A194268 -id:A194268 - cp's OEIS frontend

A014634 a(n) = (2n+1)(4*n+1).

1, 15, 45, 91, 153, 231, 325, 435, 561, 703, 861, 1035, 1225, 1431, 1653, 1891, 2145, 2415, 2701, 3003, 3321, 3655, 4005, 4371, 4753, 5151, 5565, 5995, 6441, 6903, 7381, 7875, 8385, 8911, 9453, 10011, 10585, 11175, 11781, 12403, 13041, 13695, 14365, 15051

Offset: 0

Mohammad K. Azarian, N. J. A. Sloane

Odd hexagonal numbers. Bisection of A000384. - Omar E. Pol, Apr 06 2008
Sequence found by reading the line from 1, in the direction 1, 15, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 03 2011
a(n) is also the sum of natural numbers which can be placed in a center box and expanded ones on 4 arms on N, S, E, W or NE, NW, SW, SE directions. See illustration in links. - Kival Ngaokrajang, Jul 08 2014

G. C. Greubel, Table of n, a(n) for n = 0..5000
Kival Ngaokrajang, Illustration of initial terms.
Leo Tavares, Illustration: Dual Square Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000384, A005408, A016813, A033954, A157870, A194268.

Cf. A003154, A000290.

[(2*n+1)*(4*n+1) : n in [0..50]]; // Wesley Ivan Hurt, Jul 09 2014

A014634:=n->(2*n+1)*(4*n+1); seq(A014634(k), k=0..100); # Wesley Ivan Hurt, Nov 04 2013

lst={};Do[a=(2*n+1)*(4*n+1);AppendTo[lst,a],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 10 2009 *)
Table[(2 n + 1) (4 n + 1), {n, 0, 50}] (* Wesley Ivan Hurt, Jul 09 2014 *)
LinearRecurrence[{3,-3,1},{1,15,45},50] (* Harvey P. Dale, Aug 30 2021 *)

a(n)=(2*n+1)*(4*n+1) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A157870(n)/2. - Vladimir Joseph Stephan Orlovsky, Mar 10 2009
a(n) = a(n-1) + 16*n-2 (with a(0)=1). - Vincenzo Librandi, Nov 20 2010
G.f.: (1+12*x+3*x^2)/(1-x)^3. - Colin Barker, Jan 08 2012
a(n) = A005408(n) * A016813(n). - Omar E. Pol, Nov 05 2013
a(n) = 2*A033954(n) + 1 = A194268(n) - n. - Wesley Ivan Hurt, Jul 14 2014
E.g.f.: (8*x^2 +14*x + 1)*exp(x). - G. C. Greubel, Jul 18 2017
From Amiram Eldar, Feb 28 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi/4 + log(2)/2.
Sum_{n>=0} (-1)^n/a(n) = Pi*(sqrt(2)-1)/4 + log(sqrt(2)+1)/sqrt(2). (End)
a(n) = A003154(n+1) + 2*A000290(n). - Leo Tavares, Mar 26 2022

More terms from Wesley Ivan Hurt, Jul 09 2014

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

Original entry on oeis.org

2, 7, 18, 31, 50, 71, 98, 127, 162, 199, 242, 287, 338, 391, 450, 511, 578, 647, 722, 799, 882, 967, 1058, 1151, 1250, 1351, 1458, 1567, 1682, 1799, 1922, 2047, 2178, 2311, 2450, 2591, 2738, 2887, 3042, 3199, 3362, 3527, 3698, 3871, 4050, 4231, 4418, 4607, 4802

Offset: 0

Bruno Berselli, Sep 21 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the numbers in increasing order on the vertical line containing 2 of the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

Also A077591 (without first term) and A157914 interleaved.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195605.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A077591, A157914.
Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).
Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].

[(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];

CoefficientList[Series[(2 + 3 x + 4 x^2 - x^3) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2,0,-2,1},{2,7,18,31},50] (* Harvey P. Dale, Jan 21 2017 *)

for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));

G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).
a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = A047524(A000982(n+1)).
Sum_{n>=0} 1/a(n) = 1/2 + Pi^2/16 - cot(Pi/(2*sqrt(2)))*Pi/(4*sqrt(2)). - Amiram Eldar, Mar 06 2023

A194431 a(n) = 8n^2 - 6n - 1.

Original entry on oeis.org

1, 19, 53, 103, 169, 251, 349, 463, 593, 739, 901, 1079, 1273, 1483, 1709, 1951, 2209, 2483, 2773, 3079, 3401, 3739, 4093, 4463, 4849, 5251, 5669, 6103, 6553, 7019, 7501, 7999, 8513, 9043, 9589, 10151, 10729, 11323, 11933, 12559, 13201, 13859, 14533, 15223, 15929

Offset: 1

Omar E. Pol, Sep 05 2011

Sequence found by reading the line from 1, in the direction 1, 19, ..., in the square spiral whose vertices are the triangular numbers A000217.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A014634, A051870, A069129, A139098, A194268.

[8*n^2 - 6*n - 1: n in [1..50]]; // Vincenzo Librandi, Sep 07 2011

Table[8n^2-6n-1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,19,53},50] (* Harvey P. Dale, May 29 2021 *)

a(n)=8*n^2-6*n-1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(-1 - 16*x + x^2)/(x-1)^3. - R. J. Mathar, Sep 06 2011
From Elmo R. Oliveira, Jun 04 2025: (Start)
E.g.f.: 1 + (-1 + 2*x + 8*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

Original entry on oeis.org

1, 2, 3, 23, 59, 111, 179, 263, 363, 479, 611, 759, 923, 1103, 1299, 1511, 1739, 1983, 2243, 2519, 2811, 3119, 3443, 3783, 4139, 4511, 4899, 5303, 5723, 6159, 6611, 7079, 7563, 8063, 8579, 9111, 9659, 10223, 10803, 11399, 12011, 12639, 13283, 13943

Offset: 0

Bruno Berselli, Sep 13 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the line 1, 2, 3, 23,.. in the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

This is a subsequence of A110326 (without signs) and A047838 (apart from the second term, 2).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195241.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217.

Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195605 [incomplete list].

m:=44; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+19*x^3-3*x^4)/(1-x)^3));

CoefficientList[Series[(1 - x + 19 x^3 - 3 x^4)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{3,-3,1},{1,2,3,23,59},50] (* Harvey P. Dale, Dec 04 2022 *)

makelist(coeff(taylor((1-x+19*x^3-3*x^4)/(1-x)^3, x, 0, n), x, n), n, 0, 43);

Vec((1-x+19*x^3-3*x^4)/(1-x)^3+O(x^44))

G.f.: (1-x+19*x^3-3*x^4)/(1-x)^3.

a(n) = 8*n^2-20*n+11 for n>1; a(0)=1, a(1)=2.

cp's OEIS Frontend

A014634 a(n) = (2*n+1)*(4*n+1).

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A195605 a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.

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A194431 a(n) = 8*n^2 - 6*n - 1.

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A195241 Expansion of (1-x+19*x^3-3*x^4)/(1-x)^3.

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Formula

A014634 a(n) = (2n+1)(4*n+1).

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

A194431 a(n) = 8n^2 - 6n - 1.

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.