A153125 -id:A153125 - cp's OEIS frontend

A153126 Sums of rows of the triangle in A153125.

1, 6, 18, 33, 55, 80, 112, 147, 189, 234, 286, 341, 403, 468, 540, 615, 697, 782, 874, 969, 1071, 1176, 1288, 1403, 1525, 1650, 1782, 1917, 2059, 2204, 2356, 2511, 2673, 2838, 3010, 3185, 3367, 3552, 3744, 3939, 4141, 4346, 4558, 4773, 4995, 5220, 5452

Offset: 0

Reinhard Zumkeller, Dec 20 2008

Sequence found by reading the line from 1, in the direction 1, 6,..., and the same line from 1, in the direction 1, 18,..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. Line perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000566, A001622, A033571, A153125, A153127.

LinearRecurrence[{2,0,-2,1},{1,6,18,33},50] (* Harvey P. Dale, Apr 13 2014 *)

a(n)=n*(5*n+7)/2 + 1 - n%2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = n*(5*n+7)/2 + 1 - n mod 2.
a(n) = Sum_{k=1..n+1} A153125(n+1,k).
a(2*n) = A033571(n); a(2*n+1) = A153127(n).
a(n) = A000566(n+1) - n mod 2.
From Colin Barker, Jul 07 2012: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+4*x+6*x^2-x^3)/((1-x)^3*(1+x)). (End)
Sum_{n>=0} 1/a(n) = 5/7 + 2*sqrt(1+2/sqrt(5))*Pi/21 + 2*sqrt(5)*log(phi)/21 + 5*log(5)/21 - 8*log(2)/21, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 23 2022

A016861 a(n) = 5*n + 1.

Original entry on oeis.org

1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, 81, 86, 91, 96, 101, 106, 111, 116, 121, 126, 131, 136, 141, 146, 151, 156, 161, 166, 171, 176, 181, 186, 191, 196, 201, 206, 211, 216, 221, 226, 231, 236, 241, 246, 251, 256, 261, 266, 271, 276, 281

Offset: 0

N. J. A. Sloane, Dec 11 1996

Numbers ending in 1 or 6.
Apart from initial terms, same as 5n-14.
Complement of A047203; A027445(a(n)) mod 10 = 4. - Reinhard Zumkeller, Oct 23 2006
Campbell reference shows: "A graph on n vertices with at least 4n-9 edges is intrinsically linked. A graph on n vertices with at least 5n-14 edges is intrinsically knotted." - Jonathan Vos Post, Jan 18 2007
Central terms of the triangle in A153125: a(n) = A153125(2*n+1, n+1). - Reinhard Zumkeller, Dec 20 2008
For n > 2, also the number of (not necessarily maximal) cliques in the n-Moebius ladder graph. - Eric W. Weisstein, Nov 29 2017
For n > 3, also the number of (not necessarily maximal) cliques in the n-prism graph. - Eric W. Weisstein, Nov 29 2017
For n >= 1, a(n) is the size of any hexagonal chain graph with n cells. - Christian Barrientos, Sarah Minion, Mar 07 2018
For n >= 1, a(n) is the number of possible outcomes of the summation when using n dice. - Bram Kole, Dec 24 2018
Numbers congruent to 1 (mod 5). - Muniru A Asiru, Jan 01 2019
Numbers k such that the k-th Fibonacci number, A000045(k), and the k-th Lucas number, A000032(k), end with the same decimal digit. - Amiram Eldar, Apr 15 2023

Ivan Panchenko, Table of n, a(n) for n = 0..1000
J. Campbell, T. W. Mattman, R. Ottman, J. Pyzer, M. Rodrigues and S. Williams, Intrinsic knotting and linking of almost complete graphs, arXiv:math/0701422 [math.GT], Jan 15 2007.
Tanya Khovanova, Recursive Sequences.
Eric Weisstein's World of Mathematics, Clique.
Eric Weisstein's World of Mathematics, Moebius Ladder.
Eric Weisstein's World of Mathematics, Prism Graph.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093562 ((5, 1) Pascal, column m=1).
Cf. A000566 (partial sums).
Cf. A000032, A000045, A001622, A131843, A342819.

a:=List([0..60],n->5*n+1);; Print(a); # Muniru A Asiru, Jan 01 2019

a016861 = (+ 1) . (* 5)
a016861_list = [1, 6 ..]  -- Reinhard Zumkeller, Jun 16 2013

A016861:=n->5*n+1; seq(A016861(n), n=0..100); # Wesley Ivan Hurt, May 03 2014

Range[1, 500, 5] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
LinearRecurrence[{2, -1}, {6, 11}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(1 + 4 x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=5*n+1 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: (1+4*x)/(1-x)^2.
Row sums of triangle A131843. - Gary W. Adamson, Jul 21 2007
a(n) = 2*a(n-1) - a(n-2) with a(0)=1, a(1)=6. - Vincenzo Librandi, Aug 01 2010
a(n) = A017293(n)/2 = A008587(n)+1. - Wesley Ivan Hurt, May 03 2014
E.g.f.: exp(x)*(1 + 5*x). - Stefano Spezia, Mar 23 2021
Sum_{n>=0} (-1)^n/a(n) = sqrt(2+2/sqrt(5))*Pi/10 + log(phi)/sqrt(5) + log(2)/5, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

More terms from Reinhard Zumkeller, Oct 23 2006

A047461 Numbers that are congruent to {1, 4} mod 8.

Original entry on oeis.org

1, 4, 9, 12, 17, 20, 25, 28, 33, 36, 41, 44, 49, 52, 57, 60, 65, 68, 73, 76, 81, 84, 89, 92, 97, 100, 105, 108, 113, 116, 121, 124, 129, 132, 137, 140, 145, 148, 153, 156, 161, 164, 169, 172, 177, 180, 185, 188, 193, 196, 201, 204, 209, 212, 217, 220, 225, 228, 233

Offset: 1

N. J. A. Sloane

Maximal number of squares that can be covered by a queen on an n X n chessboard. - Reinhard Zumkeller, Dec 15 2008

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004526, A004770, A010703, A017077, A017113, A047398, A047452.

Cf. A047470, A047524, A047535, A047615, A047617, A153125, A342819.

Filtered([1..250], n->n mod 8=1 or n mod 8 =4); # Muniru A Asiru, Jul 23 2018

[4*n-3 - ((n+1) mod 2): n in [1..70]]; // G. C. Greubel, Mar 15 2024

seq(coeff(series(factorial(n)*((8-exp(-x)+(8*x-7)*exp(x))/2), x,n+1),x,n),n=1..60); # Muniru A Asiru, Jul 23 2018

Flatten[(#+{1,4})&/@(8Range[0,30])] (* or *) LinearRecurrence[ {1,1,-1},{1,4,9},60] (* Harvey P. Dale, Jun 18 2013 *)
CoefficientList[ Series[(4x^2 + 3x + 1)/((x + 1) (x - 1)^2), {x, 0, 58}], x] (* Robert G. Wilson v, Jul 24 2018 *)

makelist(4*n -(7 + (-1)^n)/2, n, 1, 100); /* Franck Maminirina Ramaharo, Jul 22 2018 */

def A047461(n): return (n-1<<2)|(n&1) # Chai Wah Wu, Mar 30 2024

[4*n-3 - ((n+1)%2) for n in range(1,71)] # G. C. Greubel, Mar 15 2024

From R. J. Mathar, Oct 29 2008: (Start)
G.f.: x*(1+3*x+4*x^2)/((1+x)*(1-x)^2).
a(n) = a(n-2) + 8.
a(n) + a(n+1) = A004770(n).
a(n+1) - a(n) = A010703(n). (End)
a(n) = 8*floor((n-1)/2) + 4 - 3*(n mod 2). - Reinhard Zumkeller, Dec 15 2008
a(n) = A153125(n,n). - Reinhard Zumkeller, Dec 20 2008
a(n) = 8*n - a(n-1) - 11 (with a(1)=1). - Vincenzo Librandi, Aug 06 2010
a(n) = 4*n - (7 + (-1)^n)/2. - Arkadiusz Wesolowski, Sep 18 2012
a(1)=1, a(2)=4, a(3)=9, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Jun 18 2013
a(n) = 1 + A004526(n)*3 + A004526(n-1)*5. - Gregory R. Bryant, Apr 16 2014
From Franck Maminirina Ramaharo, Jul 22 2018: (Start)
a(n) = A047470(n) + 1.
E.g.f.: (8 - exp(-x) + (8*x - 7)*exp(x))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 + log(2)/4 + sqrt(2)*arccoth(sqrt(2))/8. - Amiram Eldar, Dec 11 2021

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A153126 Sums of rows of the triangle in A153125.

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A016861 a(n) = 5*n + 1.

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A047461 Numbers that are congruent to {1, 4} mod 8.

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