A074148 -id:A074148 - cp's OEIS frontend

A230370 Voids left after packing 3 curves coins patterns (3c3s type) into fountain of coins base n.

0, 0, 3, 6, 13, 19, 39, 54, 66, 85, 100, 123, 141, 168, 189, 220, 244, 279, 306, 345, 375, 418, 451, 498, 534, 585, 624, 679, 721, 780, 825, 888, 936, 1003, 1054, 1125, 1179, 1254, 1311, 1390, 1450, 1533, 1596, 1683

Offset: 1

Kival Ngaokrajang, Oct 17 2013

nonn

Refer to arrangement same as A005169: "A fountain is formed by starting with a row of coins, then stacking additional coins on top so that each new coin touches two in the previous row". The 3 curves coins patterns consist of a part of each coin circumference and forms a continuous area. There are total 4 distinct patterns. For selected pattern, I would like to call "3c3s" type as it cover 3 coins and symmetry. When packing 3c3s into fountain of coins base n, the total number of 3c3s is A008805, the coins left is A008795 and voids left is a(n). See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (V)

A001399, A230267, A230276 (5-curves coins patterns); A074148, A229093, A220154 (4-curves coins patterns); A008795 (3-curves coins patterns).

G.f.: x^3*(11*x^8 - 5*x^7 - 21*x^6 + 6*x^5 + 9*x^4 + x^2 + 3*x + 3)/((1-x)*(1-x^2)^2) (conjectured). Ralf Stephan, Oct 19 2013

A214075 Triangle read by rows: T(n,k) = floor(A213998(n,k) / A213999(n,k)), 0<=k<=n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 2, 0, 1, 4, 7, 6, 2, 0, 1, 5, 12, 14, 8, 2, 0, 1, 6, 17, 26, 22, 11, 2, 0, 1, 7, 24, 44, 49, 34, 13, 2, 0, 1, 8, 31, 68, 93, 83, 47, 16, 2, 0, 1, 9, 40, 100, 162, 177, 131, 64, 19, 2, 0, 1, 10, 49, 140, 263, 340, 309

Offset: 0

Reinhard Zumkeller, Jul 03 2012

T(n,0) = 1;
T(n,1) = n - 1 for n > 0, cf. A001477;
T(n,2) = A074148(n-2) for n > 2;
T(n,n-2) = A022819(n) for n > 1;
T(n,n-1) = A055980(n) for n > 0;
T(n,n) = A000007(n).

			Start of triangle preceded by triangle "A213998/A213999":
. 0:                      1                                 1
. 1:                   1   1/2                             1  0
. 2:               1    3/2   1/3                         1  1  0
. 3:            1   5/2    11/6   1/4                    1 2  1  0
. 4:        1   7/2   13/3    25/12   1/5               1 3  4  2 0
. 5:     1   9/2   47/6   77/12   137/60   1/6         1 4  7  6 2 0
. 6:  1  11/2   37/3   57/4   87/10   49/20   1/7,    1 5 12 14 8 2 0.

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened

a214075 n k = a214075_tabl !! n !! k
a214075_row n = a214075_tabl !! n
a214075_tabl = zipWith (zipWith div) a213998_tabl a213999_tabl

A230548 Twin hearts patterns packing into n X n coins.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 12, 15, 16, 24, 25, 28, 35, 40, 41, 54, 55, 60, 70, 77, 78, 96, 97, 104, 117, 126, 127, 150, 151, 160, 176, 187, 188, 216, 217, 228, 247, 260, 261, 294, 295, 308, 330, 345, 346, 384, 385, 400, 425, 442

Offset: 2

Kival Ngaokrajang, Oct 23 2013

nonn

Twin hearts (6c4a type) is one of total 17 distinct patterns appearing in 3X2 coins where each pattern consists of 6 perimeter parts from each coin and forms a continuous area.

a(n) is the number of total twin hearts patterns (6c4a type: 6-curves cover 4 coins) packing into n X n coins with rotation not allowed. The total coins left after packing twin hearts patterns into n X n coins is A230549 and voids left is A230550. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (T)

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620 (6-curves).

G.f.: x^2 * (x^10 + x^8 + 2*x^5 + 3*x^4 + 2*x^3 + 2*x^2 + x)/((1+x^3) * (1-x^3)^2 * (1-x^2)) (conjectured). - Ralf Stephan, Oct 30 2013

A230549 Coins left after packing twin hearts patterns into n X n coins.

Original entry on oeis.org

4, 5, 8, 13, 12, 21, 32, 33, 40, 57, 48, 69, 84, 85, 96, 125, 108, 141, 160, 161, 176, 217, 192, 237, 260, 261, 280, 333, 300, 357, 384, 385, 408, 473, 432, 501, 532, 533, 560, 637, 588, 669, 704, 705, 736, 825, 768, 861

Offset: 2

Kival Ngaokrajang, Oct 23 2013

nonn

Twin hearts (6c4a type) is one of total 17 distinct patterns appearing in 3X2 coins where each pattern consists of 6 perimeter parts from each coin and forms a continuous area.

a(n) is total coins left after packing twin hearts patterns (6c4a type: 6-curves cover 4 coins) into n X n coins with rotation not allowed. The total twin hearts patterns is A230548 and voids left is A230550. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (U)

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620 (6-curves).

a(n) = n^2 - 4*A230548(n).

G.f.: x^2 * (-3*x^10 - 4*x^8 + 3*x^7 + 8*x^6 + 4*x^5 - x^4 + 4*x^3 + 4*x^2 + 5*x + 4)/(1+x^3)*(1-x^3)^2*(1-x^2). (conjectured). - Ralf Stephan, Oct 30 2013

A230550 Voids left after packing twin hearts patterns into n X n coins.

Original entry on oeis.org

1, 2, 5, 10, 13, 22, 33, 40, 51, 68, 73, 94, 113, 126, 145, 174, 181, 214, 241, 260, 287, 328, 337, 382, 417, 442, 477, 530, 541, 598, 641, 672, 715, 780, 793, 862, 913, 950, 1001, 1078, 1093, 1174, 1233, 1276, 1335, 1424

Offset: 2

Kival Ngaokrajang, Oct 23 2013

nonn

Twin hearts (6c4a type) is one of total 17 distinct patterns appearing in 3X2 coins where each pattern consists of 6 perimeter parts from each coin and forms a continuous area.

a(n) is the number of total voids left after packing twin hearts patterns (6c4a type: 6-curves cover 4 coins) into n X n coins with rotation not allowed. The total twin hearts patterns packing into n X n coins is A230548 and coins left is A230549. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (V)

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620 (6-curves).

a(n) = (n-1)^2 - 2*A230548(n).

G.f.: x^2 * (-2*x^10 + x^9 + 2*x^8 + 8*x^7 + 11*x^6 + 8*x^5 + 6*x^4 + 7*x^3 + 4*x^2 + 2*x + 1)/((1+x^3)*(1-x^3)^2*(1-x^2)) (conjectured). - Ralf Stephan, Oct 30 2013

A100795 n occurs n times, as early as possible subject to the constraint that no two successive terms are identical.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12, 13, 12

Offset: 1

Amarnath Murthy, Dec 05 2004

Permutation of A002024. - Reinhard Zumkeller, Jan 17 2014

			After a(8) = 4 the next term is 5 as 3 has already occurred three times.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A061925, A074148, A100798.

a100795 n = a100795_list !! (n-1)
a100795_list = f 0 a002024_list where
   f x ws = v : f v (us ++ vs) where (us, v:vs) = span (== x) ws
-- Reinhard Zumkeller, Jan 17 2014

Corrected and extended by Ray Chandler, Dec 08 2004

A100798 n occurs n times, as early as possible subject to the constraint that two successive occurrences of n are separated by at least by n terms.

Original entry on oeis.org

1, 2, 3, 4, 2, 5, 3, 6, 4, 7, 3, 5, 8, 4, 6, 9, 10, 5, 4, 7, 11, 6, 8, 5, 12, 9, 13, 7, 6, 5, 10, 8, 11, 14, 15, 6, 7, 9, 12, 16, 8, 10, 6, 13, 7, 11, 17, 9, 14, 8, 15, 12, 7, 10, 18, 19, 16, 9, 8, 11, 7, 13, 20, 14, 10, 12, 15, 8, 9, 17, 21, 11, 22, 16, 18, 10, 8, 13, 9, 12, 14, 19, 15, 11

Offset: 1

Amarnath Murthy, Dec 05 2004

Subsidiary sequences: first (A100919) and the last (A100920) occurrences of n.

			Index of the first occurrence of 2 is 2 and that of the second occurrence is 5, separated by a(3) and a(4), two terms.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A061925, A074148, A100795.

mx = 22; lst = cnt = ConstantArray[0, mx + 1]; a = {}; Do[k = Min@Select[Range[mx + 1], lst[[#]] <= n && cnt[[#]] < # &]; AppendTo[a, k]; lst[[k]] = n + k + 1; cnt[[k]]++; If[k > mx, Break[]], {n, mx^2}]; a (* Ivan Neretin, Nov 25 2016 *)

Extended by Ray Chandler, Dec 08 2004

A138179 Wiener index of the prism graph Y_n on 2n nodes.

Original entry on oeis.org

1, 8, 21, 48, 85, 144, 217, 320, 441, 600, 781, 1008, 1261, 1568, 1905, 2304, 2737, 3240, 3781, 4400, 5061, 5808, 6601, 7488, 8425, 9464, 10557, 11760, 13021, 14400, 15841, 17408, 19041, 20808, 22645, 24624, 26677, 28880, 31161, 33600, 36121, 38808

Offset: 1

Eric W. Weisstein, Mar 04 2008

Sequence expended to a(1)-a(2) using the formula/recurrence. - Eric W. Weisstein, Sep 08 2017
Apparently a(n) = n * A074148(n), so a(n)= +2*a(n-1) +a(n-2) -4*a(n-3) +a(n-4) +2*a(n-5) -a(n-6). - R. J. Mathar, May 31 2010
From Emeric Deutsch, Sep 16 2010: (Start)
The Wiener index of a connected graph is the sum of all distances in the graph.
Y_n is also called circular ladder (= P_2 X C_n, where P_2 is the path graph on 2 nodes and C_n is the cycle graph on n nodes).
a(n) = Sum(k*A180572(n,k), k>=1).
a(n) is the derivative of the Wiener polynomial of Y_n (given in A180572) evaluated at t=1. (see the Sagan et al. reference).
(End)

			a(3) = 21 because the triangular prism has 9 distances equal to 1 (the edges) and 6 distances equal to 2 (from the vertices of the lower base to the "opposite" vertices of the upper base). - _Emeric Deutsch_, Sep 16 2010

J. Gross and J. Yellen, Graph Theory and its Applications, CRC, Boca Raton, 1999 (p. 14). - Emeric Deutsch, Sep 16 2010

Colin Barker, Table of n, a(n) for n = 1..1000 (corrected by Michel Marcus, Jan 19 2019)
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969. - _Emeric Deutsch_, Sep 16 2010
Y.-N. Yeh and I. Gutman, On the sum of all distances in composite graphs, Discrete Math., 135 (1994), 359-365 (set m=2 in the formula for W(Cyl_{m,n}) on p. 363). - _Emeric Deutsch_, Sep 16 2010
Eric Weisstein's World of Mathematics, Prism Graph
Eric Weisstein's World of Mathematics, Wiener Index
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A180572

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 8, 21, 48, 85, 144}, 40] (* Harvey P. Dale, Jul 29 2013 *)
Table[1/4 n (-1 + (-1)^n + 2 n (2 + n)), {n, 20}] (* Eric W. Weisstein, May 11 2017 *)
CoefficientList[Series[(1 + 6 x + 4 x^2 + 2 x^3 - x^4)/((-1 + x)^4 (1 + x)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 08 2017 *)

Vec((x*(1+ 6*x+4*x^2+2*x^3-x^4))/((-1+x)^4*(1+x)^2) + O(x^50)) \\ Colin Barker, Jun 23 2015; Michel Marcus, Jan 19 2019

From Emeric Deutsch, Sep 16 2010: (Start)
a(2n+1) = (2n+1)(2n^2+4*n+1); a(2n)=4n^2*(n+1).
G.f.: (z (1 + 6 z + 4 z^2 + 2 z^3 - z^4))/((-1 + z)^4 (1 + z)^2).
(End)
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).

a(1)-a(2) from Eric W. Weisstein, Sep 08 2017

A237447 Infinite square array: row 1 is the positive integers 1, 2, 3, ..., and on any subsequent row n, n is moved to the front: n, 1, ..., n-1, n+1, n+2, ...

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 3, 1, 4, 5, 4, 2, 1, 5, 6, 5, 4, 2, 1, 6, 7, 6, 5, 3, 2, 1, 7, 8, 7, 6, 5, 3, 2, 1, 8, 9, 8, 7, 6, 4, 3, 2, 1, 9, 10, 9, 8, 7, 6, 4, 3, 2, 1, 10, 11, 10, 9, 8, 7, 5, 4, 3, 2, 1, 11, 12, 11, 10, 9, 8, 7, 5, 4, 3, 2, 1, 12, 13, 12, 11, 10, 9, 8, 6, 5, 4, 3, 2, 1, 13, 14, 13, 12, 11, 10, 9, 8, 6, 5, 4, 3, 2, 1, 14, 15, 14, 13, 12, 11, 10, 9, 7, 6, 5, 4, 3, 2, 1, 15

Offset: 1

Antti Karttunen, Feb 10 2014

Row n is the lexicographically earliest permutation of positive integers beginning with n. This also holds for the reverse colexicographic order, thus A007489(n-1) gives the position of n-th row of this array (which is one-based) in zero-based arrays A195663 & A055089.
The finite n X n square matrices in sequence A237265 converge towards this infinite square array.
Rows can be constructed also simply as follows: The first row is A000027 (natural numbers, also known as positive integers). For the n-th row, n=2, ..., pick n out from the terms of A000027 and move it to the front. This will create a permutation with one cycle of length n, in cycle notation: (1 n n-1 n-2 ... 3 2), which is the inverse of (1 2 ... n-1 n).
There are A000110(n) ways to choose n permutations from the n first rows of this table so that their composition is identity (counting all the different composition orders). This comment is essentially the same as my May 01 2006 comment on A000110, please see there for more information. - Antti Karttunen, Feb 10 2014
Also, for n > 1, the whole symmetric group S_n can be generated with just two rows, row 2, which is transposition (1 2), and row n, which is the inverse of cycle (1 ... n). See Rotman, p. 24, Exercise 2.9 (iii).

			The top left 9 X 9 corner of this infinite square array:
  1 2 3 4 5 6 7 8 9
  2 1 3 4 5 6 7 8 9
  3 1 2 4 5 6 7 8 9
  4 1 2 3 5 6 7 8 9
  5 1 2 3 4 6 7 8 9
  6 1 2 3 4 5 7 8 9
  7 1 2 3 4 5 6 8 9
  8 1 2 3 4 5 6 7 9
  9 1 2 3 4 5 6 7 8
Note how this is also the 9th finite subsquare of the sequence A237265, which can be picked from its terms A237265(205) .. A237265(285), where 205 = 1+A000330(9-1), the starting offset for that 9th subsquare in A237265.

Joseph J. Rotman, An Introduction to the Theory of Groups, 4th ed., Springer-Verlag, New York, 1995. First chapter, pp. 1-19 [For a general introduction], and from chapter 2, problem 2.9, p. 24.

Transpose: A237448.
Topmost row and the leftmost column: A000027. Second column: A054977. Central diagonal: A028310 (note the different starting offsets).
Antidiagonal sums: A074148.
This array is the infinite limit of the n X n square matrices in A237265.
Cf. also A000330, A002260, A004736, A002024, A010054, A007489, A055089, A195663.

T:= proc(r,c) if c > r then c elif c=1 then r else c-1 fi end proc:
seq(seq(T(r,n-r),r=1..n-1),n=1..20); # Robert Israel, May 09 2017

Table[Function[n, If[k == 1, n, k - Boole[k <= n]]][m - k + 1], {m, 15}, {k, m, 1, -1}] // Flatten (* Michael De Vlieger, May 09 2017 *)

A237447(n,k=0)=if(k, if(k>1, k-(k<=n), n), A237447(A002260(n), A004736(n))) \\ Yields the element [n,k] of the matrix, or the n-th term of the "linearized" sequence if no k is given. - M. F. Hasler, Mar 09 2014

(define (A237447 n) (+ (* (A010054 n) (A002024 n)) (* (- 1 (A010054 n)) (- (A004736 n) (if (>= (A002260 n) (A004736 n)) 1 0)))))
;; Another variant based on Cano's A237265.
(define (A237447 n) (let* ((row (A002260 n)) (col (A004736 n)) (sss (max row col)) (sof (+ 1 (A000330 (- sss 1))))) (A237265 (+ sof (* sss (- row 1)) (- col 1)))))

When col > row, T(row,col) = col, when 1 < col <= row, T(row,col) = col-1, and when col=1, T(row,1) = row.
a(n) = A010054(n) * A002024(n) + (1-A010054(n)) * (A004736(n) - [A002260(n) >= A004736(n)]). [This gives the formula for this entry represented as a one-dimensional sequence. Here the expression inside Iverson brackets results 1 only when the row index (A002260) is greater than or equal to the column index (A004736), otherwise zero. A010054 is the characteristic function for the triangular numbers, A000217.]
T(row,col) = A237265((A000330(max(row,col)-1)+1) + (max(row,col)*(row-1)) + (col-1)). [Takes the infinite limit of n X n matrices of A237265.]
G.f. as array: g(x,y) = (1 - 4*x*y + 3*x*y^2 + x^2*y - x*y^3)*x*y/((1-x*y)*(1-x)^2*(1-y)^2). - Robert Israel, May 09 2017

A231056 The maximum number of X patterns that can be packed into an n X n array of coins.

Original entry on oeis.org

0, 1, 1, 2, 4, 5, 8, 10, 13, 16, 20, 24, 29, 34, 40, 45, 51, 58, 65, 73, 80, 88, 97, 106, 116, 125, 135, 146, 157, 169, 180, 192, 205, 218, 232, 245, 259, 274, 289, 305, 320, 336, 353, 370, 388, 405, 423, 442, 461, 481, 500, 520, 541, 562, 584, 605, 627, 650, 673, 697, 720, 744, 769, 794

Offset: 2

Kival Ngaokrajang, Nov 03 2013

nonn

The X pattern (8c5s2 type) is a pattern in which 8 curves cover 5 coins, and is one of a total of 13 such distinct patterns that appear in a tightly-packed 3 X 3 square array of coins of identical size; each of the 8 curves is a circular arc lying along the edge of one of the 5 coins, and the 8 curves are joined end-to-end to form a continuous area.
a(n) is the maximum number of X patterns that can be packed into an n X n array of coins. The total coins left after packing X patterns into an n X n array of coins is A231064 and voids left is A231065.
a(n) is also the maximum number of "+" patterns (8c5s1 type) that can be packed into an n X n array of coins. See illustration in links.

Kival Ngaokrajang, Illustration of initial terms (X and +)

Cf. A008795, A230370 (3-curves); A074148, A227906, A229093, A229154 (4-curves); A001399, A230267, A230276 (5-curves); A229593, A228949, A229598, A002620, A230548, A230549, A230550 (6-curves).

Empirical g.f.: -x^3*(x^15 -2*x^14 +x^13 -x^12 +2*x^11 -2*x^10 +2*x^9 -x^8 +x^5 -x^4 +x^3 +x^2 -x +1) / ((x -1)^3*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Nov 27 2013

cp's OEIS Frontend

A230370 Voids left after packing 3 curves coins patterns (3c3s type) into fountain of coins base n.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A214075 Triangle read by rows: T(n,k) = floor(A213998(n,k) / A213999(n,k)), 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Programs

Haskell

A230548 Twin hearts patterns packing into n X n coins.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A230549 Coins left after packing twin hearts patterns into n X n coins.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A230550 Voids left after packing twin hearts patterns into n X n coins.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A100795 n occurs n times, as early as possible subject to the constraint that no two successive terms are identical.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Extensions

A100798 n occurs n times, as early as possible subject to the constraint that two successive occurrences of n are separated by at least by n terms.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A138179 Wiener index of the prism graph Y_n on 2n nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A237447 Infinite square array: row 1 is the positive integers 1, 2, 3, ..., and on any subsequent row n, n is moved to the front: n, 1, ..., n-1, n+1, n+2, ...

Views