A125714 -id:A125714 - cp's OEIS frontend

A125781 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 - 2 (m>=2).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 8, 4, 1, 1, 24, 23, 14, 5, 1, 1, 77, 76, 52, 21, 6, 1, 1, 295, 294, 217, 91, 29, 7, 1, 1, 1329, 1328, 1033, 433, 141, 39, 8, 1, 1, 6934, 6933, 5604, 2307, 739, 216, 50, 9, 1, 1, 41351, 41350, 34416, 13804, 4276, 1274, 306, 62, 10, 1, 1

Offset: 0

Paul D. Hanna, Dec 09 2006

Generated by a method similar to Moessner's factorial triangle (A125714).

			Rows are partial sums excluding terms in columns k = {1,4,8,13,...}:
row 2 = partial sums of [1, 3,4, 6,7,8, 10,11,12,13, ...];
row 3 = partial sums of [1, 8,14, 29,39,50, 75,90,106,123, ...];
row 4 = partial sums of [1, 23,52, 141,216,306, 535,695,876,1079,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
1,(1), 1, 1,(1), 1, 1, 1,(1), 1, 1, 1, 1,(1), 1, 1, 1, ...;
1,(2), 3, 4,(5), 6, 7, 8,(9), 10, 11, 12, 13,(14), 15, 16, 17, ...;
1,(4), 8, 14,(21), 29, 39, 50,(62), 75, 90, 106, 123,(141), 160, 181,.;
1,(9), 23, 52,(91), 141, 216, 306,(412), 535, 695, 876, 1079,(1305),..;
1,(24), 76, 217,(433), 739, 1274, 1969,(2845), 3924, 5479, 7335,...;
1,(77), 294, 1033,(2307), 4276, 8200, 13679,(21014), 30534, 45528,...;
1,(295), 1328, 5604,(13804), 27483, 58017, 103545,(167868), 255305,...;
1,(1329), 6933, 34416,(92433), 195978, 451283, 855463,(1454823),...;
1,(6934), 41350, 237328,(688611), 1544074, 3847960, 7700971,...;
1,(41351), 278679, 1822753,(5670713), 13371684, 35818351, 75299744,...;
1,(278680), 2101433, 15473117,(51291468), 126591212, 362337006,...;
1,(2101434), 17574551, 144165763,(506502769), 1303252476,...;
1,(17574552), 161740315, 1464992791,(5430460072), 14517950305,...;
Column 1 of this table equals column 1 of triangle A091351;
triangle A091351 begins:
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 9, 9, 4, 1;
1, 24, 30, 16, 5, 1;
1, 77, 115, 70, 25, 6, 1;
1, 295, 510, 344, 135, 36, 7, 1;
1, 1329, 2602, 1908, 805, 231, 49, 8, 1; ...
where column k of A091351 = row sums of matrix power A091351^k for k>=0.

Cf. A091351, A091352; columns: A125782, A125783, A125784, A125785, A125786; diagonals: A125787, A125788; A125789 (antidiagonal sums), A125714.

{T(n,k)=local(A=0,b=2,c=0,d=0);if(n==0,A=1, until(d>k,if(c==b*(b+1)/2-2,b+=1,A+=T(n-1,c);d+=1);c+=1));A}

Surprisingly, column 1 equals A091352 = column 1 of triangle A091351, in which column k equals row sums of the matrix power A091351^k. Column 3 of this table also equals column 1 of matrix power A091351^2.

A136212 Triple factorial array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {[m*(m+5)/6], m >= 0} and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 28, 10, 3, 1, 280, 80, 18, 4, 1, 3640, 880, 162, 28, 5, 1, 58240, 12320, 1944, 280, 39, 6, 1, 1106560, 209440, 29160, 3640, 418, 52, 7, 1, 24344320, 4188800, 524880, 58240, 5714, 600, 66, 8, 1, 608608000, 96342400, 11022480, 1106560, 95064

Offset: 0

Paul D. Hanna, Dec 22 2007

This is the triple factorial variant of Moessner's factorial array described by A125714 and also of the double factorial array A135876. Another very interesting variant is A136217.

			Square array begins:
(1),(1),(1),1,(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,1,(1),1,1,1,...;
(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),19,20,21,..;
(4),(10),(18),28,(39),52,(66),82,(99),118,138,(159),182,206,(231),258,286,..;
(28),(80),(162),280,(418),600,(806),1064,(1350),1696,2074,(2485),2966,3484,..;
(280),(880),(1944),3640,(5714),8680,(12164),16840,(22194),29080,36824,(45474),.;
(3640),(12320),(29160),58240,(95064),151200,(219108),315440,(428652),581680,...;
(58240),(209440),(524880),1106560,(1864456),3082240,...;
where terms in parenthesis are at positions {[m*(m+5)/6], m>=0}
and are removed before taking partial sums to obtain the next row.
To generate the array, start with all 1's in row 0; from then on,
obtain row n+1 from row n by first removing terms in row n at
positions {[m*(m+5)/6], m>=0} and then taking partial sums.
For example, to generate row 2 from row 1:
[(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),...],
remove terms at positions [0,1,2,4,6,8,11,14,17,...] to get:
[4, 6, 8, 10,11, 13,14, 16,17, 19,20,21, 23,24,25, 27,28,29, ...]
then take partial sums to obtain row 2:
[4, 10, 18, 28,39, 52,66, 82,99, 118,138,159, 182,206,231, ...].
Continuing in this way will generate all the rows of this array.

Cf. columns: A007559, A008544, A032031, A024216, A024395; related tables: A136213, A125714, A135876, A127054, A125781; A136217.

t[n_, k_] := t[n, k] = Module[{a = 0, m = 0, c = 0, d = 0}, If[n == 0, a = 1, While[d <= k, If[c == Quotient[(m*(m + 5)), 6], m += 1, a += t[n - 1, c]; d += 1]; c += 1]]; a]; Table[t[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013, translated from Pari *)

{T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==(m*(m+5))\6, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}

Columns 0, 1 and 2 form the triple factorials A007559, A008544 and A032031, respectively. Column 4 equals A024216; column 6 equals A024395.

A346004 If n even then n otherwise ((n+1)/2)^2.

Original entry on oeis.org

0, 1, 2, 4, 4, 9, 6, 16, 8, 25, 10, 36, 12, 49, 14, 64, 16, 81, 18, 100, 20, 121, 22, 144, 24, 169, 26, 196, 28, 225, 30, 256, 32, 289, 34, 324, 36, 361, 38, 400, 40, 441, 42, 484, 44, 529, 46, 576, 48, 625, 50, 676, 52, 729, 54, 784, 56, 841, 58, 900, 60, 961, 62, 1024, 64, 1089

Offset: 0

N. J. A. Sloane, Jul 25 2021

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. Sequence can be seen in the circled numbers at foot of page 63.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A125714, A346005-A346007, A346595.

A346004[n_] := If[OddQ[n], (n+1)^2/4, n]; Array[A346004, 100, 0] (* or *)
Riffle[#-2, #^2/4] & [Range[2, 100, 2]] (* Paolo Xausa, Aug 28 2024 *)

def A346004(n): return ((n+1)//2)**2 if n % 2 else n # Chai Wah Wu, Jul 25 2021

G.f.: x*(-1-2*x-x^2+2*x^3) / ( (x-1)^3*(1+x)^3 ). - R. J. Mathar, Aug 05 2021

a(n) = ((n^2 + 6*n + 1) - (n-1)^2*(-1)^n)/8. - Aaron J Grech, Aug 27 2024

A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 34, 33, 15, 5, 1, 1, 154, 153, 65, 23, 6, 1, 1, 874, 873, 339, 119, 32, 7, 1, 1, 5914, 5913, 2103, 719, 186, 42, 8, 1, 1, 46234, 46233, 15171, 5039, 1230, 267, 54, 9, 1, 1, 409114, 409113, 124755, 40319, 9258, 1891, 380

Offset: 0

Paul D. Hanna, Jan 04 2007

Variant of table A125781. Generated by a method similar to Moessner's factorial triangle (A125714).

			Rows are partial sums excluding terms in columns k = {1,3,6,10,...}:
row 2 = partial sums of [1, 3, 5,6, 8,9,10, 12,13,14,15, ...];
row 3 = partial sums of [1, 9, 23,32, 54,67,81, 113,131,150,170, ...];
row 4 = partial sums of [1, 33, 119,186, 380,511,661, 1045,1283,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
1,(1), 1, (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, ...;
1,(2), 3, (4), 5, 6, (7), 8, 9, 10, (11), 12, 13, 14, 15, (16), ...;
1,(4), 9, (15), 23, 32, (42), 54, 67, 81, (96), 113, 131, 150, ...;
1,(10), 33, (65), 119, 186, (267), 380, 511, 661, (831), 1045, ...;
1,(34), 153, (339), 719, 1230, (1891), 2936, 4219, 5765, (7600), ...;
1,(154), 873, (2103), 5039, 9258, (15023), 25148, 38203, 54625, ..;
1,(874), 5913, (15171), 40319, 78522, (133147), 238124, 379339, ...;
1,(5914), 46233, (124755), 362879, 742218, (1305847), 2477468, ...;
1,(46234), 409113, (1151331), 3628799, 7742058, (14059423), ...;
1,(409114), 4037913, (11779971), 39916799, 88369098, (164977399),...;
Columns include:
k=1: A003422 (Left factorials: !n = Sum k!, k=0..n-1);
k=2: A007489 (Sum of k!, k=1..n);
k=3: A097422 (Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j);
k=4: A033312 (n! - 1);
k=5: Partial sums of A001705;
k=6: partial sums of A000399 (Stirling numbers of first kind s(n,3)).

Cf. variants: A125781, A125714; antidiagonal sums: A127055; diagonal: A127056; columns: A003422, A007489, A097422, A033312.

{T(n,k)=local(A=0,b=2,c=0,d=0);if(n==0,A=1, until(d>k,if(c==b*(b-1)/2,b+=1,A+=T(n-1,c);d+=1);c+=1));A}

A135876 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+2)^2/4 - 1] for m>=0 and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 105, 48, 15, 4, 1, 945, 384, 105, 23, 5, 1, 10395, 3840, 945, 176, 33, 6, 1, 135135, 46080, 10395, 1689, 279, 44, 7, 1, 2027025, 645120, 135135, 19524, 2895, 400, 57, 8, 1, 34459425, 10321920, 2027025, 264207, 35685, 4384, 561

Offset: 0

Paul D. Hanna, Dec 14 2007

This is the double factorial analog of Moessner's factorial array (A125714). Compare to triangle A135877 which is generated by a complementary process. A very interesting variant is square array A135878.

			Square array begins:
(1),(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,1,(1),1,1,1,(1),1,1,1,1,(1),...;
(1),(2),3,(4),5,(6),7,8,(9),10,11,(12),13,14,15,(16),17,18,19,(20),...;
(3),(8),15,(23),33,(44),57,71,(86),103,121,(140),161,183,206,(230),..;
(15),(48),105,(176),279,(400),561,744,(950),1206,1489,(1800),2171,..;
(105),(384),945,(1689),2895,(4384),6555,9129,(12139),16161,20763,..;
(945),(3840),10395,(19524),35685,(56448),89055,129072,(177331),245778,...;
(10395),(46080),135135,(264207),509985,(836352),1381905,2071215,(2924172),.;
(135135),(645120),2027025,(4098240),8294895,(14026752),24137505,...; ...
where terms in parenthesis are removed before taking partial sums.
For example, to generate row 2 from row 1, remove terms at positions
{[(m+2)^2/4-1], m>=0} = [0,1,3,5,8,11,15,19,24,29,35,...] to obtain:
[3, 5, 7,8, 10,11, 13,14,15, 17,18,19, 21,22,23,24, 25,26,27,28, ...]
then take partial sums to get row 2:
[3, 8, 15,23, 33,44, 57,71,86, 103,121,140, 161,183,206,230, ...].
Repeating this process will generate all the rows of the triangle,
where column 0 will be the odd double factorials (A001147)
and column 1 will be the even double factorials (A000165).

Cf. columns: A001147, A000165, A004041, A129890; variants: A135878, A125714.

{T(n, k)=local(A=0, b=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==floor((b+2)^2/4)-1, b+=1, A+=T(n-1, c); d+=1); c+=1)); A}

T(n,0) = (2n)!/n!/2^n; T(n,1) = 2^n*n!; T(n,2) = (2n+1)!/n!/2^n; T(n,3) = A004041(n) = (2n+1)!/n!/2^n * Sum_{k=0..n} 1/(2k+1). T(n,4) = A129890(n) = 2^(n+1)*(n+1)! - (2n+1)!/n!/2^n = T(n+1,1)-T(n+1,0).

A135878 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+3)^2/4 - 2] for m>=0 and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 25, 25, 12, 4, 1, 138, 138, 63, 19, 5, 1, 970, 970, 421, 113, 28, 6, 1, 8390, 8390, 3472, 832, 190, 38, 7, 1, 86796, 86796, 34380, 7420, 1560, 283, 50, 8, 1, 1049546, 1049546, 399463, 78406, 15250, 2502, 411, 63, 9, 1, 14563135, 14563135

Offset: 0

Paul D. Hanna, Dec 14 2007

Column 0 is A135881 which equals column 0 of triangle A135879 and also equals column 0 of triangle A135880. Compare to triangle A135879, which is generated by a complementary process. An interesting variant is square array A135876, in which column 0 equals the double factorials (A001147).

			Square array begins:
(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,1,(1),1,1,1,(1),1,1,1,1,(1),...;
(1),2,(3),4,(5),6,7,(8),9,10,(11),12,13,14,(15),16,17,18,(19),20,...;
(2),6,(12),19,(28),38,50,(63),77,93,(110),128,148,169,(191),214,...;
(6),25,(63),113,(190),283,411,(559),728,942,(1181),1446,1766,2116,...;
(25),138,(421),832,(1560),2502,3948,(5714),7830,10740,(14130),18036,...;
(138),970,(3472),7420,(15250),25990,44026,(67112),95918,138343,(189598),..;
(970),8390,(34380),78406,(174324),312667,(563287),897471,1329234,2003240,..;
(8390),86796,(399463),962750,(2291984),4295224,8168819,(13523882),20656067,.;
(86796),1049546,(5344770),13513589,(34169656),66534382,132787852,(227380975),.;
(1049546),14563135,(81097517),213885369,(570682050),1149537869,2395865161,..;
(14563135),228448504,(1377986373),3773851534,(10568874312),21945438536,...;
where terms in parenthesis are removed before taking partial sums.
For example, to generate row 2 from row 1, remove terms at positions
{[(m+3)^2/4-2], m>=0} = [0,2,4,7,10,14,18,23,28,34,...] to obtain:
[2, 4, 6,7, 9,10, 12,13,14, 16,17,18, 20,21,22,23, ...]
then take partial sums to get row 2:
[2, 6, 12,19, 28,38, 50,63,77, 93,110,128, 148,169,191,214, ...].
Repeating this process will generate all the rows of the triangle.
Triangle A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and is generated by matrix powers of itself.

Cf. A135881, A135879, A135880; variants: A135876, A125714.

{T(n, k)=local(A=0, b=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==floor((b+3)^2/4)-2, b+=1, A+=T(n-1, c); d+=1); c+=1)); A}

A346005 Successive numbers arising from the Moessner construction of the cubes on page 64 of Conway-Guy's "Book of Numbers".

Original entry on oeis.org

0, 1, 3, 3, 8, 12, 6, 27, 27, 9, 64, 48, 12, 125, 75, 15, 216, 108, 18, 343, 147, 21, 512, 192, 24, 729, 243, 27, 1000, 300, 30, 1331, 363, 33, 1728, 432, 36, 2197, 507, 39, 2744, 588, 42, 3375, 675, 45, 4096, 768, 48, 4913, 867, 51, 5832, 972, 54, 6859, 1083, 57, 8000, 1200

Offset: 0

N. J. A. Sloane, Jul 25 2021

nonn

a(3*k+1) = (k+1)^3 for k >= 0.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. Sequence can be seen in the successive circled numbers at the top of page 64.

Cf. A125714, A346004, A346006-A346007, A346595.

f:=proc(n) if (n mod 3) = 0 then n
elif (n mod 3) = 1 then ((n+2)/3)^3;
else (n+1)^2/3; fi; end;
[seq(f(n),n=0..100)];

def A346005(n): return n if n % 3 == 0 else ((n+2)//3)**3 if n % 3 == 1 else (n+1)**2//3 # Chai Wah Wu, Jul 25 2021

a(n) = n if n mod 3 = 0, = ((n+2)/3)^3 if n mod 3 = 1, and otherwise (n+1)^2/3.

A346007 Let b=5. If n == -i (mod b) for 0 <= i < b, then a(n) = binomial(b,i+1)*((n+i)/b)^(i+1).

Original entry on oeis.org

0, 1, 5, 10, 10, 5, 32, 80, 80, 40, 10, 243, 405, 270, 90, 15, 1024, 1280, 640, 160, 20, 3125, 3125, 1250, 250, 25, 7776, 6480, 2160, 360, 30, 16807, 12005, 3430, 490, 35, 32768, 20480, 5120, 640, 40, 59049, 32805, 7290, 810, 45, 100000, 50000, 10000, 1000, 50

Offset: 0

N. J. A. Sloane, Jul 25 2021

nonn

These are the numbers that would arise if the Moessner construction on page 64 of Conway-Guy's "Book of Numbers" were extended to the fifth powers.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. See pp. 63-64.

Setting b = 2, 3, or 4 gives A346004, A346005, and A346006.

Cf. A125714, A346595.

f:=proc(n,b) local i;
for i from 0 to b-1 do
if ((n+i) mod b) = 0 then return(binomial(b,i+1)*((n+i)/b)^(i+1)); fi;
od;
end;
[seq(f(n,5),n=0..80)];

from sympy import binomial
def A346007(n):
    i = (5-n)%5
    return binomial(5,i+1)*((n+i)//5)**(i+1) # Chai Wah Wu, Jul 25 2021

A346595 Successive numbers arising from the Moessner construction of the sequence A010790 (n!*(n+1)!) on pages 64, 65 of Conway-Guy's "Book of Numbers".

Original entry on oeis.org

1, 2, 5, 4, 12, 40, 51, 31, 9, 144, 564, 904, 769, 376, 106, 16, 2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25, 86400, 408960, 840216, 991276, 748530, 381065, 133848, 32523, 5370, 575, 36, 3628800, 18299520, 40691952, 52965360, 45165064, 26726896, 11323991, 3487055, 782187, 126483, 14357, 1085, 49

Offset: 1

N. J. A. Sloane, Jul 25 2021

The circled numbers 5, 31, 106, 270, 575, 1085, ... in the second row of the display at the foot of page 64 are (essentially) A212523.

This sequence can also be represented as a triangle of numbers where the rows have lengths 1, 3, 5, 7, ... - Jinyuan Wang, Aug 06 2021

			As a triangle, this is:
1,
2, 5, 4,
12, 40, 51, 31, 9,
144, 564, 904, 769, 376, 106, 16,
2880, 12576, 23300, 24080, 15345, 6273, 1650, 270, 25,
86400, 408960, 840216, 991276, 748530, 381065, 133848, 32523, 5370, 575, 36,
...

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. Sequence can be obtained by reading the successive circled numbers in the tableau at the foot of page 64.

Cf. A010790, A125714, A212523, A346004, A346005, A346006, A346007.

More terms from Jinyuan Wang, Aug 06 2021

A346006 Successive numbers arising from the Moessner construction of the sequence of fourth powers on page 64 of Conway-Guy's "Book of Numbers".

Original entry on oeis.org

0, 1, 4, 6, 4, 16, 32, 24, 8, 81, 108, 54, 12, 256, 256, 96, 16, 625, 500, 150, 20, 1296, 864, 216, 24, 2401, 1372, 294, 28, 4096, 2048, 384, 32, 6561, 2916, 486, 36, 10000, 4000, 600, 40, 14641, 5324, 726, 44, 20736, 6912, 864, 48, 28561, 8788, 1014, 52, 38416, 10976, 1176, 56, 50625, 13500, 1350, 60

Offset: 0

N. J. A. Sloane, Jul 25 2021

nonn

a(4*k+1) = (k+1)^2 for k >= 0.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. Sequence can be obtained by reading the successive circled numbers in the second display on page 64.

Cf. A125714, A346004, A346005, A346595.

f:=proc(n,b) local i;
for i from 0 to b-1 do
if ((n+i) mod b) = 0 then return(binomial(b,i+1)*((n+i)/b)^(i+1)); fi;
od;
end;
[seq(f(n,3),n=0..60)];

from sympy import binomial
def A346006(n):
    i = (4-n)%4
    return binomial(4,i+1)*((n+i)//4)**(i+1) # Chai Wah Wu, Jul 25 2021

Let b=4. If n == -i (mod b) for 0 <= i < b, then a(n) = binomial(b,i+1)*((n+i)/b)^(i+1).

cp's OEIS Frontend

A125781 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 - 2 (m>=2).

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A136212 Triple factorial array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {[m*(m+5)/6], m >= 0} and then taking partial sums, starting with all 1's in row 0.

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Mathematica

PARI

Formula

A346004 If n even then n otherwise ((n+1)/2)^2.

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Mathematica

Python

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A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

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Maple

A135876 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+2)^2/4 - 1] for m>=0 and then taking partial sums, starting with all 1's in row 0.

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PARI

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A135878 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+3)^2/4 - 2] for m>=0 and then taking partial sums, starting with all 1's in row 0.

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PARI

A346005 Successive numbers arising from the Moessner construction of the cubes on page 64 of Conway-Guy's "Book of Numbers".

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Maple

Python

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A346007 Let b=5. If n == -i (mod b) for 0 <= i < b, then a(n) = binomial(b,i+1)*((n+i)/b)^(i+1).

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