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.

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A195827 Triangle read by rows with T(n,k) = n - A085787(k), n>=1, k>=1, if (n - A085787(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 0, 4, 1, 5, 2, 6, 3, 0, 7, 4, 1, 8, 5, 2, 9, 6, 3, 10, 7, 4, 11, 8, 5, 12, 9, 6, 0, 13, 10, 7, 1, 14, 11, 8, 2, 15, 12, 9, 3, 16, 13, 10, 4, 17, 14, 11, 5, 0, 18, 15, 12, 6, 1, 19, 16, 13, 7, 2, 20, 17, 14, 8, 3, 21, 18, 15, 9, 4, 22, 19, 16, 10, 5
Offset: 1

Views

Author

Omar E. Pol, Sep 24 2011

Keywords

Comments

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A085787(k).
This sequence is related to the generalized heptagonal numbers A085787, A195837 and A036820 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

Examples

			Written as a triangle:
.  0;
.  1;
.  2;
.  3,  0;
.  4,  1;
.  5,  2;
.  6,  3,  0;
.  7,  4,  1;
.  8,  5,  2;
.  9,  6,  3;
. 10,  7,  4;
. 11,  8,  5;
. 12,  9,  6,  0;
. 13, 10,  7,  1;
. 14, 11,  8,  2;

Crossrefs

Cf. A036820, A085787, A195310, A195825, A195826, A195828, A195829, A195830, A195831, A195832, A195833, A195837.

A210944 Triangle read by rows with T(n,k) = n - A195818(k), n>=1, k>=1, if (n - A195818(k))>=0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 11, 1, 12, 2, 13, 3, 0, 14, 4, 1, 15, 5, 2, 16, 6, 3, 17, 7, 4, 18, 8, 5, 19, 9, 6, 20, 10, 7, 21, 11, 8, 22, 12, 9, 23, 13, 10, 24, 14, 11, 25, 15, 12, 26, 16, 13, 27, 17, 14, 28, 18, 15, 29, 19, 16, 30, 20, 17, 31, 21, 18
Offset: 1

Views

Author

Omar E. Pol, Jun 16 2012

Keywords

Comments

Also triangle read by rows in which column k lists the nonnegative integers A001477 starting at the row A195818(k).
This sequence is related to the generalized 14-gonal numbers A195818, A210954 and A210964 in the same way as A195310 is related to the generalized pentagonal numbers A001318, A175003 and A000041. See comments in A195825.

Examples

			Written as an irregular triangle:
0;
1;
2;
3;
4;
5;
6;
7;
8;
9;
10, 0;
11, 1;
12, 2;
13, 3,  0;
14, 4,  1;
15, 5,  2;
16, 6,  3;
17, 7,  4;
18, 8,  5;
19, 9,  6;

Crossrefs

Cf. A195310, A195825, A195826, A195827, A195828, A195829, A195830, A195831, A195832, A195833, A195818, A210954, A210964.

A195309 Row sums of an irregular triangle read by rows in which row n lists the next A026741(n+1) natural numbers A000027.

Original entry on oeis.org

1, 9, 11, 45, 39, 126, 94, 270, 185, 495, 321, 819, 511, 1260, 764, 1836, 1089, 2565, 1495, 3465, 1991, 4554, 2586, 5850, 3289, 7371, 4109, 9135, 5055, 11160, 6136, 13464, 7361, 16065, 8739, 18981, 10279, 22230, 11990, 25830, 13881
Offset: 1

Views

Author

Omar E. Pol, Sep 21 2011

Keywords

Comments

The integers in same rows of the source triangle have a property related to Euler's Pentagonal Theorem.
Note that the column 1 of the mentioned triangle gives the positive terms of A001318 (see example).

Examples

			a(1) = 1
a(2) = 2+3+4 = 9
a(3) = 5+6 = 11
a(4) = 7+8+9+10+11 = 45
a(5) = 12+13+14 = 39
a(6) = 15+16+17+18+19+20+21 = 126
a(7) = 22+23+24+25 = 94
a(8) = 26+27+28+29+30+31+32+33+34 = 270
a(9) = 35+36+37+38+39 = 185

Links

Crossrefs

Cf. A026741, A195310, A195311, A004188 (bisection).

Programs

  • Maple
    A195309 := proc(n)
        (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32
end proc:
seq(A195309(n),n=1..60) ; # R. J. Mathar, Oct 08 2011
  • Mathematica
    LinearRecurrence[{0,4,0,-6,0,4,0,-1},{1,9,11,45,39,126,94,270},80] (* Harvey P. Dale, Jun 22 2015 *)

Formula

a(n) = (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32 . - R. J. Mathar, Oct 08 2011
G.f. x*(1+9*x+7*x^2+9*x^3+x^4) / ( (x-1)^4*(1+x)^4 ). - R. J. Mathar, Oct 08 2011

A190138 Final number of terms obtained with Euler's recurrence formula when computing the sum of divisors of n.

Original entry on oeis.org

1, 2, 3, 5, 9, 15, 27, 46, 80, 138, 238, 413, 713, 1235, 2136, 3695, 6393, 11057, 19130, 33091, 57246, 99032, 171315, 296365, 512682, 886902, 1534266, 2654154, 4591475, 7942870, 13740526, 23769981, 41120131, 71134474, 123056829, 212878289, 368262059, 637063333
Offset: 1

Views

Author

Michel Marcus, Dec 19 2012

Keywords

Comments

It appears that a(n) is the number of compositions of n whose parts are pentagonal numbers. See Neville link. - Michel Marcus, Jul 28 2017

Examples

			For n=5, start with row 5 of A195310: [4, 3, 0]. Then replace 4 by row 4: [3, 2], replace 3 by row 3: [2, 1]. The row is now [3, 2, 2, 1, 0].
Repeat process until all terms are 0: [4, 3, 0], [3, 2, 2, 1, 0], [2, 1, 1, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0].
The final array has 9 items, hence a(5) = 9.

Links

Crossrefs

Cf. A000203, A001318, A195310.

Programs

  • Mathematica
    rows = 30;
gpenta[n_] := If[EvenQ[n], n(3n/2+1)/4, (n+1)(3n+1)/8];
T[n_, k_] := n - gpenta[k];
Do[row[n] = DeleteCases[Table[T[n, k], {k, n}], _?Negative], {n, rows}];
a[n_] := a[n] = row[n] //. j_?Positive :> Sequence @@ row[j] // Length;
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, rows}] (* Jean-François Alcover, Sep 22 2018 *)
  • PARI
    A001318(n) = { return((3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8);}
A195310(n) = {if (n == 0, return ([0])); nb = 1; vec = vector(0); nn = n; while (nn >=0, nn = n - A001318(nb); if (nn >=0, vec = concat(vec, nn)); nb++;); return(vec);}
A190138(m) = { vval = vector(m); for (n=1, m, vec = A195310(n); svec = 0; for (k=1, length(vec), if (vec[k] == 0, svec += 1, svec += vval[vec[k]]);); vval[n] = svec;); for (n=1, m, print1(vval[n], ", "););}
Previous Showing 21-24 of 24 results.

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