A008587 -id:A008587 - cp's OEIS frontend

A011558 Expansion of (x + x^3)/(1 + x + ... + x^4) mod 2.

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0

Offset: 0

N. J. A. Sloane

Multiplicative with a(5^e) = 0, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005
Characteristic function of numbers coprime to 5. - Reinhard Zumkeller, Nov 30 2009
From R. J. Mathar, Jul 15 2010: (Start)
The sequence is the principal Dirichlet character mod 5. (The other real character mod 5 is A080891.)
Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 = 1.5791367... = (psi'(1/5) + psi'(2/5) + psi'(3/5) + psi'(4/5))/25 or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.192440... = -(psi''(1/5) + psi''(2/5) + psi''(3/5) + psi''(4/5))/250, where psi' and psi'' are the trigamma and tetragamma functions. (End)
a(n) is for n >= 1 also the characteristic function for rational g-adic integers (+n/5)A047201).%20See%20the%20definition%20in%20the%20Mahler%20reference,%20p.%207%20and%20also%20p.%2010.%20-%20_Wolfdieter%20Lang">g and also (-n/5)_g for all integers g >= 2 without a factor of 5 (A047201). See the definition in the Mahler reference, p. 7 and also p. 10. - _Wolfdieter Lang, Jul 11 2014
Conjecture: a(n+1) is the number of ways of partitioning n into distinct parts of A084215. - R. J. Mathar, Mar 01 2023

			G.f. = x + x^2 + x^3 + x^4 + x^6 + x^7 + x^8 + x^9 + x^11 + x^12 + ...

Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10).
K. Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Gilleland, Some Self-Similar Integer Sequences
R. Gold, Characteristic linear sequences and their coset functions, J. SIAM Applied. Math., 14 (1966), 980-985.
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A000035, A011655, A109720 coprimality with 2, 3, 7, respectively.
Cf. A168185, A145568, A168184, A168182, A168181, A097325, A166486.
Cf. A080891, A100047.

seq(n&^4 mod 5, n=0..50); # Gary Detlefs, Mar 20 2010

Mod[#,2]&/@CoefficientList[Series[(x+x^3)/(1+x+x^2+x^3+x^4) ,{x,0,100}], x] (* or *) Flatten[Table[{0,1,1,1,1},{30}]] (* Harvey P. Dale, May 15 2011 *)
a[ n_] := Sign@Mod[ n, 5]; (* Michael Somos, May 24 2015 *)

a(n)=!!(n%5) \\ Charles R Greathouse IV, Sep 23 2012

{a(n) = n%5>0}; /* Michael Somos, May 24 2015 */

(define (A011558 n) (if (zero? (modulo n 5)) 0 1)) ;; Antti Karttunen, Dec 21 2017

O.g.f.: x*(1+x+x^2+x^3)/(1-x^5). - Wolfdieter Lang, Feb 05 2009
From Reinhard Zumkeller, Nov 30 2009: (Start)
a(n) = 1 - A079998(n).
a(A047201(n))=1, a(A008587(n))=0.
A033437(n) = Sum_{k=0..n} a(k)*(n-k). (End)
a(n) = n^4 mod 5. - Gary Detlefs, Mar 20 2010
Sum_{n>=1} a(n)/n^s = L(s,chi) = (1-1/5^s)*Riemann_zeta(s), s > 1. - R. J. Mathar, Jul 31 2010
For the general case. The characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = sgn(n mod 5). - Wesley Ivan Hurt, Jun 30 2013
Euler transform of length 5 sequence [ 1, 0, 0, -1, 1]. - Michael Somos, May 24 2015
Moebius transform is length 5 sequence [ 1, 0, 0, 0, -1]. - Michael Somos, May 24 2015
G.f.: f(x) - f(x^5) where f(x) := x / (1 - x). - Michael Somos, May 24 2015
|a(n)| = |A080891(n)| = |A100047(n)|. - Michael Somos, May 24 2015

More terms from Antti Karttunen, Dec 21 2017

A195020 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The edges of the spiral have length A195019.

Original entry on oeis.org

0, 3, 7, 13, 21, 30, 42, 54, 70, 85, 105, 123, 147, 168, 196, 220, 252, 279, 315, 345, 385, 418, 462, 498, 546, 585, 637, 679, 735, 780, 840, 888, 952, 1003, 1071, 1125, 1197, 1254, 1330, 1390, 1470, 1533, 1617, 1683, 1771, 1840, 1932, 2004, 2100

Offset: 0

Omar E. Pol, Sep 07 2011 - Sep 12 2011

Zero together with the partial sums of A195019.
The spiral contains infinitely many Pythagorean triples in which the hypotenuses on the main diagonal are the positives A008587. The vertices on the main diagonal are the numbers A024966 = (3+4)*A000217 = 7*A000217, where both 3 and 4 are the first two edges in the spiral. The distance "a" between nearest edges that are perpendicular to the initial edge of the spiral is 3, while the distance "b" between nearest edges that are parallel to the initial edge is 4, so the distance "c" between nearest vertices on the same axis is 5 because from the Pythagorean theorem we can write c = (a^2+b^2)^(1/2) = sqrt(3^2+4^2) = sqrt(9+16) = sqrt(25) = 5.
Let an array have m(0,n)=m(n,0)=n*(n-1)/2 and m(n,n)=n*(n+1)/2. The first n+1 terms in row(n) are the numbers in the closed interval m(0,n) to m(n,n). The terms in column(n) are the same from m(n,0) to m(n,n). The first few antidiagonals are 0; 0,0; 1,1,1; 3,2,2,3; 6,4,3,4,6; 10,7,5,5,7,10. a(n) is the difference between the sum of the terms in the n+1 X n+1 matrices and those in the n X n matrices. - J. M. Bergot, Jul 05 2013 [The first five rows are: 0,0,1,3,6; 0,1,2,4,7; 1,2,3,5,8; 3,4,5,6,9; 6,7,8,9,10]

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Ron Knott, Pythagorean triangles and triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A024966, A008585, A008586, A001106, A022264, A024966, A033572, A144555, A158482, A158485, A195018, A195032, A195034, A195036.

[(2*n*(7*n+13)+(2*n-5)*(-1)^n+5)/16: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

With[{r = Range[50]}, Join[{0}, Accumulate[Riffle[3*r, 4*r]]]] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 3, 7, 13, 21}, 100] (* Paolo Xausa, Feb 09 2024 *)

From Bruno Berselli, Oct 13 2011: (Start)
G.f.: x*(3+4*x)/((1+x)^2*(1-x)^3).
a(n) = (1/2)*A004526(n+2)*A047335(n+1) = (2*n*(7*n+13) + (2*n-5)*(-1)^n+5)/16.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) - a(n-2) = A047355(n+1). (End)

A249674 a(n) = 30*n.

Original entry on oeis.org

0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 480, 510, 540, 570, 600, 630, 660, 690, 720, 750, 780, 810, 840, 870, 900, 930, 960, 990, 1020, 1050, 1080, 1110, 1140, 1170, 1200, 1230, 1260, 1290, 1320, 1350, 1380, 1410, 1440

Offset: 0

Kaylan Purisima, Nov 03 2014

Numbers divisible by 2, 3 and 5. - Robert Israel, Nov 19 2014

a(n) is the maximum score of a 10-pin n-frame bowling game and the maximum score of an n-pin 10-frame bowling game, given the rules: a strike is worth the number of pins in each frame plus the number of pins knocked down by the next two balls (except in the last frame), a spare is worth the number of pins in each frame plus the number of pins knocked down by the next ball (except in the last frame), and if a strike or spare is earned in the last frame then the player must continue to throw balls until they have thrown 3 balls in the last frame. - Iain Fox, Mar 02 2018

			a(7) = 7 * 30 = 210.

Iain Fox, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Bowling.
Wikipedia, Ten-pin bowling.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A008585, A008587, A008588, A008592, A008597, A050519, A169823.

[30*n : n in [0..50]]; // Wesley Ivan Hurt, Nov 18 2014

A249674:=n->30*n: seq(A249674(n), n=0..50); # Wesley Ivan Hurt, Nov 18 2014

30*Range[0, 59] (* Alonso del Arte, Nov 18 2014 *)

vector(100,n,30*(n-1)) \\ Derek Orr, Nov 18 2014

first(n) = Vec(30*x/(x-1)^2 + O(x^n), -n) \\ Iain Fox, Mar 02 2018

G.f.: 30*x/(x-1)^2; a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Nov 18 2014
a(n) = 2*A008597(n) = 3*A008592(n) = 5*A008588(n) = 6*A008587(n) = 10*A008585(n) = 15*A005843(n). - Omar E. Pol, Nov 24 2014
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 30*x*exp(x).
a(n) = A169823(n)/2. (End)

A195019 Multiples of 3 and of 4 interleaved: a(2n-1) = 3n, a(2n) = 4n.

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 12, 16, 15, 20, 18, 24, 21, 28, 24, 32, 27, 36, 30, 40, 33, 44, 36, 48, 39, 52, 42, 56, 45, 60, 48, 64, 51, 68, 54, 72, 57, 76, 60, 80, 63, 84, 66, 88, 69, 92, 72, 96, 75, 100, 78, 104, 81, 108, 84, 112, 87, 116, 90, 120, 93, 124, 96, 128

Offset: 1

Omar E. Pol, Sep 07 2011, Sep 12 2011

First differences of A195020.

a(n) is also the length of the n-th edge of a square spiral in which the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The spiral contains infinitely many Pythagorean triples in which the hypotenuses are the positives A008587. Zero together with partial sums give A195020; the vertices of the spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean triangles and triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008585, A008586, A026741, A195020-A195029, A195031, A195033, A195035.

[((n-3)*(-1)^n+7*n+3)/4: n in [1..60]]; // Vincenzo Librandi, Sep 12 2011

Table[((n-3)*(-1)^n + 7*n + 3)/4, {n,1,50}] (* G. C. Greubel, Aug 19 2017 *)

a(n)=(n+1)\2*(4-n%2)  \\ M. F. Hasler, Sep 08 2011

pair(3*n, 4*n).
a(2*n-1) = 3*n, a(2*n) = 4*n. - M. F. Hasler, Sep 08 2011
G.f.: x*(3+4*x) / ( (x-1)^2*(1+x)^2 ). - R. J. Mathar, Sep 09 2011
From Bruno Berselli, Sep 12 2011: (Start)
a(n) = ((n-3)*(-1)^n + 7*n + 3)/4.
a(n) + a(n+1) = A047355(n+2). (End)
E.g.f.: (1/4)*((3 + 7*x)*exp(x) - (3 + x)*exp(-x)). - G. C. Greubel, Aug 19 2017

A195140 Multiples of 5 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 5, 3, 10, 5, 15, 7, 20, 9, 25, 11, 30, 13, 35, 15, 40, 17, 45, 19, 50, 21, 55, 23, 60, 25, 65, 27, 70, 29, 75, 31, 80, 33, 85, 35, 90, 37, 95, 39, 100, 41, 105, 43, 110, 45, 115, 47, 120, 49, 125, 51, 130, 53, 135, 55, 140, 57, 145, 59, 150, 61, 155, 63

Offset: 0

Omar E. Pol, Sep 10 2011

This is 5*n/2 if n is even, n if n is odd.
Partial sums give the generalized enneagonal numbers A118277.
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized enneagonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

A008587 and A005408 interleaved.
Column 5 of A195151.
Cf. Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, this sequence, zero together with A165998, A195159, A195161, A195312.
Cf. A047336, A118277.

&cat[[5*n,2*n+1]: n in [0..31]]; // Bruno Berselli, Sep 27 2011

With[{nn=40},Riffle[5*Range[0,nn],Range[1,2nn+1,2]]] (* or *) LinearRecurrence[ {0,2,0,-1},{0,1,5,3},80] (* Harvey P. Dale, Dec 15 2014 *)

a(n)=(7+3*(-1)^n)*n/4 \\ Charles R Greathouse IV, Oct 07 2015

a(2n) = 5n, a(2n+1) = 2n+1.
G.f.: x*(1+5*x+x^2) / ((x-1)^2*(x+1)^2). - Alois P. Heinz, Sep 26 2011
From Bruno Berselli, Sep 27 2011: (Start)
a(n) = (7+3*(-1)^n)*n/4.
a(n) = -a(-n) = a(n-2)*n/(n-2) = 2*a(n-2)-a(n-4).
a(n) + a(n-1) = A047336(n). (End)
Multiplicative with a(2^e) = 5*2^(e-1), a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 3/2^s). - Amiram Eldar, Oct 25 2023

Corrected and edited by Alois P. Heinz, Sep 25 2011

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 3, 1, 1, 0, 0, 3, 2, 1, 0, 5, 2, 3, 3, 1, 0, 0, 5, 4, 3, 4, 1, 0, 7, 3, 5, 6, 3, 5, 1, 0, 0, 7, 6, 5, 8, 3, 6, 1, 0, 9, 4, 7, 9, 5, 10, 3, 7, 1, 0, 0, 9, 8, 7, 12, 5, 12, 3, 8, 1, 0, 11, 5, 9, 12, 7, 15, 5, 14, 3, 9, 1, 0, 0, 11, 10, 9, 16, 7

Offset: 0

Omar E. Pol, Sep 14 2011

Also square array T(n,k) read by antidiagonals in which column k lists the multiples of k and the odd numbers interleaved, n>=0, k>=0. Also square array T(n,k) read by antidiagonals in which if n is even then row n lists the multiples of (n/2), otherwise if n is odd then row n lists a constant sequence: the all n's sequence. Partial sums of the numbers of column k give the column k of A195152. Note that if k >= 1 then partial sums of the numbers of the column k give the generalized m-gonal numbers, where m = k + 4.

All columns are multiplicative. - Andrew Howroyd, Jul 23 2018

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,...
.  3,   3,   3,   3,   3,   3,   3,   3,   3,   3,...
.  0,   2,   4,   6,   8,  10,  12,  14,  16,  18,...
.  5,   5,   5,   5,   5,   5,   5,   5,   5,   5,...
.  0,   3,   6,   9,  12,  15,  18,  21,  24,  27,...
.  7,   7,   7,   7,   7,   7,   7,   7,   7,   7,...
.  0,   4,   8,  12,  16,  20,  24,  28,  32,  36,...
.  9,   9,   9,   9,   9,   9,   9,   9,   9,   9,...
.  0,   5,  10,  15,  20,  25,  30,  35,  40,  45,...
...

Rows: A000004, A000012, A001477, A010701, A005843, A010716, A008585, A010727, A008586, A010734, A008587.

Columns k: A026741 (k=1), A001477 (k=2), zero together with A080512 (k=3), A022998 (k=4), A195140 (k=5), zero together with A165998 (k=6), A195159 (k=7), A195161 (k=8), A195312 k=(9), A195817 (k=10), A317311 (k=11), A317312 (k=12), A317313 (k=13), A317314 k=(14), A317315 (k=15), A317316 (k=16), A317317 (k=17), A317318 (k=18), A317319 k=(19), A317320 (k=20), A317321 (k=21), A317322 (k=22), A317323 (k=23), A317324 k=(24), A317325 (k=25), A317326 (k=26).

T(n,k) = n*((k-2)*(-1)^n+k+2)/4 \\ Andrew Howroyd, Jul 23 2018

A063260 Sextinomial (also called hexanomial) coefficient array.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 126, 205, 305, 420, 540, 651, 735, 780

Offset: 0

Wolfdieter Lang, Jul 24 2001

The sequence of step width of this staircase array is [1,5,5,...], hence the degree sequence for the row polynomials is [0,5,10,15,...]=A008587.
The column sequences (without leading zeros) are for k=0..5 those of the lower triangular array A007318 (Pascal) and for k=6..9: A062989, A063262-4. Row sums give A000400 (powers of 6). Central coefficients give A063419; see also A018901.
This can be used to calculate the number of occurrences of a given roll of n six-sided dice, where k is the index: k=0 being the lowest possible roll (i.e., n) and n*6 being the highest roll.

			The irregular table T(n, k) begins:
n\k 0 1 2  3  4  5  6  7  8  9 10 11 12 13 14 15
1:  1
2:  1 1 1  1  1  1
3:  1 2 3  4  5  6  5  4  3  2  1
4:  1 3 6 10 15 21 25 27 27 25 21 15 10  6  3  1
...reformatted - _Wolfdieter Lang_, Oct 31 2015

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77,78.

T. D. Noe, Rows n = 0..25, flattened
Steven R. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle, arXiv:0802.2654 [math.NT], 2008.
Iain G. Johnston, Optimal strategies in the Fighting Fantasy gaming system: influencing stochastic dynamics by gambling with limited resource, arXiv:2002.10172 [cs.AI], 2020.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 4.

The q-nomial arrays for q=2..5 are: A007318 (Pascal), A027907, A008287, A035343 and for q=7: A063265, A171890, A213652, A213651.
Columns for k=0..9 (with some shifts) are: A000012, A000027, A000217, A000292, A000332, A000389, A062989, A063262, A063263, A063264.
Cf. A000400, A008587, A018901, A063261, A063419.

#Define the r-nomial coefficients for r = 1, 2, 3, ...
rnomial := (r,n,k) -> add((-1)^i*binomial(n,i)*binomial(n+k-1-r*i,n-1), i = 0..floor(k/r)):
#Display the 6-nomials as a table
r := 6:  rows := 10:
for n from 0 to rows do
seq(rnomial(r,n,k), k = 0..(r-1)*n)
end do;
# Peter Bala, Sep 07 2013

Flatten[Table[CoefficientList[(1 + x + x^2 + x^3 + x^4 + x^5)^n, x], {n, 0, 25}]] (* T. D. Noe, Apr 04 2011 *)

concat(vector(5,k,Vec(sum(j=0,5,x^j)^k)))  \\ M. F. Hasler, Jun 17 2012

G.f. for row n: (Sum_{j=0..5} x^j)^n.
G.f. for column k: (x^(ceiling(k/5)))*N6(k, x)/(1-x)^(k+1) with the row polynomials from the staircase array A063261(k, m) and with N6(6,x) = 5 - 10*x + 10*x^2 - 5*x^3 + x^4.
T(n, k) = 0 if n=-1 or k<0 or k >= 5*n + 1; T(0, 0)=1; T(n, k) = Sum_{j=0..5} T(n-1, k-j) else.
T(n, k) = Sum_{i = 0..floor(k/6)} (-1)^i*binomial(n,i)*binomial(n+k-1-6*i,n-1) for n >= 0 and 0 <= k <= 5*n. - Peter Bala, Sep 07 2013
T(n, k) = Sum_{i = max(0,ceiling((k-2*n)/3)).. min(n,k/3)} binomial(n,i)*trinomial(n,k-3*i) for n >= 0 and 0 <= k <= 5*n. - Matthew Monaghan, Sep 30 2015

More terms and corrected recurrence from Nicholas M. Makin (NickDMax(AT)yahoo.com), Sep 13 2002

A123865 a(n) = n^4 - 1.

Original entry on oeis.org

0, 15, 80, 255, 624, 1295, 2400, 4095, 6560, 9999, 14640, 20735, 28560, 38415, 50624, 65535, 83520, 104975, 130320, 159999, 194480, 234255, 279840, 331775, 390624, 456975, 531440, 614655, 707280, 809999, 923520, 1048575, 1185920, 1336335

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 5 = 0 iff n mod 5 > 0: a(A008587(n)) = 4; a(A047201(n)) = 0; a(n) mod 5 = 4*(1-A079998(n)).

A129292(n) = number of divisors of a(n) that are not greater than n. - Reinhard Zumkeller, Apr 09 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000583, A005563, A024002, A123866, A123867, A123868, A219086.

List([1..40], n-> n^4 -1); # G. C. Greubel, Aug 08 2019

[n^4 - 1: n in [1..40]]; // Vincenzo Librandi, May 01 2011

seq(n^4 -1, n=1..40); # G. C. Greubel, Aug 08 2019

Table[n^4-1,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

vector(40, n, n^4 -1) \\ G. C. Greubel, Aug 08 2019

[n^4 -1 for n in (1..40)] # G. C. Greubel, Aug 08 2019

G.f.: x^2*(15 + 5*x + 5*x^2 - x^3)/(1-x)^5. - Colin Barker, Jan 10 2012
-4*a(n+1) = -4*n*(n+2)*(n^2+2*n+2) = (n+n*i)*(n+2+n*i)*(n+(n+2)*i)*(n+2+(n+2)*i), where i is the imaginary unit. - Jon Perry, Feb 05 2014
From Vaclav Kotesovec, Feb 14 2015: (Start)
Sum_{n>=2} 1/a(n) = 7/8 - Pi*coth(Pi)/4 = A256919.
Sum_{n>=2} (-1)^n / a(n) = 1/8 - Pi/(4*sinh(Pi)). (End)
a(n) = A005563(A005563(n)). - Bruno Berselli, May 28 2015
E.g.f.: 1 + (-1 + x + 7*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Aug 08 2019
Product_{n>=2} (1 + 1/a(n)) = 4*Pi*csch(Pi). - Amiram Eldar, Jan 20 2021

A022267 a(n) = n(9n + 1)/2.

Original entry on oeis.org

0, 5, 19, 42, 74, 115, 165, 224, 292, 369, 455, 550, 654, 767, 889, 1020, 1160, 1309, 1467, 1634, 1810, 1995, 2189, 2392, 2604, 2825, 3055, 3294, 3542, 3799, 4065, 4340, 4624, 4917, 5219, 5530, 5850, 6179

Offset: 0

N. J. A. Sloane

From Floor van Lamoen, Jul 21 2001: (Start)
Write 0, 1, 2, 3, 4, ... in a triangular spiral; then a(n) is the sequence found by reading the line from 0 in the direction 0, 5, ... . The spiral begins:
.
            15
            / \
          16  14
          /     \
        17   3  13
        /   / \   \
      18   4   2  12
      /   /     \   \
    19   5   0---1  11
    /   /             \
  20   6---7---8---9--10
.
(End)
a(n) is the sum of n consecutive integers starting from 4*n+1: (5), (9+10), (13+14+15), ... - Klaus Purath, Jul 07 2020
a(n) with n>0 are the numbers with the periodic length 3 in the Bulgarian and Mancala solitaire. - Paul Weisenhorn, Jan 29 2022

Lei Zhou, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
Leo Tavares, Illustration: X Triangles
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A051682, A110449, A235332.
Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A033429, A022268, A049452, A186030, A135703, A152734, A139273.
Cf. similar sequences listed in A254963.
Cf. similar sequences listed in A022289.
Cf. A060544, A016813.

seq(binomial(9*n+1,2)/9, n=0..37); # Zerinvary Lajos, Jan 21 2007

Table[ n (9 n + 1)/2, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 19}, 40] (* Harvey P. Dale, Jul 01 2013 *)

vector(100,n,(n-1)*(9*n-8)/2) \\ Derek Orr, Feb 06 2015

a(n) = A110449(n, 4) for n>3.
From Bruno Berselli, Feb 11 2011: (Start)
G.f.: x*(5 + 4*x)/(1 - x)^3.
a(n) = 4*A000217(n) + A000566(n). (End)
a(n) = 9*n + a(n-1) - 4 with n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
a(n) = A218470(9*n+4). - Philippe Deléham, Mar 27 2013
a(n) = A000217(5*n) - A000217(4*n). - Bruno Berselli, Oct 13 2016
E.g.f.: (1/2)*(9*x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017
a(n) = A060544(n+1) - A016813(n). - Leo Tavares, Mar 20 2022

A009976 Powers of 32.

Original entry on oeis.org

1, 32, 1024, 32768, 1048576, 33554432, 1073741824, 34359738368, 1099511627776, 35184372088832, 1125899906842624, 36028797018963968, 1152921504606846976, 36893488147419103232, 1180591620717411303424, 37778931862957161709568, 1208925819614629174706176, 38685626227668133590597632

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 32), L(1, 32), P(1, 32), T(1, 32). Essentially same as Pisot sequences E(32, 1024), L(32, 1024), P(32, 1024), T(32, 1024). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 32-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

			a(6) = 32^6 = 1073741824.

T. D. Noe, Table of n, a(n) for n = 0..100
Philip DeOrsey, Stephen G. Hartke, and Jason Williford, A Classification of Hyperfocused 12-Arcs, arXiv:2105.08300 [math.CO], 2021.
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (32).

Subsequence of A000079.

Cf. A001025, A008587, A008776, A089357.

[32^n: n in [0..100]]; // Vincenzo Librandi, Nov 21 2010

A009976:=n->32^n; seq(A009976(k), k=0..50); # Wesley Ivan Hurt, Nov 04 2013

Table[32^n, {n,0,50}] (* Wesley Ivan Hurt, Nov 04 2013 *)

A009976(n):=32^n$
makelist(A009976(n),n,0,30); /* Martin Ettl, Nov 07 2012 */

a(n)=32^n \\ Charles R Greathouse IV, Sep 24 2015

def A009976(n): return 1<<5*n # Chai Wah Wu, Nov 10 2022

[lucas_number1(n,32,0) for n in range(1, 17)] # Zerinvary Lajos, Nov 07 2009

G.f.: 1/(1-32*x). - Philippe Deléham, Nov 24 2008
a(n) = 32^n; a(n) = 32*a(n-1) for n > 0, a(0)=1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(32*x).
a(n) = 2^A008587(n) = A000079(n)*A001025(n) = A089357(n)/A000079(n). (End)

cp's OEIS Frontend

A011558 Expansion of (x + x^3)/(1 + x + ... + x^4) mod 2.

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A195020 Vertex number of a square spiral in which the length of the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The edges of the spiral have length A195019.

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A249674 a(n) = 30*n.

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A195019 Multiples of 3 and of 4 interleaved: a(2*n-1) = 3*n, a(2*n) = 4*n.

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A195140 Multiples of 5 and odd numbers interleaved.

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A195151 Square array read by antidiagonals upwards: T(n,k) = n*((k-2)*(-1)^n+k+2)/4, n >= 0, k >= 0.

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A063260 Sextinomial (also called hexanomial) coefficient array.

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A195019 Multiples of 3 and of 4 interleaved: a(2n-1) = 3n, a(2n) = 4n.

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

A022267 a(n) = n(9n + 1)/2.