A104246 -id:A104246 - cp's OEIS frontend

A338482 Least number of centered triangular numbers that sum to n.

1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5

Offset: 1

Ilya Gutkovskiy, Oct 29 2020

nonn

It appears that a(n) = 3 for n == 0 (mod 3), 1 <= a(n) <= 4 for n == 1 (mod 3), and 2 <= a(n) <= 5 for n == 2 (mod 3). - Robert Israel, Nov 13 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers

Cf. A005448, A061336, A101412, A104246, A234533, A338484.

f:= proc(n) option remember; local r,i;
    r:= sqrt(24*n-15)/6+1/2;
    if r::integer then return 1 fi;
    1+min(seq(procname(n-(3*i*(i-1)/2+1)),i=1..floor(r)))
end proc:
map(f, [$1..200]); # Robert Israel, Nov 13 2020

f[n_] := f[n] = Module[{r}, r = Sqrt[24n-15]/6+1/2; If[IntegerQ[r], Return[1]]; 1+Min[Table[f[n-(3i*(i-1)/2+1)], {i, 1, Floor[r]}]]];
Map[f, Range[200]] (* Jean-François Alcover, Sep 16 2022, after Robert Israel *)

A338493 Least number of square pyramidal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 2, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 5, 3, 4, 3, 4, 2, 3, 4, 4, 5, 3, 4, 5, 5, 6, 4, 1, 2, 3, 3, 4, 2, 3, 4, 4, 5, 3, 4, 5, 5, 2, 3, 4, 4, 5, 3, 4, 5, 5, 6, 4, 5, 6, 6, 3, 4, 2, 3, 4, 4, 5, 3, 1, 2, 3, 4, 4, 2, 3, 4, 3, 4, 3

Offset: 1

Ilya Gutkovskiy, Oct 30 2020

nonn

Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A002828, A104246, A338484, A338494, A338495.

A102801 Let f(n) be the minimal number of distinct nonzero tetrahedral numbers that add to n (or -1 if n is not a sum of distinct tetrahedral numbers); sequence gives numbers n for which f(n) = 2.

Original entry on oeis.org

5, 11, 14, 21, 24, 30, 36, 39, 45, 55, 57, 60, 66, 76, 85, 88, 91, 94, 104, 119, 121, 124, 130, 140, 155, 166, 169, 175, 176, 185, 200, 204, 221, 224, 230, 240, 249, 255, 276, 285, 287, 290, 296, 304, 306, 321, 340, 342, 365, 368, 370, 374, 384

Offset: 1

Jud McCranie, Feb 26 2005

nonn

			680 (of A034404) is a sum of two distinct positive tetrahedral numbers but not in the list because it is also a tetrahedral number itself. - _R. J. Mathar_, Jun 05 2025

R. J. Mathar, Table of n, a(n) for n = 1..966

Cf. A000292, A104246, A102795, etc.

A283370 Minimal number of terms required to write n as sum of numbers in A000389 = { C(k,5); k=1,2,3,... } (with repetitions allowed).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6

Offset: 0

M. F. Hasler, Mar 06 2017

nonn

Analog, for A000389 = {C(n,5)}, of A061336 (for triangular numbers A000217 = {C(n,2)}), A104246 (for tetrahedral numbers A000292 = {C(n,3)}) and A283365 (for A000332 = {C(n,4)}).

R. J. Mathar, Table of n, a(n) for n = 0..10000
Hyun Kwang Kim, On regular polytope numbers, Proc. Amer. Math. Soc. 131 (2003), p. 65-75. DOI:10.1090/S0002-9939-02-06710-2.

Cf. A000332 = {C(n,4)}; A061336 (analog for A000217), A104246 (analog for A000292), A283365 (analog for A000332).

{a(n,k=5,M=9e9,N=n) = n>k || return(n); for(m=k,M,binomial(m,k)>n && (M=m) && break); M-- <= k && return(n); my(b=binomial(M,k),c=binomial(M-1,k),NN); forstep( nn=n\b,0,-1, if(N>NN=nn+a(n-nn*b,k,M,N),N=NN); n-(nn-1)*b >= (N-nn+1)*c && break); N}

a(n) <= 10 = a(220) for all n, according to Kim (2003, p. 74, first row of table "d = 5"), but this "numerical result" has no "* denoting exact values" (see Remark at end of paper), so it could be incorrect. [Disclaimer added by M. F. Hasler, Sep 22 2022]

A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 0, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 0, 1, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 3, 2, 0, 0, 1, 12, 55, 120, 135, 112, 112, 60, 15, 12, 3, 2, 0, 0

Offset: 0

Ilya Gutkovskiy, Jul 31 2017

A(n,k) is the number of ways of writing n as a sum of k tetrahedral (or triangular pyramidal) numbers (A000292).

			Square array begins:
1,  1,  1,  1,   1,   1,  ...
0,  1,  2,  3,   4,   5,  ...
0,  0,  1,  3,   6,  10,  ...
0,  0,  0,  1,   4,  10,  ...
0,  1,  2,  3,   5,  10,  ...
0,  0,  2,  6,  12,  21,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Cf. A000292, A104246, A286180, A290054, A290430.
Cf. A000007 (column 0), A023533 (column 1), A282172 (column 5).
Main diagonal gives A303170.
Similar to, but different from, A045847.

Table[Function[k, SeriesCoefficient[Sum[x^(i (i + 1) (i + 2)/6), {i, 0, n}]^k, {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

A102802 Let f(n) be the minimal number of distinct nonzero tetrahedral numbers that add to n (or -1 if n is not a sum of distinct tetrahedral numbers); sequence gives numbers n for which f(n) = 3.

Original entry on oeis.org

15, 25, 31, 34, 40, 46, 49, 59, 61, 65, 67, 70, 77, 80, 86, 89, 92, 95, 98, 101, 105, 108, 111, 114, 123, 125, 129, 131, 134, 139, 141, 144, 150, 156, 159, 160, 170, 177, 179, 180, 186, 189, 195, 196, 201, 205, 208, 210, 211, 214, 222, 225, 231

Offset: 1

Jud McCranie, Feb 26 2005

nonn

Cf. A000292, A104246, A102795, etc.

A102803 Let f(n) be the minimal number of distinct nonzero tetrahedral numbers that add to n (or -1 if n is not a sum of distinct tetrahedral numbers); sequence gives numbers n for which f(n) = 4.

Original entry on oeis.org

50, 69, 71, 81, 87, 90, 96, 99, 102, 109, 112, 115, 118, 133, 135, 143, 145, 149, 151, 154, 161, 164, 181, 187, 190, 197, 199, 206, 209, 212, 215, 218, 226, 228, 232, 235, 242, 243, 245, 251, 254, 257, 261, 263, 264, 266, 270, 273, 279, 281

Offset: 1

Jud McCranie, Feb 26 2005

nonn

Cf. A000292, A104246, A102795, etc.

A330031 Sums of two nonzero tetrahedral numbers (A000292).

Original entry on oeis.org

2, 5, 8, 11, 14, 20, 21, 24, 30, 36, 39, 40, 45, 55, 57, 60, 66, 70, 76, 85, 88, 91, 94, 104, 112, 119, 121, 124, 130, 140, 155, 166, 168, 169, 175, 176, 185, 200, 204, 221, 224, 230, 240, 249, 255, 276, 285, 287, 290, 296, 304, 306, 321, 330, 340, 342, 365

Offset: 1

Peter Kagey, Mar 07 2020

nonn

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000292, A102798, A104246.

Analogs for other figurate numbers are A000404 (squares), A003325 (cubes), A051533 (triangular numbers), A286636 (centered square numbers), A287960 (centered triangular numbers), A288631 (square pyramidal numbers), A332987 (pentagonal numbers).

A338458 Least number of heptagonal pyramidal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 1, 2, 3, 4, 5, 6, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 3, 4, 5, 6, 7, 8, 6, 7, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 1, 2, 3, 4, 5, 6, 6, 7, 2, 3, 4, 5, 6, 7, 7, 8, 3, 4, 3, 4, 5, 6, 7, 8, 4, 5, 2, 3, 4, 5, 6, 7, 5, 6, 3, 4, 5, 6, 7, 8, 6, 7

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

nonn

Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A002413, A104246, A338480, A338492, A338493, A338494, A338495, A338496.

A338496 Least number of octagonal pyramidal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 1, 2, 3, 4, 5, 6, 4, 5, 6, 2, 3, 4, 5, 6, 7, 5, 6, 7, 3, 4, 5, 6, 7, 8, 6, 7, 8, 4, 5, 6, 2, 3, 4, 5, 6, 7, 5, 6, 7, 3, 1, 2, 3, 4, 5, 6, 7, 8, 4, 2, 3, 4, 5, 6, 7, 8, 9, 5, 3, 4, 3, 4, 5, 6, 7, 8, 6, 4, 5, 4, 2, 3

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

nonn

Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A002414, A104246, A338458, A338481, A338493, A338494, A338495.

cp's OEIS Frontend

A338482 Least number of centered triangular numbers that sum to n.

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A338493 Least number of square pyramidal numbers needed to represent n.

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A102801 Let f(n) be the minimal number of distinct nonzero tetrahedral numbers that add to n (or -1 if n is not a sum of distinct tetrahedral numbers); sequence gives numbers n for which f(n) = 2.

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A283370 Minimal number of terms required to write n as sum of numbers in A000389 = { C(k,5); k=1,2,3,... } (with repetitions allowed).

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A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j*(j+1)*(j+2)/6))^k.

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A102802 Let f(n) be the minimal number of distinct nonzero tetrahedral numbers that add to n (or -1 if n is not a sum of distinct tetrahedral numbers); sequence gives numbers n for which f(n) = 3.

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A102803 Let f(n) be the minimal number of distinct nonzero tetrahedral numbers that add to n (or -1 if n is not a sum of distinct tetrahedral numbers); sequence gives numbers n for which f(n) = 4.

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A330031 Sums of two nonzero tetrahedral numbers (A000292).

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A338458 Least number of heptagonal pyramidal numbers needed to represent n.

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A338496 Least number of octagonal pyramidal numbers needed to represent n.

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A290429 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Sum_{j>=0} x^(j(j+1)(j+2)/6))^k.