A213547 -id:A213547 - cp's OEIS frontend

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90

Offset: 1

Clark Kimberling, Jun 14 2012

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1:  (1,2,3,...)**(1,2,3,...) = A000292.
Row 2:  (1,2,3,...)**(2,3,4,...) = A005581.
Row 3:  (1,2,3,...)**(3,4,5,...) = A006503.
Row 4:  (1,2,3,...)**(4,5,6,...) = A060488.
Row 5:  (1,2,3,...)**(5,6,7,...) = A096941.
Row 6:  (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples:  let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)

t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017

def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n:  x*(n - (n - 1)*x)/(1 - x)^4.

A098359 Multiplication table of the square numbers read by antidiagonals.

Original entry on oeis.org

1, 4, 4, 9, 16, 9, 16, 36, 36, 16, 25, 64, 81, 64, 25, 36, 100, 144, 144, 100, 36, 49, 144, 225, 256, 225, 144, 49, 64, 196, 324, 400, 400, 324, 196, 64, 81, 256, 441, 576, 625, 576, 441, 256, 81, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 121, 400, 729, 1024, 1225, 1296, 1225, 1024, 729, 400, 121

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k-1) = n. See A003991. - Charlie Marion, Apr 22 2013

			Square array A(n,k) begins:
   1,   4,   9,  16,   25,   36,   49, ...
   4,  16,  36,  64,  100,  144,  196, ...
   9,  36,  81, 144,  225,  324,  441, ...
  16,  64, 144, 256,  400,  576,  784, ...
  25, 100, 225, 400,  625,  900, 1225, ...
  36, 144, 324, 576,  900, 1296, 1764, ...
  49, 196, 441, 784, 1225, 1764, 2401, ...

Cf. A000290, A003991, A098358, A098360, A098361, A213547.
Antidiagonal sums give A033455.
Main diagonal gives A000583.

A:= (n,k)-> (n*k)^2:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, May 19 2025

A(n,k) = n^2*k^2.
G.f.: [xy(1+x)(1+y)] / [(1-x)^3 * (1-y)^3 ]. - Ralf Stephan, Oct 27 2004
Sum_{j=1..n} A(j,1+n-j)*j = A213547(n). - Alois P. Heinz, May 19 2025

Offset corrected by Alois P. Heinz, May 19 2025

A213505 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 8, 4, 34, 25, 9, 104, 88, 52, 16, 259, 234, 170, 89, 25, 560, 524, 424, 280, 136, 36, 1092, 1043, 899, 674, 418, 193, 49, 1968, 1904, 1708, 1384, 984, 584, 260, 64, 3333, 3252, 2996, 2555, 1979, 1354, 778, 337, 81, 5368, 5268, 4944, 4368, 3584

Offset: 1

Clark Kimberling, Jun 16 2012

Principal diagonal: A213546.
Antidiagonal sums: A213547.
Row 1, (1,4,9,...)**(1,4,9,...): A033455.
Row 2, (1,4,9,...)**(4,9,16,...): (k^5 + 10*k^4 + 40*k^3 + 50*k^2 +19*k)/30.
Row 3, (1,4,9,...)**(9,16,25,...):  (k^5 + 15*k^4 + 90*k^3 + 120*k^2+44*k)/30.
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....8.....34....104...259....560
4....25....88....234...524....1043
9....52....170...424...899....1708
16...89....280...674...1384...2555
25...136...418...984...1979...3584
...
T(5,1) = (1)**(25) = 25
T(5,2) = (1,4)**(25,36) = 1*36+4*25 = 136
T(5,3) = (1,4,9)**(25,36,49) = 1*49+4*36+9*25 = 418

Clark Kimberling, Antidiagonals n = 1..60, flattened
Henri Muehle, Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details, arXiv preprint arXiv:1301.1654, 2013.

Cf. A213500.

b[n_] := n^2; c[n_] := n^2
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213505 *)
d = Table[t[n, n], {n, 1, 40}] (* A213546 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213547 *)

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) - T(n,k-6).

G.f. for row n: f(x)/g(x), where f(x) = n^2 - (n^2 - 2*n - 1)*x - (n^2 - 2)*x^2 - ((n - 1)^2)*x^3 and g(x) = (1 - x)^6.

A213555 Rectangular array: (row n) = bc, where b(h) = h^3, c(h) = n-1+h, n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 10, 2, 46, 19, 3, 146, 82, 28, 4, 371, 246, 118, 37, 5, 812, 596, 346, 154, 46, 6, 1596, 1253, 821, 446, 190, 55, 7, 2892, 2380, 1694, 1046, 546, 226, 64, 8, 4917, 4188, 3164, 2135, 1271, 646, 262, 73, 9, 7942, 6942, 5484, 3948, 2576, 1496, 746

Offset: 1

Clark Kimberling, Jun 17 2012

Principal diagonal: A213556.
Antidiagonal sums: A213547.
Row 1,  (1,8,27,...)**(1,2,3,...):  A024166.
Row 2,  (1,8,27,...)**(2,3,4,...): (3*k^5 + 30*k^4 + 55*k^3 + 30*k^2 + 2*k)/60.
Row 3,  (1,8,27,...)**(3,4,5,...): (3*k^5 + 45*k^4 + 85*k^3 + 45*k^2 + 2*k)/60.
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1...10...46....146...371....812
2...19...82....246...596....1253
3...28...118...346...821....1694
4...37...154...446...1046...2135
5...46...190...546...1271...2576
6...55...226...646...1496...3017

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500, A213553.

b[n_] := n^3; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213555 *)
d = Table[t[n, n], {n, 1, 40}] (* A213556 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213547 *)

T(n,k) = 6*T(n,k-1) - 15*T(n,k-2) + 20*T(n,k-3) - 15*T(n,k-4) + 6*T(n,k-5) -T(n,k-6).

G.f. for row n: f(x)/g(x), where f(x) = n + (3*n + 1)*x - (3*n - 4)*x^2 - (n - 1)*x^3 and g(x) = (1 - x)^6.

A138451 a(n) = (prime(n)^6-prime(n)^2)/60.

Original entry on oeis.org

1, 12, 260, 1960, 29524, 80444, 402288, 784092, 2467256, 9913708, 14791712, 42762084, 79168376, 105356020, 179653552, 369405972, 703008836, 858672844, 1507639628, 2135004648, 2522237016, 4051457488, 5449006108, 8283021384

Offset: 1

Artur Jasinski, Mar 19 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A213547.

[( NthPrime(n)^6-NthPrime(n)^2 )/60: n in [1..30]]; // Vincenzo Librandi, Sep 11 2014

seq((ithprime(n)^6-ithprime(n)^2)/60, n=1..100); # Robert Israel, Sep 11 2014

a = {}; Do[p = Prime[n]; AppendTo[a, (p^6 - p^2)/60], {n, 1, 50}]; a
(#^6-#^2)/60&/@Prime[Range[30]] (* Harvey P. Dale, Sep 11 2014 *)

forprime(p=2,1e3,print1((p^6-p^2)/60", ")) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A213547(prime(n)-1). - Robert Israel, Sep 11 2014

A231548 Numbers n such that 2*n - 1 < sigma(n) - sigma(n-2).

Original entry on oeis.org

1680, 2520, 3360, 3780, 3960, 4200, 4680, 5040, 6300, 6720, 7560, 7920, 8820, 9240, 9360, 10080, 10800, 10920, 11340, 11760, 11880, 12600, 13440, 13860, 14040, 15120, 15840, 15960, 16380, 16800, 17280, 17640, 18480, 18900, 19800, 20160, 20520, 21000, 21420

Offset: 1

Jaroslav Krizek, Nov 12 2013

nonn

Also numbers n such that antisigma(n) < antisigma(n-2), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n.
Sequence contains anomalous increased frequency of values ending with digit 0.
Conjecture: there are no numbers n such that antisigma(n) < antisigma(n-3).

			1680 is in sequence because antisigma(1680) = 1406088 < antisigma(1678) = 1406161.

Jaroslav Krizek, Table of n, a(n) for n = 1..228, all terms < 10^5

Cf. A024816, A213547 (numbers n such that antisigma(n) < antisigma(n-1)).

A195166 Numbers expressible as 2^a - 2^b, with 0 <= b < a, such that n^a - n^b is divisible by 2^a - 2^b for all n.

Original entry on oeis.org

1, 2, 6, 12, 30, 24, 60, 120, 252, 240, 504, 16380, 32760, 65520

Offset: 1

Michel Marcus, Dec 21 2012

= 2^1  - 2^0. (n^1  - n^0)/1     : A000027
= 2^2  - 2^1. (n^2  - n^1)/2     : A000217
= 2^3  - 2^1. (n^3  - n^1)/6     : A000292
= 2^4  - 2^2. (n^4  - n^2)/12    : A002415
= 2^5  - 2^1. (n^5  - n^1)/30    : A033455
= 2^5  - 2^3. (n^5  - n^3)/24    : A006414
= 2^6  - 2^2. (n^6  - n^2)/60    : A213547
= 2^7  - 2^3. (n^7  - n^3)/120   : A114239
= 2^8  - 2^2. (n^8  - n^2)/252   :
= 2^8  - 2^4. (n^8  - n^4)/240   : A078876
= 2^9  - 2^3. (n^9  - n^3)/504   :
= 2^14 - 2^2. (n^14 - n^2)/16380 :
= 2^15 - 2^3. (n^15 - n^3)/32760 :
= 2^16 - 2^4. (n^16 - n^4)/65520 :
Comment from Tomohiro Yamada, Oct 05 2022: (Start)
"The Mod Set Stanford University" and Carl Pomerance independently noted that the completeness of this sequence follows from a result of Schinzel on primitive prime factors of sequences a^n - b^n in the remark to a problem of Harry Ruderman asking whether if 2^a - 2^b divides 3^a - 3^b, then 2^a - 2^b belongs to this sequence.
The Mod Set verified that if a > b, 2^a - 2^b divides 3^a - 3^b, but 2^a - 2^b does not belong to this sequence, then a - b > 1900. Ram Murty and Kumar Murty proved that there are only finitely many natural numbers a, b such that 2^a - 2^b divides 3^a - 3^b.
Qi Sun and Ming Zhi Zhang also showed that if a > b and n^a - n^b is divisible by 2^a - 2^b for all n, then 2^a - 2^b belongs to this sequence. (End)

			a(3) = 6 belongs to this sequence since (n^3 - n)/6 = C(n+1, 3) = A000292(n-1).

M. Ram Murty and V. Kumar Murty, On a Problem of Ruderman, Amer. Math. Monthly 118 (2011), 644-650, available from the first author's website.
Harry Ruderman, Problem E2468, Amer. Math. Monthly 81 (1974), p. 405.
A. Schinzel, On primitive prime factors of a^n - b^n, Proc. Cambridge Phil. Soc. 58 (1962), 556-562.
Qi Sun and Ming Zhi Zhang, Pairs where 2^a-2^b divides n^a-n^b for all n, Proc. Amer. Math. Soc. 93 (1985), 218-220.
The Mod Set Stanford University and Carl Pomerance, When 2^m - 2^n divides 3^m - 3^n, remarks to Problem E2468*, Amer. Math. Monthly 84 (1977), 59-60.
W. Y. Velez, When 2^m - 2^n divides 3^m - 3^n, remarks to Problem E2468, Amer. Math. Monthly 83 (1976), 288-289.

cp's OEIS Frontend

A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A098359 Multiplication table of the square numbers read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A213505 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A213555 Rectangular array: (row n) = b**c, where b(h) = h^3, c(h) = n-1+h, n>=1, h>=1, and ** = convolution.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A138451 a(n) = (prime(n)^6-prime(n)^2)/60.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A231548 Numbers n such that 2*n - 1 < sigma(n) - sigma(n-2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A195166 Numbers expressible as 2^a - 2^b, with 0 <= b < a, such that n^a - n^b is divisible by 2^a - 2^b for all n.

Views

Author

Keywords

Comments

Examples

Links

A213500 Rectangular array T(n,k): (row n) = bc, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and = convolution.

A213505 Rectangular array: (row n) = bc, where b(h) = h^2, c(h) = (n-1+h)^2, n>=1, h>=1, and = convolution.

A213555 Rectangular array: (row n) = bc, where b(h) = h^3, c(h) = n-1+h, n>=1, h>=1, and = convolution.