, {3, 13, 39, 89}, 100] (* G. C. Greubel, May 28 2016 *)

a(n)=n*(4*n^2+12*n+14)/3+3 \\ Charles R Greathouse IV, Dec 21 2011

First differences: a(n)-a(n-1) = 2+4*n+4*n^2 = 1+(1+2n)^2 = 1 + A016754(n+1) = A069894(n+1).
Second differences: a(n) - 2*a(n-1) + a(n-2) = 8*n = A008590(n+2).
Third differences: a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 8.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (3 + x + 5*x^2 - x^3)/(1-x)^4.
a(n) = A166464(n) + 2*(n+1)^2 = A166464(n) + A001105(n+1).
E.g.f.: (1/3)*(9 + 30*x + 24*x^2 + 4*x^3)*exp(x). - G. C. Greubel, May 28 2016

Edited and extended by R. J. Mathar, Mar 02 2010

A244082 a(n) = 32*n^2.

Original entry on oeis.org

0, 32, 128, 288, 512, 800, 1152, 1568, 2048, 2592, 3200, 3872, 4608, 5408, 6272, 7200, 8192, 9248, 10368, 11552, 12800, 14112, 15488, 16928, 18432, 20000, 21632, 23328, 25088, 26912, 28800, 30752, 32768, 34848, 36992, 39200, 41472, 43808, 46208, 48672, 51200

Offset: 0

Wesley Ivan Hurt, Jun 19 2014

Geometric connections of a(n) to the area and perimeter of a square.
Area:
. half the area of a square with side 8n (cf. A008590);
. area of a square with diagonal 8n (cf. A008590);
. twice the area of a square with side 4n (cf. A008586);
. four times the area of a square with diagonal 4n (cf. A008586);
. eight times the area of a square with side 2n (cf. A005843);
. sixteen times the area of a square with diagonal 2n (cf. A005843);
. thirty two times the area of a square with side n (cf. A001477);
. sixty four times the area of a square with diagonal n (cf. A001477).
Perimeter:
. perimeter of a square with side 8n^2 (cf. A139098);
. twice the perimeter of a square with side 4n^2 (cf. A016742);
. four times the perimeter of a square with side 2n^2 (cf. A001105);
. eight times the perimeter of a square with side n^2 (cf. A000290).
Sequence found by reading the line from 0, in the direction 0, 32, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, May 10 2018

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Pentasection of A077221, A181900.
Cf. A000290, A001105, A010021, A016742, A016802, A139098.
Cf. A001477, A005843, A008586, A008590, A139098, A174312.

Magma
```
[32*n^2 : n in [0..50]];
```

A244082:=n->32*n^2; seq(A244082(n), n=0..50);

32 Range[0, 50]^2 (* or *)
Table[32 n^2, {n, 0, 50}] (* or *)
CoefficientList[Series[32 x (1 + x)/(1 - x)^3, {x, 0, 30}], x]

a(n)=32*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 32*x*(1+x)/(1-x)^3.
a(n) =  2 * A016802(n).
a(n) =  4 * A139098(n).
a(n) =  8 * A016742(n).
a(n) = 16 * A001105(n).
a(n) = 32 * A000290(n).
a(n) = A010021(n) - 2 for n > 0. - Bruno Berselli, Jun 24 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Nov 19 2021
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 32*x*(1 + x)*exp(x).
a(n) = n*A174312(n) = A139098(2*n). (End)

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121

Offset: 0

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n > 0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007
A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007
Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m) = m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008

			Array begins:
  1  4  9 16 25 36  49  64  81 100 ...
  3  8 15 24 35 48  63  80  99 120 ...
  5 12 21 32 45 60  77  96 117 140 ...
  7 16 27 40 55 72  91 112 135 160 ...
  9 20 33 48 65 84 105 128 153 180 ...
  ...
Triangle begins:
   1;
   3,  4;
   5,  8,  9;
   7, 12, 15, 16;
   9, 16, 21, 24, 25;
  11, 20, 27, 32, 35, 36;
  13, 24, 33, 40, 45, 48, 49;
  15, 28, 39, 48, 55, 60, 63, 64;
  17, 32, 45, 56, 65, 72, 77, 80, 81;
  19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.
Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.
Diagonals give A033428, A045944, A067725.
Cf. A000384, A002412, A024352, A105020, A128076.

[(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023

t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]]
(* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *)
Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)

def A105020(n,k): return (k+1)*(2*n-k+1)
flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023

a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021
a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022
From G. C. Greubel, Mar 15 2023: (Start)
Sum_{k=0..n} T(n, k) = A002412(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)

More terms from Robert G. Wilson v, Jul 11 2005

cp's OEIS Frontend

A047521 Numbers that are congruent to {0, 7} mod 8.

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A085131 Multiples of 8 which are 7-smooth.

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A087410 Multiples of 8 with digits grouped in pairs and leading zeros omitted.

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A144433 Multiples of 8 interleaved with the sequence of odd numbers >= 3.

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A225358 Partition numbers of the form 8k.

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A098502 a(n) = 16*n - 4.

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A164897 a(n) = 4*n*(n+1) + 3.

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A164897 a(n) = 4n(n+1) + 3.

A166911 a(n) = (9 + 14n + 12n^2 + 4*n^3)/3.