cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A002417 4-dimensional figurate numbers: a(n) = n*binomial(n+2, 3).

Original entry on oeis.org

1, 8, 30, 80, 175, 336, 588, 960, 1485, 2200, 3146, 4368, 5915, 7840, 10200, 13056, 16473, 20520, 25270, 30800, 37191, 44528, 52900, 62400, 73125, 85176, 98658, 113680, 130355, 148800, 169136, 191488, 215985, 242760, 271950, 303696, 338143, 375440, 415740
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(n) is 1/6 the number of colorings of a 2 X 2 hexagonal array with n+2 colors. - R. H. Hardin, Feb 23 2002
a(n) is the sum of all numbers that cannot be written as t*(n+1) + u*(n+2) for nonnegative integers t,u (see Schuh). - Floor van Lamoen, Oct 09 2002
a(n) is the total number of rectangles (including squares) contained in a stepped pyramid of n rows (or of base 2n-1) of squares. A stepped pyramid of squares of base 2*6 - 1 = 11, for instance, has the following vertices:
..........X.X
........X.X.X.X
......X.X.X.X.X.X
....X.X.X.X.X.X.X.X
..X.X.X.X.X.X.X.X.X.X
X.X.X.X.X.X.X.X.X.X.X.X
X.X.X.X.X.X.X.X.X.X.X.X - Lekraj Beedassy, Sep 02 2003
Partial sums of A002412. - Jonathan Vos Post, Mar 16 2006
a(n) equals -1 times the coefficient of x^3 of the characteristic polynomial of the (n + 2) X (n + 2) matrix with 2's along the main diagonal and 1's everywhere else (see Mathematica code below). - John M. Campbell, May 28 2011
a(n) is the n-th antidiagonal sum of the convolution array A213750. - Clark Kimberling, Jun 20 2012
Convolution of A000027 with A000384 (excluding 0). - Bruno Berselli, Dec 06 2012
The sequence is the binomial transform of (1, 7, 15, 13, 4, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015
Also the number of 3-cycles in the (n+2)-triangular graph. - Eric W. Weisstein, Aug 14 2017

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
  • K. -W. Lau, Solution to Problem 2495, Journal of Recreational Mathematics 2002-3 31(1) 79-80.
  • Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Bisection of A002624.
a(n) = A093561(n+3, 4).
Cf. A000027, A000384, A002412, A062196, A213750, A000332.
Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
Cf. A151974 (number of 4-cycles in the triangular graph), A290939 (5-cycles), A290940 (6-cycles).

Programs

  • GAP
    List([1..40], n-> n^2*(n+1)*(n+2)/6 ); # G. C. Greubel, Jul 03 2019
  • Magma
    /* A000027 convolved with A000384 (excluding 0): */ A000384:=func; [&+[(n-i+1)*A000384(i): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Dec 06 2012
  • Magma
    [n*Binomial(n+2,3):n in [1..40]]; // Vincenzo Librandi, Aug 02 2015
  • Maple
    seq(n^2*(n+1)*(n+2)/6, n=1..50);
  • Mathematica
    Table[n Binomial[n+2, 3], {n, 40}]
Table[-Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + 1 &, {n+2, n+2}], x], x^3], {n, 40}] (* John M. Campbell, May 28 2011 *)
Nest[Accumulate, Range[1, 170, 4], 3] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 8, 30, 80, 175}, 40] (* Harvey P. Dale, Jan 11 2014 *)
Table[n Pochhammer[n, 3]/6, {n, 40}] (* or *) CoefficientList[Series[ (1+3x)/(1-x)^5, {x,0,40}], x] (* Eric W. Weisstein, Aug 14 2017 *)
  • PARI
    a(n)=n^2*(n+1)*(n+2)/6 \\ Charles R Greathouse IV, Jun 10 2011
  • Sage
    [n*binomial(n+2,3) for n in (1..40)] # G. C. Greubel, Jul 03 2019

Formula

a(n) = n^2*(n+1)*(n+2)/6.
G.f.: x*(1+3*x)/(1-x)^5. - Simon Plouffe in his 1992 dissertation
a(n) = C(n+2, 2)*n^2/3. - Paul Barry, Jun 26 2003
a(n) = C(n+3, n)*C(n+1, 1). - Zerinvary Lajos, Apr 27 2005
a(n) = (binomial(n+3,n-1) - binomial(n+2,n-2))*(binomial(n+1,n-1) - binomial(n,n-2)). - Zerinvary Lajos, May 12 2006
a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5), n>5. - Wesley Ivan Hurt, Aug 01 2015
G.f.: x*hypergeometric2F1(2,4;1;x). - R. J. Mathar, Aug 09 2015
a(n) = A080852(4,n-1). - R. J. Mathar, Jul 28 2016
Sum_{n>=1} 1/a(n) = Pi^2/2 - 15/4. - Jaume Oliver Lafont, Jul 13 2017
E.g.f.: x*(6 + 18*x + 9*x^2 + x^3)*exp(x)/3!. - G. C. Greubel, Jul 03 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi^2 + 27 - 48*log(2))/4. - Amiram Eldar, Jun 28 2020
a(n) = A000332(n+3) + 3*A000332(n+2). - Mircea Dan Rus, Jul 29 2020

Extensions

Edited and extended by Floor van Lamoen, Oct 09 2002

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.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jun 14 2012

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Cf. A000027.

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • Python
    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

Formula

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.

A051925 a(n) = n*(2*n+5)*(n-1)/6.

Original entry on oeis.org

0, 0, 3, 11, 26, 50, 85, 133, 196, 276, 375, 495, 638, 806, 1001, 1225, 1480, 1768, 2091, 2451, 2850, 3290, 3773, 4301, 4876, 5500, 6175, 6903, 7686, 8526, 9425, 10385, 11408, 12496, 13651, 14875, 16170, 17538, 18981, 20501, 22100, 23780
Offset: 0

Views

Author

N. J. A. Sloane, Dec 19 1999

Keywords

Comments

Related to variance of number of inversions of a random permutation of n letters.
Zero followed by partial sums of A005563. - Klaus Brockhaus, Oct 17 2008
a(n)/12 is the variance of the number of inversions of a random permutation of n letters. See evidence in Mathematica code below. - Geoffrey Critzer, May 15 2010
The sequence is related to A033487 by A033487(n-1) = n*a(n) - Sum_{i=0..n-1} a(i) = n*(n+1)*(n+2)*(n+3)/4. - Bruno Berselli, Apr 04 2012
Deleting the two 0's leaves row 2 of the convolution array A213750. - Clark Kimberling, Jun 20 2012
For n>=4, a(n-2) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0110 (the first n-4 zeros), or, the same, a(n-2) is up-down coefficient {n,6} (see comment in A060351). - Vladimir Shevelev, Feb 15 2014
Minimum sum of the bottom row of a triangular array A filled with the integers [0..binomial(n, 2) - 1] that obeys the rule A[i, j] + 1 <= A[i+1, j] and A[i, j] + 1 <= A[i, j-1]. - C.S. Elder, Oct 13 2023
The preceding statement can be extended: a(n) is the minimum sum of the main antidiagonal of a n X n square array A filled eith the integers [0..n^2-1] that is increasing on each row from left to right, and on each column from top to bottom. - Yifan Xie, Dec 19 2024

References

  • V. N. Sachkov, Probabilistic Methods in Combinatorial Analysis, Cambridge, 1997.

Links

Crossrefs

Cf. A000330, A005563, A033487.

Programs

  • Magma
    I:=[0, 0, 3, 11]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Apr 27 2012
  • Mathematica
    f[{x_, y_}] := 2 y - x^2; Table[f[Coefficient[ Series[Product[Sum[Exp[i t], {i, 0, m}], {m, 1, n - 1}]/n!, {t, 0, 2}], t, {1, 2}]], {n, 0, 41}]*12 (* Geoffrey Critzer, May 15 2010 *)
CoefficientList[Series[x^2*(3-x)/(1-x)^4,{x,0,50}],x] (* Vincenzo Librandi, Apr 27 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,0,3,11},50] (* Harvey P. Dale, Sep 07 2024 *)
  • PARI
    {print1(a=0, ","); for(n=0, 42, print1(a=a+(n+1)^2-1, ","))} \\ Klaus Brockhaus, Oct 17 2008

Formula

a(n) = A000330(n) - n. - Andrey Kostenko, Nov 30 2008
G.f.: x^2*(3-x)/(1-x)^4. - Colin Barker, Apr 04 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Apr 27 2012
E.g.f.: (x^2/6)*(2*x + 9)*exp(x). - G. C. Greubel, Jul 19 2017
From Amiram Eldar, Nov 10 2023: (Start)
Sum_{n>=2} 1/a(n) = 62/1225 + 24*log(2)/35.
Sum_{n>=2} (-1)^n/a(n) = 6*Pi/35 + 72*log(2)/35 - 2078/1225. (End)

A007585 10-gonal (or decagonal) pyramidal numbers: a(n) = n*(n + 1)*(8*n - 5)/6.

Original entry on oeis.org

0, 1, 11, 38, 90, 175, 301, 476, 708, 1005, 1375, 1826, 2366, 3003, 3745, 4600, 5576, 6681, 7923, 9310, 10850, 12551, 14421, 16468, 18700, 21125, 23751, 26586, 29638, 32915, 36425, 40176, 44176, 48433, 52955, 57750
Offset: 0

Views

Author

N. J. A. Sloane, R. K. Guy

Keywords

Comments

Binomial transform of [1, 10, 17, 8, 0, 0, 0, ...] = (1, 11, 38, 90, ...). - Gary W. Adamson, Mar 18 2009
This sequence is related to A000384 by a(n) = n*A000384(n) - Sum_{i=0..n-1} A000384(i) and this is the case d=4 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/6. - Bruno Berselli, Apr 21 2010
For n>0, (a(n)) is the principal diagonal of the convolution array A213750. - Clark Kimberling, Jun 20 2012
From Ant King, Oct 30 2012: (Start)
The partial sums of the figurate decagonal numbers A001107.
For n>1, the digital roots of this sequence A010888(A007585(n)) form the purely periodic 27-cycle {1,2,2,9,4,4,8,6,6,7,8,8,6,1,1,5,3,3,4,5,5,3,7,7,2,9,9}.
For n>1, the units’ digits of this sequence A010879(A007585(n)) form the purely periodic 20-cycle {1,1,8,0,5,1,6,8,5,5,6,6,3,5,0,6,1,3,0,0}.
(End)

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
  • E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000384.
Cf. A093565 ((8, 1) Pascal, column m=3). Partial sums of A001107.
Cf. similar sequences listed in A237616.

Programs

Formula

a(n) = (8*n-5)*binomial(n+1, 2)/3.
G.f.: x*(1+7*x)/(1-x)^4.
a(n) = (8*n^3 + 3*n^2 - 5*n)/6. - Vincenzo Librandi, Aug 01 2010
a(0)=0, a(1)=1, a(2)=11, a(3)=38, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Dec 20 2011
From Ant King, Oct 30 2012: (Start)
a(n) = a(n-1) + n*(4*n-3).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 8.
a(n) = (n+1)*(2*A001107(n) + n)/6.
a(n) = A000292(n) + 7*A000292(n-1).
a(n) = A007584(n) + A000292(n-1).
a(n) = A000217(n) + 8*A000292(n-1).
a(n) = binomial(n+2,3) + 7*binomial(n+1,3).
Sum_{n>=1} 1/a(n) = 6*(4*pi*(sqrt(2)-1) + 4*(8-sqrt(2))*log(2) + 8*sqrt(2)*log(2-sqrt(2))-5)/65 = 1.145932345...
(End)
a(n) = Sum_{i=0..n-1} (n-i)*(8*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(6 + 27*x + 8*x^2)*exp(x)/6. - Ilya Gutkovskiy, May 12 2017
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