A047622 -id:A047622 - cp's OEIS frontend

A234902 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3) after n rotations.

2, 9, 13, 17, 24, 26, 33, 37, 41, 48, 50, 57, 61, 65, 72, 74, 81, 85, 89, 96, 98, 105, 109, 113, 120, 122, 129, 133, 137, 144, 146, 153, 157, 161, 168, 170, 177, 181, 185, 192, 194, 201, 205, 209, 216, 218, 225, 229, 233, 240, 242, 249, 253, 257

Offset: 1

Kival Ngaokrajang, Jan 01 2014

Let points 1, 2 & 3 be placed on a straight line at intervals of 1 unit. At point 1, make a half unit circle; then, at point 2, make another half circle and maintain continuity of circumferences. Continue using this procedure at points 3, 1, 2 and so on. The form of the spiral is a non-expanded loop.
The sequence will be A047622 if the second radius = 2; if the second radius = 0, the sequence is a(n).
See illustration in links.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A014105*Pi (total spiral length, 2 inline center points).

LinearRecurrence[{1,0,0,0,1,-1},{2,9,13,17,24,26},60] (* Harvey P. Dale, May 21 2021 *)

G.f.: x*(7*x^4 + 4*x^3 + 4*x^2 + 7*x + 2)/((1-x)*(1-x^5)). - Ralf Stephan, Jan 20 2014

A047408 Numbers that are congruent to {1, 4, 6} mod 8.

Original entry on oeis.org

1, 4, 6, 9, 12, 14, 17, 20, 22, 25, 28, 30, 33, 36, 38, 41, 44, 46, 49, 52, 54, 57, 60, 62, 65, 68, 70, 73, 76, 78, 81, 84, 86, 89, 92, 94, 97, 100, 102, 105, 108, 110, 113, 116, 118, 121, 124, 126, 129, 132, 134, 137, 140, 142, 145, 148, 150, 153, 156, 158

Offset: 1

N. J. A. Sloane

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

Cf. A047622.

[n : n in [0..150] | n mod 8 in [1, 4, 6]]; // Wesley Ivan Hurt, Jun 10 2016

A047408:=n->3*n-floor(n/3)-2; seq(A047408(k), k=1..100); # Wesley Ivan Hurt, Nov 07 2013

Table[3n-Floor[n/3]-2, {n, 100}] (* Wesley Ivan Hurt, Nov 07 2013 *)

G.f.: x*(1+3*x+2*x^2+2*x^3)/((1+x+x^2)*(x-1)^2). - R. J. Mathar, Dec 05 2011
a(n) = 3n - 2 - floor(n/3). - Wesley Ivan Hurt, Nov 07 2013
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = (24*n-15-3*cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-2, a(3k-1) = 8k-4, a(3k-2) = 8k-7. (End)
E.g.f.: 2 + exp(x)*(8*x - 5)/3 - exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Mar 30 2023

A236535 a(n)*Pi is the total length of irregular spiral (center points: 2, 3, 1; pattern 1) after n rotations.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 42, 45, 48, 50, 53, 56, 58, 61, 64, 66, 69, 72, 74, 77, 80, 82, 85, 88, 90, 93, 96, 98, 101, 104, 106, 109, 112, 114, 117, 120, 122, 125, 128, 130, 133, 136, 138, 141, 144, 146, 149, 152, 154, 157, 160, 162, 165, 168, 170, 173, 176, 178, 181, 184, 186, 189

Offset: 1

Kival Ngaokrajang, Jan 28 2014

nonn

Let points 2, 3, & 1 be placed on a straight line at intervals of 1 unit. At point 1 make a half unit circle then at point 2 make another half circle; by selecting radius point on the left hand side of point 1 (pattern 1); at point 3 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 1, 2, 3, ... and so on.
Conjecture: All forms of 3 center points are non-expanded loops.
There are other sets of center points that give the same sequence, e.g.: [2,3,1,4]; [3,2,4,1]; [3,2,4,1,5]; [2,3,1,4,5,7,6]; [2,3,1,7,4,6,5]; [3,4,2,5,1,6,7]; [4,3,5,6,2,7,1]; [4,5,3,2,1,6,7]; [5,4,6,3,2,7,1].
Also, there are some similar patterns that give difference sequences, e.g.:
   A047622: [1,2,7,3,4,6,5]; [1,2,7,6,3,5,4]...
   A047399: [1,2,7,3,6,4,5]; [1,2,7,6,5,3,4]...
   A047395: [2,3,1,4 7,5,6]; [2,3,1,7,6,4,5]...
   A047464: [4,5,3,6,2,7,1]; [1,8,2,7,3,6,4,5];
            [9,1,8,2,7,3,6,4,5].
See illustration in links.
Appears to be basically a duplicate of A047618. - R. J. Mathar, Feb 03 2014

Kival Ngaokrajang, Illustration of irregular spirals (Center points 3..9)

Cf. A014105 (2 center points); A234902, A234903, A234904 (3 center points); A235088, A235089 (4 center points); A236326, A236327 (5 center points).

Conjecture from Colin Barker, Jul 12 2014: (Start)
a(n) = a(n-1)+a(n-3)-a(n-4).
G.f.: x*(3*x^2+3*x+2) / ((x-1)^2*(x^2+x+1)). (End)

A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k(4k + 1)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 3, 1, 3, 3, 2, 6, 3, 4, 8, 4, 9, 9, 6, 15, 10, 12, 20, 12, 22, 23, 18, 35, 26, 30, 46, 32, 51, 54, 45, 76, 62, 71, 99, 76, 111, 117, 104, 160, 136, 154, 205, 167, 230, 244, 223, 319, 286, 319, 406, 349, 456, 484, 458, 619, 570, 632, 779, 695

Offset: 0

Ilya Gutkovskiy, May 24 2019

nonn

Number of partitions of n into parts congruent to {0, 3, 5} mod 8.

Convolution inverse of A244465.

			For n=23 the a(23)=6 solutions are 3+3+3+3+3+3+5, 3+3+3+3+3+8, 3+3+3+3+11, 3+5+5+5+5, 5+5+5+8, and 5+5+13.

Ludovic Schwob, Table of n, a(n) for n = 0..10000

Cf. A000009, A000041, A006950, A007742, A047622, A074378, A195850, A244465, A308400.

nmax = 68; CoefficientList[Series[1/Sum[(-x)^(k (4 k + 1)), {k, -nmax, nmax}], {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[Product[1/((1 - x^(8 k - 5)) (1 - x^(8 k - 3)) (1 - x^(8 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 68; CoefficientList[Series[Sum[PartitionsP[k] (-x)^k, {k, 0, nmax}]/Sum[PartitionsQ[2 k] (-x)^k, {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: 1 / Sum_{k>=0} (-x)^A074378(k).
G.f.: Product_{k>=1} 1 / ((1 - x^(8*k - 5)) * (1 - x^(8*k - 3)) * (1 - x^(8*k))).
G.f.: ( Sum_{k>=0} A000041(k)*(-x)^k ) / ( Sum_{k>=0} A000009(2*k)*(-x)^k ).
a(n) ~ sqrt(sqrt(2) - 1) * exp(sqrt(n)*Pi/2) / (2^(9/4)*n). - Vaclav Kotesovec, May 25 2019
a(n) = a(n-3) + a(n-5) - a(n-14) - a(n-18) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 5, 14, 18, ... is the sequence A074378. - Ludovic Schwob, Aug 04 2021

A063285 Dimension of the space of weight n cuspidal newforms for Gamma_1( 12 ).

Original entry on oeis.org

-1, 0, 3, 5, 5, 8, 8, 14, 11, 17, 13, 22, 16, 26, 19, 31, 21, 34, 24, 40, 27, 43, 29, 48, 32, 52, 35, 57, 37, 60, 40, 66, 43, 69, 45, 74, 48, 78, 51, 83, 53, 86, 56, 92, 59, 95, 61, 100, 64, 104, 67, 109, 69, 112, 72, 118, 75, 121, 77, 126, 80, 130

Offset: 2

N. J. A. Sloane, Jul 14 2001

sign

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_1(N))
William A. Stein, The modular forms database
Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 2, -1, 0, 1, -1).

Cf. A047622 (bisection).

g.f.: -x^2*(x^8-5*x^7+x^6-5*x^5-2*x^4-x^3-3*x^2-x+1) / ((x-1)^2*(x+1)^2*(x^2-x+1)*(x^2+1)). - Colin Barker, Feb 24 2015

cp's OEIS Frontend

A234902 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3) after n rotations.

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A047408 Numbers that are congruent to {1, 4, 6} mod 8.

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A236535 a(n)*Pi is the total length of irregular spiral (center points: 2, 3, 1; pattern 1) after n rotations.

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A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k*(4*k + 1)).

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A063285 Dimension of the space of weight n cuspidal newforms for Gamma_1( 12 ).

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A308399 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k(4k + 1)).