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 1-10 of 12 results. Next

A307597 Number of partitions of n into 2 distinct positive triangular numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 3, 0, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2019

Keywords

Comments

The greedy inverse (positions of first occurrence of n) starts 0, 4, 16, 81, 471, 2031, 1381, 11781, 6906, 17956, ... - R. J. Mathar, Apr 28 2020

Examples

			a(16) = 2 because we have [15, 1] and [10, 6].
		

Crossrefs

Formula

a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^(k*(k+1)/2)).
a(n) = Sum_{k=1..floor((n-1)/2)} c(k) * c(n-k), where c = A010054. - Wesley Ivan Hurt, Jan 06 2024

A063993 Number of ways of writing n as an unordered sum of exactly 3 nonzero triangular numbers.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 2, 1, 1, 2, 2, 2, 0, 3, 2, 2, 2, 2, 2, 1, 3, 1, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 3, 1, 2, 5, 1, 2, 1, 2, 5, 3, 3, 1, 4, 2, 3, 2, 2, 4, 4, 2, 1, 4, 3, 3, 3, 2, 4, 3, 3, 3, 4, 2, 1, 6, 1, 5, 3, 3, 5, 2, 2, 2, 5, 2, 5, 4, 2, 4, 5, 3, 1
Offset: 0

Views

Author

N. J. A. Sloane, Sep 18 2001

Keywords

Comments

a(A002097(n)) = 0; a(A111638(n)) = 1; a(A064825(n)) = 2. - Reinhard Zumkeller, Jul 20 2012

Examples

			5 = 3 + 1 + 1, so a(5) = 1.
		

Crossrefs

Cf. A053604, A008443, A002636, A064181 (greedy inverse), A307598 (3 distinct positive).
Column k=3 of A319797.

Programs

  • Haskell
    a063993 n = length [() | let ts = takeWhile (< n) $ tail a000217_list,
                        x <- ts, y <- takeWhile (<= x) ts,
                        let z = n - x - y, 0 < z, z <= y, a010054 z == 1]
    -- Reinhard Zumkeller, Jul 20 2012
    
  • Maple
    A063993 := proc(n)
        local a,t1idx,t2idx,t1,t2,t3;
        a := 0 ;
        for t1idx from 1 do
            t1 := A000217(t1idx) ;
            if 3*t1 > n then
                break;
            end if;
            for t2idx from t1idx do
                t2 := A000217(t2idx) ;
                if t1+t2 > n then
                    break;
                end if;
                t3 :=  n-t1-t2 ;
                if t3 >= t2 then
                    if isA000217(t3) then
                        a := a+1 ;
                    end if;
                end if ;
            end do:
        end do:
        a ;
    end proc: # R. J. Mathar, Apr 28 2020
  • Mathematica
    a = Table[ n(n + 1)/2, {n, 1, 15} ]; b = {0}; c = Table[ 0, {100} ]; Do[ b = Append[ b, a[ [ i ] ] + a[ [ j ] ] + a[ [ k ] ] ], {k, 1, 15}, {j, 1, k}, {i, 1, j} ]; b = Delete[ b, 1 ]; b = Sort[ b ]; l = Length[ b ]; Do[ If[ b[ [ n ] ] < 100, c[ [ b[ [ n ] ] + 1 ] ]++ ], {n, 1, l} ]; c
  • PARI
    trmx(n)=my(k=sqrtint(8*n+1)\2);if(k^2+k>2*n,k-1,k)
    trmn(n)=trmx(ceil(n)-1)+1
    a(n)=if(n<3, return(0)); sum(a=trmn(n/3),trmx(n-2),my(t=n-a*(a+1)/2);sum(b=trmn(t/2),min(trmx(t-1),a), ispolygonal(t-b*(b+1)/2,3))) \\ Charles R Greathouse IV, Jul 07 2022

Extensions

More terms from Robert G. Wilson v, Sep 20 2001

A350405 a(n) is the smallest number which can be represented as the sum of n distinct nonzero n-gonal numbers in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

37, 142, 285, 536, 911, 1268, 1909, 2713, 3876, 5179, 6891, 8901, 11190, 14384, 18087, 21697, 27055, 32166, 39111, 46560, 53892, 64412, 73949, 86778, 98202, 113635, 130088, 148051, 167505, 190968, 214955, 240143, 269775, 297615, 331201, 367429, 409179, 451340, 497830
Offset: 3

Views

Author

Ilya Gutkovskiy, Dec 29 2021

Keywords

Examples

			For n = 3: 37 = 1 + 15 + 21 = 3 + 6 + 28 = 6 + 10 + 21.
		

Crossrefs

Programs

  • Mathematica
    Do[i=1;While[b=PolygonalNumber[n,Range@i++];!IntegerQ[t=Min[First/@Select[Tally[Select[Total/@Subsets[b,{n}],#<=Max@b&]],Last@#==n&]]]];Print@t,{n,3,10}] (* Giorgos Kalogeropoulos, Dec 30 2021 *)

Formula

a(n) >= A006484(n). - David A. Corneth, Dec 30 2021

Extensions

a(10)-a(31) from Michael S. Branicky, Dec 29 2021
More terms from David A. Corneth, Dec 30 2021

A341021 Number of partitions of n into 4 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 2, 0, 1, 1, 0, 2, 1, 3, 0, 1, 2, 0, 2, 2, 1, 3, 2, 0, 2, 2, 2, 1, 3, 0, 4, 3, 1, 3, 2, 2, 3, 2, 1, 5, 3, 3, 2, 4, 1, 2, 5, 1, 5, 3, 2, 5, 3, 3, 4, 4, 3, 4, 6, 0, 6, 4, 2, 7, 4, 3, 5, 4, 3, 5, 5, 5, 4, 5, 5, 5, 8, 2, 6, 5, 1, 10, 5, 4, 7, 7, 4
Offset: 20

Views

Author

Ilya Gutkovskiy, Feb 02 2021

Keywords

Crossrefs

A341027 Number of partitions of n into 10 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 1, 0, 4, 0, 2, 1, 0, 2, 2, 1, 2, 1, 0, 2, 2, 2, 2, 2, 1, 2, 3, 0, 2, 4, 1, 3, 2, 4, 1, 3, 2, 3, 4, 0, 4, 4, 1, 5, 5, 3
Offset: 220

Views

Author

Ilya Gutkovskiy, Feb 02 2021

Keywords

Crossrefs

A341022 Number of partitions of n into 5 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 3, 0, 2, 2, 0, 1, 3, 2, 3, 2, 1, 2, 3, 2, 0, 5, 1, 3, 3, 2, 3, 2, 4, 2, 5, 2, 2, 6, 1, 4, 6, 1, 5, 6, 3, 4, 4, 4, 4, 5, 3, 5, 7, 6, 4, 8, 2, 5, 7, 3, 7, 7, 7, 5, 7, 6, 4, 12, 5, 4, 10, 3, 11, 9, 5, 9, 8, 5
Offset: 35

Views

Author

Ilya Gutkovskiy, Feb 02 2021

Keywords

Crossrefs

A341023 Number of partitions of n into 6 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 0, 0, 2, 1, 2, 0, 2, 0, 2, 2, 0, 3, 1, 1, 3, 2, 1, 3, 2, 1, 3, 3, 1, 2, 3, 3, 5, 2, 2, 5, 1, 2, 5, 3, 4, 3, 5, 2, 4, 7, 1, 6, 4, 2, 8, 4, 5, 7, 4, 4, 6, 6, 4, 9, 6, 4, 8, 5, 6, 11, 8, 4, 9, 5, 8, 8, 9, 8, 9, 11
Offset: 56

Views

Author

Ilya Gutkovskiy, Feb 02 2021

Keywords

Crossrefs

A341024 Number of partitions of n into 7 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 2, 0, 0, 2, 1, 1, 0, 2, 1, 1, 3, 0, 1, 1, 1, 3, 2, 2, 2, 2, 1, 2, 4, 0, 2, 4, 0, 3, 4, 3, 4, 3, 1, 4, 3, 3, 4, 4, 4, 2, 6, 3, 6, 6, 1, 6, 3, 3, 5, 9, 4, 4, 8, 2, 6, 9, 3, 9, 7, 4, 9, 6, 6, 10, 8, 5
Offset: 84

Views

Author

Ilya Gutkovskiy, Feb 02 2021

Keywords

Crossrefs

A341025 Number of partitions of n into 8 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 3, 1, 1, 3, 1, 1, 2, 1, 2, 4, 2, 2, 2, 0, 2, 5, 2, 1, 5, 2, 2, 4, 3, 4, 2, 3, 4, 3, 5, 1, 8, 4, 3, 8, 1, 4, 7, 2, 5, 7, 5, 6, 6, 4, 4, 10, 6
Offset: 120

Views

Author

Ilya Gutkovskiy, Feb 02 2021

Keywords

Crossrefs

A341026 Number of partitions of n into 9 distinct nonzero triangular numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 2, 1, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 0, 0, 3, 1, 0, 4, 1, 2, 1, 1, 1, 2, 3, 1, 2, 3, 0, 4, 3, 2, 3, 2, 2, 1, 4, 1, 6, 3, 1, 5, 2, 2, 7, 4, 2, 3, 4, 5, 5, 5, 4, 4, 6
Offset: 165

Views

Author

Ilya Gutkovskiy, Feb 02 2021

Keywords

Crossrefs

Showing 1-10 of 12 results. Next