A008590 -id:A008590 - cp's OEIS frontend

A181598 Numbers m with divisor 8 | m and abundance sigma(m)-2*m = 8.

56, 368, 11096, 17816, 77744, 128768, 2087936, 2291936, 13174976, 35021696, 45335936, 381236216, 4856970752, 6800228816, 8589344768, 1461083549696, 1471763808896, 2199013818368, 19502341651712, 118123076415296, 933386556194816, 144141575952121856, 417857739454939136

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

a(19) > 10^13. - Giovanni Resta, Apr 02 2014

Cf. A008590, A088833, A097498, A181595, A181596, A181597, A118372, A045768, A000396, A005101, A153501, A005820.

isok(n) = !(n % 8) && (sigma(n) - 2*n == 8); \\ Michel Marcus, Feb 08 2016

A088833 INTERSECT A008590. - R. J. Mathar, Nov 04 2010

Definition rephrased by R. J. Mathar, Nov 04 2010
a(16)-a(17) from Donovan Johnson, Dec 08 2011
a(18) from Giovanni Resta, Apr 02 2014
a(19)-a(23) from the b-file at A088833 added by Amiram Eldar, Mar 11 2024

A283122 a(n) = sigma(8*n).

Original entry on oeis.org

15, 31, 60, 63, 90, 124, 120, 127, 195, 186, 180, 252, 210, 248, 360, 255, 270, 403, 300, 378, 480, 372, 360, 508, 465, 434, 600, 504, 450, 744, 480, 511, 720, 558, 720, 819, 570, 620, 840, 762, 630, 992, 660, 756, 1170, 744, 720, 1020, 855, 961

Offset: 1

Seiichi Manyama, Mar 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), this sequence (k=8), A283123 (k=9).

Cf. A008590.

Table[DivisorSigma[1, 8 n], {n, 50}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sigma(8*n) \\ Amiram Eldar, Dec 16 2022

a(n) = A000203(8*n).

Sum_{k=1..n} a(k) = (23*Pi^2/24) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A073577 a(n) = 4n^2 + 4n - 1.

Original entry on oeis.org

7, 23, 47, 79, 119, 167, 223, 287, 359, 439, 527, 623, 727, 839, 959, 1087, 1223, 1367, 1519, 1679, 1847, 2023, 2207, 2399, 2599, 2807, 3023, 3247, 3479, 3719, 3967, 4223, 4487, 4759, 5039, 5327, 5623, 5927, 6239, 6559, 6887, 7223, 7567, 7919, 8279, 8647

Offset: 1

M. N. Deshpande (dpratap(AT)nagpur.dot.net.in), Aug 27 2002

The sum of the squares of two consecutive terms multiplied (or divided) by 2 is always a perfect square. In general, numbers represented by the quadratic form a(n) = (2*i*n + j)^2 - 2*i^2 for any i and j have 2*(a(n)^2 + a(n+1)^2) and (a(n)^2 + a(n+1)^2)/2 as perfect squares: in this case, i=j=1.
The terms of this sequence may be seen to be 2 less than the odd squares. As such they run parallel to those in the square spiral as well as the Ulam square spiral. - Stuart M. Ellerstein (ellerstein(AT)aol.com), Oct 01 2002
Primes in the sequence are in A028871. - Russ Cox, Aug 26 2019
The continued fraction expansion of sqrt(4*a(n)) is [4n+1; {1, n-1, 2, 2n, 2, n-1, 1, 8n+2}]. For n=1, this collapses to [5; {3, 2, 3, 10}]. - Magus K. Chu, Sep 12 2022

			a(2) = 8*2 + 7 = 23;
a(3) = 8*3 + 23 = 47;
a(4) = 8*4 + 47 = 79. - _Vincenzo Librandi_, Aug 08 2010

G. C. Greubel, Table of n, a(n) for n = 1..5000
Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, Expositiones Mathematicae, Vol. 38, No. 4 (2020), pp. 430-479; arXiv preprint, arXiv:1807.08899 [math.NT], 2018-2019. See 6.6.7, p. 36 (p. 35 in the preprint).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008590, A028871, A214345.

List([1..50],n->4*n^2+4*n-1); # Muniru A Asiru, Nov 01 2018

[4*n^2 + 4*n - 1: n in [1..50]]; // Wesley Ivan Hurt, Apr 18 2016

seq(4*n^2+4*n-1,n=1..100); # Robert Israel, Jan 13 2015

Table[4*n^2+4*n-1,{n,60}] (* Vladimir Joseph Stephan Orlovsky, Nov 18 2009 *)
LinearRecurrence[{3,-3,1},{7,23,47},50] (* Harvey P. Dale, Dec 04 2018 *)

A073577(n):=4*n^2+4*n-1$
makelist(A073577(n),n,1,30); /* Martin Ettl, Nov 01 2012 */

vector(50, n, 4*n^2 + 4*n - 1) \\ Michel Marcus, Jan 14 2015

for n in range(1,50): print(4*n**2+4*n-1, end=', ') # Stefano Spezia, Nov 01 2018

a(n) = FrobeniusNumber(2*n+1, 2*n+3). - Darrell Minor, Jul 29 2008
a(n) = 8*n + a(n-1) (with a(1)=7). - Vincenzo Librandi, Aug 08 2010
G.f.: x*(7+2*x-x^2)/(1-x)^3. - Robert Israel, Jan 13 2015
E.g.f.: 1 - (1-8*x-4*x^2)*exp(x). - Robert Israel, Jan 13 2015
a(n+1) = a(n) + A008590(n+1), a(1) = 7. - Altug Alkan, Sep 28 2015
a(n) = (2*n+1)+(2*n-1) + (2*n+1)*(2*n-1). - J. M. Bergot, Apr 17 2016
a(n) = (2*n+1)^2 - 2. - Zhandos Mambetaliyev, Jun 13 2017
From Stefano Spezia, Nov 04 2018: (Start)
L.g.f.: 4*x*(2+x)/(1+x)^2-log(1+x).
L.h.g.f.: -4*(-2+x)*x/(-1+x)^2+log(1-x).
(End)
Sum_{n>=1} 1/a(n) = 1 + sqrt(2)*Pi*tan(Pi/sqrt(2))/8. - Amiram Eldar, Jan 03 2021

Edited and extended by Henry Bottomley, Oct 10 2002

A185359 Numbers k such that {m^m mod k: m >= 1} is not purely periodic.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 162, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 243, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 324, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400

Offset: 1

José María Grau Ribas, Jan 21 2012

nonn

k is a term if and only if k = Product_{i=1..t} p_i^e_i with e_i > p_i for some i.
A182938(a(n)) = 0. - Reinhard Zumkeller, Feb 18 2012
The asymptotic density of this sequence is 1 - Product_{p prime} 1 - 1/p^(p+1) = 0.13585792767780221591... - Amiram Eldar, Nov 24 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. Hampel, The length of the shortest period of rests of numbers n^n, Ann. Polon. Math. 1 (1955), 360-366.

Cf. A027748, A124010, A008590 (subsequence), A185358, A207481 (complement).

a185359 n = a185359_list !! (n-1)
a185359_list = [x | x <- [1..], or $ zipWith (<)
                    (a027748_row x) (map toInteger $ a124010_row x)]
-- Reinhard Zumkeller, Feb 18 2012

j[p_,e_]:=e>p;j[n_]:={False}==Union@Module[{fa=FactorInteger[n]},Table[j[fa[[i,1]],fa[[i,2]]],{i,1,Length[fa]}]];Select[Range[1000],!j[#]&]

A045643 Palindromic and divisible by 8.

Original entry on oeis.org

0, 8, 88, 232, 272, 424, 464, 616, 656, 696, 808, 848, 888, 2112, 2552, 2992, 4224, 4664, 6336, 6776, 8008, 8448, 8888, 21112, 21312, 21512, 21712, 21912, 23032, 23232, 23432, 23632, 23832, 25152, 25352, 25552, 25752, 25952, 27072, 27272, 27472

Offset: 1

Jeff Burch

Paolo Xausa, Table of n, a(n) for n = 1..2000

Intersection of A002113 and A008590.

Cf. A029951, A045638, A045639, A043040, A043041, A045642, A045644, A083852.

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]];Select[ 8Range[0,3500],palQ] (* Harvey P. Dale, Jun 06 2011 *)
Select[8*Range[0, 3500],PalindromeQ] (* Harvey P. Dale, Feb 17 2023 *)

Edited to include a(1) = 0 by Paolo Xausa, Jul 07 2025

A047592 Numbers that are congruent to {1, 2, 3, 4, 5, 6, 7} mod 8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81

Offset: 1

N. J. A. Sloane

Or, numbers that are not multiples of 8. - Benoit Cloitre, Jul 11 2009
More generally the sequence of numbers not divisible by some fixed integer m >= 2 is given by a(n, m) = n - 1 + floor((n+m-2)/(m-1)). - Benoit Cloitre, Jul 11 2009
Complement of A008590. - Reinhard Zumkeller, Nov 30 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A008590, A168181, A207481.

[ n: n in [0..100] | n mod 8 in {1, 2, 3, 4, 5, 6, 7} ]; // Vincenzo Librandi, Dec 25 2010

A047592:=n->8*floor(n/7)+[1, 2, 3, 4, 5, 6, 7][(n mod 7)+1]: seq(A047592(n), n=0..100); # Wesley Ivan Hurt, Jul 20 2016

Complement[Range[88], 8Range[11]] (* Harvey P. Dale, Jan 22 2011 *)
CoefficientList[Series[(1 + x)*(1 + x^2)*(1 + x^4)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x - 1)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Jan 06 2013 *)

a(n)=n-1+floor((n+6)/7) \\ Benoit Cloitre, Jul 11 2009

a(n) = n - 1 + floor((n+6)/7). - Benoit Cloitre, Jul 11 2009
A168181(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009
From R. J. Mathar, Mar 08 2011: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n>8.
G.f.: x*(1+x)*(1+x^2)*(1+x^4) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). (End)
a(n) = A207481(n) for n <= 70. - Reinhard Zumkeller, Feb 18 2012
From Wesley Ivan Hurt, Jul 20 2016: (Start)
a(n) = (56*n - 28 + (n mod 7) + ((n+1) mod 7) + ((n+2) mod 7) + ((n+3) mod 7) + ((n+4) mod 7) + ((n+5) mod 7) - 6*((n+6) mod 7))/49.
a(7k) = 8k-1, a(7k-1) = 8k-2, a(7k-2) = 8k-3, a(7k-3) = 8k-4, a(7k-4) = 8k-5, a(7k-5) = 8k-6, a(7k-6) = 8k-7. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (4*sqrt(sqrt(2)+2) - 2*sqrt(2) - 1)*Pi/16. - Amiram Eldar, Dec 28 2021

A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 2, 0, 3, 4, 3, 0, 4, 6, 6, 4, 0, 5, 8, 9, 8, 5, 0, 6, 10, 12, 12, 10, 6, 0, 7, 12, 15, 16, 15, 12, 7, 0, 8, 14, 18, 20, 20, 18, 14, 8, 0, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0, 12

Offset: 1

Reinhard Zumkeller, May 31 2004

T(n,k) = A003991(n-1,k) for 1 <= k < n;
T(n,k) = T(n,n-1-k) for k < n;
T(n,1) = n-1; T(n,n) = 0;          T(n,2) = A005843(n-2) for n > 1;
T(n,3) = A008585(n-3) for n>2;     T(n,4) = A008586(n-4) for n > 3;
T(n,5) = A008587(n-5) for n>4;     T(n,6) = A008588(n-6) for n > 5;
T(n,7) = A008589(n-7) for n>6;     T(n,8) = A008590(n-8) for n > 7;
T(n,9) = A008591(n-9) for n>8;     T(n,10) = A008592(n-10) for n > 9;
T(n,11) = A008593(n-11) for n>10;  T(n,12) = A008594(n-12) for n > 11;
T(n,13) = A008595(n-13) for n>12;  T(n,14) = A008596(n-14) for n > 13;
T(n,15) = A008597(n-15) for n>14;  T(n,16) = A008598(n-16) for n > 15;
T(n,17) = A008599(n-17) for n>16;  T(n,18) = A008600(n-18) for n > 17;
T(n,19) = A008601(n-19) for n>18;  T(n,20) = A008602(n-20) for n > 19;
Row sums give A000292; triangle sums give A000332;
All numbers m > 0 occur A000005(m) times;
A002378(n) = T(A005408(n),n+1) = n*(n+1).
k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) =  =  = i  = -i . Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016
T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020

			From _M. F. Hasler_, Feb 02 2013: (Start)
Triangle begins:
  0;
  1, 0;
  2, 2, 0;
  3, 4, 3, 0;
  4, 6, 6, 4, 0;
  5, 8, 9, 8, 5, 0;
  (...)
If an additional 0 was added at the beginning, this would become:
  0;
  0, 1;
  0, 2, 2;
  0, 3, 4; 3;
  0, 4, 6, 6, 4;
  0, 5, 8, 9, 8, 5;
  ... (End)

W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.

J_3: A114327; J_1^2, J_2^2: A141387, A268759.
Cf. A000005, A002378, A003991, A005408.
Cf. A000292 (row sums), A000332 (triangle sums).
T(n,k) for values of k:
  A005843 (k=2), A008585 (k=3), A008586 (k=4), A008587 (k=5), A008588 (k=6), A008589 (k=7), A008590 (k=8), A008591 (k=9), A008592 (k=10), A008593 (k=11), A008594 (k=12), A008595 (k=13), A008596 (k=14), A008597 (k=15), A008598 (k=16), A008599 (k=17), A008600 (k=18), A008601 (k=19), A008602 (k=20).

/* As triangle */ [[k*(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 30 2016

Flatten[Table[(j - m) (j + m + 1), {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

{for(n=1, 13, for(k=1, n, print1(k*(n - k)," ");); print(););} \\ Indranil Ghosh, Mar 12 2017

A119457 Triangle read by rows: T(n, 1) = n, T(n, 2) = 2*(n-1) for n>1 and T(n, k) = T(n-1, k-1) + T(n-2, k-2) for 2 < k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 5, 5, 8, 9, 10, 8, 6, 10, 12, 15, 16, 13, 7, 12, 15, 20, 24, 26, 21, 8, 14, 18, 25, 32, 39, 42, 34, 9, 16, 21, 30, 40, 52, 63, 68, 55, 10, 18, 24, 35, 48, 65, 84, 102, 110, 89, 11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144, 12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233

Offset: 1

Reinhard Zumkeller, May 20 2006

			Triangle begins as:
   1;
   2,  2;
   3,  4,  3;
   4,  6,  6,  5;
   5,  8,  9, 10,  8;
   6, 10, 12, 15, 16, 13;
   7, 12, 15, 20, 24, 26,  21;
   8, 14, 18, 25, 32, 39,  42,  34;
   9, 16, 21, 30, 40, 52,  63,  68,  55;
  10, 18, 24, 35, 48, 65,  84, 102, 110,  89;
  11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144;
  12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Fibonacci Number

Main diagonal: A023607(n).
Sums: A001891 (row), A355020 (signed row).
Columns: A000027(n) (k=1), A005843(n-1) (k=2), A008585(n-2) (k=3), A008587(n-3) (k=4), A008590(n-4) (k=5), A008595(n-5) (k=6), A008603(n-6) (k=7).
Diagonals: A000045(n+1) (k=n), A006355(n+1) (k=n-1), A022086(n-1) (k=n-2), A022087(n-2) (k=n-3), A022088(n-3) (k=n-4), A022089(n-4) (k=n-5), A022090(n-5) (k=n-6).
Cf. A023652, A112469.

A119457:= func< n,k | (n-k+1)*Fibonacci(k+1) >;
[A119457(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 16 2025

(* First program *)
T[n_, 1] := n;
T[n_ /; n > 1, 2] := 2 n - 2;
T[n_, k_] /; 2 < k <= n := T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)
(* Second program *)
A119457[n_,k_]:= (n-k+1)*Fibonacci[k+1];
Table[A119457[n,k], {n,13}, {k,n}]//Flatten (* G. C. Greubel, Apr 16 2025 *)

def A119457(n,k): return (n-k+1)*fibonacci(k+1)
print(flatten([[A119457(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Apr 16 2025

T(n, k) = (n-k+1)*T(k,k) for 1 <= k < n, with T(n, n) = A000045(n+1).
From G. C. Greubel, Apr 15 2025: (Start)
T(n, k) = (n-k+1)*Fibonacci(k+1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A023652(floor((n+1)/2)) + (1+(-1)^n)*A001891(floor(n/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n)*A112469(floor((n-1)/2)) + (1+(-1)^n)*A355020(floor((n-2)/2)). (End)

A212703 Main transitions in systems of n particles with spin 4.

Original entry on oeis.org

8, 144, 1944, 23328, 262440, 2834352, 29760696, 306110016, 3099363912, 30993639120, 306837027288, 3012581722464, 29372671794024, 284688972772848, 2745215094595320, 26354064908115072, 252010745683850376, 2401514164751985936, 22814384565143866392, 216136274827678734240

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=9 (see formula), corresponding to spin S=(b-1)/2=4.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (18,-81).

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212702, A212704 (b = 2, 3, 4, 5, 6, 7, 8, 10).

Cf. A001019, A008590, A053540.

LinearRecurrence[{18,-81},{8,144},30] (* Harvey P. Dale, Jun 28 2017 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212703.txt", n, " ", mtrans(n, 9)))

Vec(8*x/(9*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n)=8*n*9^(n-1) \\ Charles R Greathouse IV, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=9.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 18*a(n-1) - 81*a(n-2) for n > 2.
G.f.: 8*x/(9*x-1)^2. (End)
From Elmo R. Oliveira, May 13 2025: (Start)
E.g.f.: 8*x*exp(9*x).
a(n) = 8*A053540(n) = A008590(n)*A001019(n-1). (End)

A047393 Numbers that are congruent to {0, 1} mod 8.

Original entry on oeis.org

0, 1, 8, 9, 16, 17, 24, 25, 32, 33, 40, 41, 48, 49, 56, 57, 64, 65, 72, 73, 80, 81, 88, 89, 96, 97, 104, 105, 112, 113, 120, 121, 128, 129, 136, 137, 144, 145, 152, 153, 160, 161, 168, 169, 176, 177, 184, 185, 192, 193, 200, 201, 208, 209, 216, 217, 224, 225, 232

Offset: 1

N. J. A. Sloane

Numbers k such that floor(k/2) = 4*floor(k/8). - Bruno Berselli, Oct 05 2017

David Lovler, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A008590 and A017077.

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+8 od: seq(a[n], n=0..58); # Zerinvary Lajos, Mar 16 2008

Riffle[#, # + 1] & [Range[0, 400, 8]] (* Paolo Xausa, Mar 11 2025 *)

forstep(n=0,200,[1,7],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

a(n) = 4*n - 11/2 - 3*(-1)^n/2; \\ David Lovler, Jul 25 2022

def A047393(n): return n-1<<2 if n&1 else (n<<2)-7 # Chai Wah Wu, Mar 11 2025

a(n) = 8*n - a(n-1) - 15 for n>1, a(1)=0. - Vincenzo Librandi, Aug 05 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 4*n - 11/2 - 3*(-1)^n/2.
G.f.: x^2*(1+7*x) / ( (1+x)*(x-1)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*A146541(k). - Philippe Deléham, Oct 17 2011
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)+1)*Pi/16 + log(2)/2 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021
E.g.f.: ((8*x - 3)*exp(x) + 3*exp(-x))/2 = 4*x*exp(x) - 3*sinh(x). - David Lovler, Aug 02 2022

More terms from James Sellers, Jun 20 2000

cp's OEIS Frontend

A181598 Numbers m with divisor 8 | m and abundance sigma(m)-2*m = 8.

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A283122 a(n) = sigma(8*n).

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A073577 a(n) = 4*n^2 + 4*n - 1.

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A185359 Numbers k such that {m^m mod k: m >= 1} is not purely periodic.

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A045643 Palindromic and divisible by 8.

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A047592 Numbers that are congruent to {1, 2, 3, 4, 5, 6, 7} mod 8.

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A094053 Triangle read by rows: T(n,k) = k*(n-k), 1 <= k <= n.

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A073577 a(n) = 4n^2 + 4n - 1.