A346595 -id:A346595 - cp's OEIS frontend

A125714 Alfred Moessner's factorial triangle.

1, 2, 3, 6, 11, 6, 24, 50, 35, 10, 120, 274, 225, 85, 15, 720, 1764, 1624, 735, 175, 21, 5040, 13068, 13132, 6769, 1960, 322, 28, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 3628800

Offset: 1

Gary W. Adamson, Dec 01 2006

Successive numbers arising from the Moessner construction of the factorial numbers. - N. J. A. Sloane, Jul 27 2021
Row sums of the triangle = 1, 5, 23, 119, 719, ...(matching the terms 0, 0, 1, 5, 23, 119, 719, ...; of A033312).
The name of the triangle derives from the fact that A125714(A000124(n)) = A000142(n) for n > 0. Moessner's method uses only additions to compute the factorial n!. - Peter Luschny, Jan 27 2009
If n = (m^2+m+2)/2 then a(n) = (m+1)!. For example, taking m = 3, n = 7, and indeed a(7) = 4! = 24. - N. J. A. Sloane, Jul 27 2021

			An "x" prefaced before each term will indicate the term following the x being circled.
x1 2 x3 4 5 x6 7 8 9 x10 11 12 13 14 x15 ...
__x2 6 x11 18 26 x35 46 58 71 x85 ...
_____________x6 24 x50 96 154 x225 ...
_________________________x24 120 x274 ...
___________________________________________x120 ...
...
i.e., circle the triangular terms in row 1. In row 2, take partial sums of the uncircled terms and circle the terms offset one place to the left of the triangular terms in row 1. Continue in subsequent rows with analogous operations. The triangle consists of the infinite set of terms prefaced with the x (circled on page 64 of "The Book of Numbers").

J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996. Sequence can be seen by reading the successive circled numbers in the "factorial" section on page 64 (based on the work of Alfred Moessner).

Joshua Zucker, Table of n, a(n) for n = 1..66
G. S. Kazandzidis, On a conjecture of Moessner and a general problem, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
Dexter Kozen and Alexandra Silva, On Moessner's theorem, Amer. Math. Monthly 120(2) (2013), 131-139.
R. Krebbers, L. Parlant, and A. Silva, Moessner's theorem: an exercise in coinductive reasoning in Coq, Theory and practice of formal methods, 309-324, Lecture Notes in Comput. Sci., 9660, Springer, 2016.
Calvin T. Long, Strike it out--add it up, Math. Gaz. 66 (438) (1982), 273-277.
Alfred Moessner, Eine Bemerkung über die Potenzen der natürlichen Zahlen, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
Ivan Paasche, Ein neuer Beweis des Moessnerschen Satzes S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 1-5 (1953). [Two years are listed at the beginning of the journal issue.]
Ivan Paasche, Beweis des Moessnerschen Satzes mittels linearer Transformationen, Arch. Math. (Basel) 6 (1955), 194-199.
Ivan Paasche, Eine Verallgemeinerung des Moessnerschen Satzes, Compositio Math. 12 (1956), 263-270.
Hans Salié, Bemerkung zu einem Satz von A. Moessner, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 7-11 (1953). [Two years are listed at the beginning of the journal issue.]
Oskar Perron, Beweis des Moessnerschen Satzes, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.

Cf. A000217, A003056, A008275, A033312, A346004-A346007, A346595.

a := proc(n) local s,m,k,i; s := array(0..n); s[0] := 1;
for m from 1 to n do s[m] := 0; for k from m by - 1 to 1 do
for i from 1 to k do s[i] := s[i] + s[i - 1] od; lprint(s[k]);
if k = n then return(s[n]) fi od; lprint("-") od end: # Peter Luschny, Jan 27 2009
with(combinat);
s:=stirling1;
A003056 := proc(n) floor((sqrt(1+8*n)-1)/2) ; end proc:
T:=n->n*(n+1)/2; # A000217
g:=proc(n) local i,j,t; global T,A003056;
j:=A003056(n-1)+1;
t:=T(j);
for i from 0 to t-1 do
if ((n+i) mod t) = 0 then return(abs(s(j+1,j-i))); fi;
od;
end;
[seq(g(n),n=1..80)]; # N. J. A. Sloane, Jul 27 2021

n = 10; A125714 = Reap[ ClearAll[s]; s[0] = 1; For[m = 1, m <= n, m++, s[m] = 0; For[k = m, k >= 1, k--, For[i = 1, i <= k, i++, s[i] = s[i] + s[i-1]]; Sow[s[k]]; If[k == n, Print[n, "! = ", s[n]]; Break[]]]]][[2, 1]] (* Jean-François Alcover, Jun 29 2012, after Peter Luschny *)

T(n, k)={ my( s=vector(n)); for( m=1, n, forstep( j=m,1,-1, s[1]++; for( i=2, j, s[i] += s[i-1]));
k<0 && print(vecextract(s,Str(m"..1"))));
if( k>0,s[n+1-k],vecextract(s,"-1..1"))} /* returns T[n,k], or the whole n-th row if k is not given, prints row 1...n of the triangle if k<0 */ \\ M. F. Hasler, Dec 03 2010

Starting with the natural numbers, circle each triangular number. Underneath, take partial sums of the uncircled terms and circle the terms in this row which are offset one place to the left of the circled 1, 3, 6, 10, ... in the first row. Repeat with analogous operations in succeeding rows. The circled terms in the infinite set become the triangle.

Given n, let j = A003056(n-1)+1 and set t = j*(j+1)/2. Then, for 0 <= i < t, if n == -i (mod t), a(n) = abs(Stirling_1(j+1,j-i)). - N. J. A. Sloane, Jul 27 2021

More terms from Joshua Zucker, Jun 17 2007

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

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

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).

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A125714 Alfred Moessner's factorial triangle.

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

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

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